MODULE 1:   Introduction
Proof-Writing Guidelines
1
Introduction

A (mathematical) proof of a statement is simply an argument, presented as an essay, that convinces the reader of the validity of the statement. Therefore, whether a proof is acceptable or not (e.g. whether it has the right level of detail, whether it is written in good style, etc.) is a subjective matter without very strict rules. This makes the task of writing good proofs (i.e. proofs that are correct and easily understood and verified) a difficult one.

As you will notice when you read this document, there are a lot of similarities between writing a correct computer program with good style and writing a correct mathematical proof with good style. Unfortunately, however, writing a good proof is arguably harder since there is no compiler to point out all the compile-time errors you have (e.g. syntax errors or type-checking errors). Furthermore, you cannot run a mathematical proof on a computer to see if it produces the expected output. All the errors in a mathematical proof must be caught by you, the author of the proof, and this requires an extra level of attention to detail.

Our goal in this document is to give you suggestions on how to write correct and clearly presented proofs, and help you more easily catch logical flaws or gaps in your arguments. Most of our suggestions are stylistic in nature, because a proof that is written using good style is a proof that exposes its bugs (if there are any). So by following our guidelines, you will make it easier for yourself and everyone else to understand and validate your proof.

Mathematics, sciences and arts all have deep, complicated and beautiful ideas. Our rate of progress as a species depends on clear communication of those ideas so they can quickly spread and evolve. We hope that you will seek clarity of thought and effective communication of ideas, not just in the proofs you write, but in all of your endeavors in life.

2
Guideline Points

This section contains the guideline points (10 of them) that we ask you to follow when you write your proofs. Before we start, below is an example of a “proof” that violates almost all the guideline points. Feel free to refer back to this example as you go through this section.

Exercise ($$2^n > n$$)

Show that $$2^n > n$$ for all integers $$n \geq 1$$.

Proposed solution

(Line numbers are added for easy referencing.)

1. $$F_n =$$$$2^n > n$$

2. $$F_1 =$$$$2 > 1$$$$\checkmark$$

3. $$F_n \implies F_{n+1}:$$

4. $$2^{n+1} = 2 \cdot 2^n > 2\cdot n \text{ (induction) } \geq n+1$$ because $$n \geq 1$$

5. Therefore proved.

Keep in mind that the above example is really a toy example. Some of the points we make below have more meaning and value in the context of more sophisticated proofs, which are the kinds of proofs you will be writing in CS251.

2.1
Is the basic structure right?

A proof is not a calculation or a sequence of mathematical symbols. A proof is an essay! It is an argument written in some human language (which, in this course, is English). This means that your proof should consist of paragraphs, and every paragraph should consist of full English sentences. Every word and mathematical notation must be part of a complete sentence. The only purpose of mathematical notation/symbols is to make your arguments clear and concise. Do not equate mathematical notation with rigor or formalism. You can write a completely rigorous and formal proof using just English words.

Pictures and/or diagrams are highly encouraged when they help clarify your argument. However, a picture or a diagram does not replace an actual argument that needs to be clearly specified in English.

2.2
Are you starting the right way?

Often, the most important step in coming up with a proof is making sure that you understand exactly what assumptions are given to you and what statement needs to be derived. To make these things absolutely clear to you (the author of the proof) and the reader, we expect the first paragraph of your proof to include a restatement of the assumptions given and what needs to be derived. Unpack the definitions of technical terms if appropriate.

Another important component of the first paragraph is stating your general proof strategy (e.g. proof by contradiction, proof by induction, etc.) For proofs by contradiction, explicitly negate the statement that you are trying to prove and assume it. For proofs by contrapositive, the contrapositive of the statement should be explicitly laid out. For proofs by induction, the parameter being inducted on should be clear.

2.3

When someone reads your proof, they will read it like they read any other essay. Therefore your proof should have a good flow and should be easy to read out loud even with the mathematical notation interspersed in the sentences. In particular, the mathematical notation you use or the diagrams you draw should not break the flow.

2.4
Is the purpose of every sentence clear?

The purpose of every sentence should be clear as you are reading it. Otherwise, the sentence can be very confusing for the reader and break the flow of the proof. A common example that violates this point is to make a statement without clarifying whether it follows from the previous statements or assumptions, or whether it is a statement that will be proved later on. With this in mind, whenever you write a sentence, make sure that it is clear if the sentence is (i) an assumption, (ii) a statement that follows from or combines previously established statements or assumptions, (iii) a claim that will be proved later, (iv) a sentence setting up a goal, (v) a sentence introducing a new variable, terminology, definition etc. (vi) part of an example or illustration, (vii) a digression, or (viii) something else.

A strategy that helps a lot is to set explicit and clear goals, and say what you will do before doing it.

2.5
Are you giving the right level of detail?

What is the right level of detail to provide when writing a proof? This is an important question whose answer depends on the audience of the proof. It is possible to write a proof that has all the right ingredients, but does not have the proper justifications for each step. So then the reader has to check the details themselves and verify that the proof is indeed correct. These kinds of proofs are actually not uncommon in, say, computer science or mathematics publications. The author may knowingly give just enough detail in a proof so that the gaps can be figured out and verified by an expert in the field. In CS251, however, we do not accept such proofs. No extra effort/verification from the reader should be necessary. For this reason, it is better to err on the side of caution, and spell things out as much as you feel is reasonable.

To help with this, our recommendation is that you view your audience as a classmate who does not know how to prove the statement you are presenting the proof for. Keep your audience in mind when presenting your proofs and provide the appropriate level of detail in your arguments. Your classmate should be able to read your proof from start to finish once, and be convinced that your argument is correct.

One thing that can be hard to appreciate is that when you think hard on a problem, you develop a lot of intuition in the process, and certain things that were not too trivial in the beginning start becoming obvious to you. This is great, because it signals that you are acquiring a deeper understanding of the problem. However, when it is time to write down your proof, you have to keep in mind that the reader has not gone through the mental process that you have gone through trying to come up with the proof. So the intuitions you have built are not necessarily accessible to your reader. And the things that are obvious to you may not be obvious to the reader. It is not always easy, but try to put yourself in the shoes of your reader and don’t skip over details that can be crucial to understanding your argument.

Related to the point above, if in your proof you are using a word that is synonymous to “obvious”, double check (with your audience in mind) that it is justified. In particular, an obvious statement should be such that a proof of it springs to mind immediately with no effort. When in doubt about whether a statement qualifies as obvious or not, ask the course staff.

2.6
Should you break up your proof?

When you first learn programming, one thing that will be emphasized over and over again is the value of breaking up your problem into smaller parts and using lots of helper functions. If you write a function that is too long or does more than one task, that is a good indication that you should consider breaking it up into smaller helper functions. Liberal use of helper functions makes your program easier to understand and debug.

In the above sense, mathematical proofs are similar to computer programs. Your proof should be divided up into lemmas and/or claims when appropriate. This will make the proof much easier to understand and it will be easier to spot and fix any potential errors. Even if you are not defining new lemmas/claims, each component of your argument should be separated into different paragraphs, and the high level organization should be easy to identify.

2.7

In programming, you are used to the idea that every object (piece of data) has a type/class. For instance, many programming languages have an integer type and a string type. The type of the data determines what kinds of operations you can apply on the data. You can, for example, multiply two integers, but you cannot multiply two strings. If you try to do any operation that conflicts with the data type, then you will get a compile-time error, and your code will not run.

Another way to get a compile-time error is by trying to access a variable that has not been defined and initialized. And in statically typed programming languages, you need to explicitly specify the type of a variable when you declare it. This way, the compiler can type-check your program before running it.

Similar to programming, every mathematical object has a type (e.g. it could be an integer, a set, a function, etc.). And as in programming, the type of an object determines what kind of operations you can apply to the object. For example, you can add an element to a set, but you cannot add an element to a function. If you do not respect these constraints, we say that your argument does not type-check. A type-checking error often leads to a nonsensical sentence (even though the author may not realize this).

With these in mind, make sure that your proofs always type-check. You must act as a compiler to find anything that might violate type constraints. In order to make this easier, whenever you learn about a new definition, make a note of the type of the object being defined. This is one of the most important parts of a definition. And when referencing an object in your proof, be aware of its type and make sure that your proof type-checks.

In addition to the above, remember that all the variables in your proofs should be introduced properly. In particular, it should be clear if a variable represents an arbitrary object (e.g. in the context of a “for all” quantifier), a specific object known to exist (e.g. in the context of a “there exists” quantifier), or something else (e.g. a specific value). In all cases, the variable must have a specific type, and it should be clear what the type is.

Here is a suggestion to keep in mind. If in a sentence you are referring to a variable, consider preceding the variable with its type if you feel that this makes things clearer and does not add too much redundancy. For example, if x and y are variables referring to strings, you can consider changing a sentence like “Concatenating x and y...” to “Concatenating string x and string y...”.

2.8

As you develop an argument in a proof, you will want to refer back to (and use) a previously established statement or an assumption. Implicitly using previous statements or assumptions can make it hard to follow the logic of the proof and put unnecessary burden on the reader. Therefore, avoid implicit references and make sure it is clear which part(s) of the proof you are using to establish the current step of your proof. Whenever you make a reference, you want it to be very easy for the reader to spot the thing that is being referred to. For example, if there is an important equation that will be referenced later on in the proof, it is a good idea to put that equation on a separate line and label it. This way, you can use the equation’s label to refer to it, and the fact that it is on a separate line by itself will make it very easy for the reader to identify it.

In addition to the above, please be careful when you use a word like “it”, “this”, or “these” in your proof. While you are writing your proof, when you use such a word, you know perfectly well what that word refers to. However, ask yourself whether it would also be perfectly clear to the reader. In general, it is a good idea to try to avoid such words as much as possible, even if this means a certain level of repetitiveness is introduced in the used words.

2.9
Where are you using the assumptions?

When you are asked to write a proof, you are usually given certain assumptions that you take as true (which is your starting point), and you have a target statement that you want to derive. As pointed out in the “Are you starting the right way?” section above, it helps you and the reader to explicitly lay out the given assumptions as well as the statement that needs to be derived. Once your proof is complete, ask yourself where in the proof the assumptions are being used. If it turns out that you are not using a given assumption, you should raise your alertness level and check: is the statement even true without that assumption? (There may be some exceptions, but almost always, the answer will be no.) If it is not true, and you do not use the assumption in your proof, then your proof is wrong.

When someone reads your proof to verify it, they will be trying to spot where the assumptions are used in the proof. This is a quick sanity check that the proof is not missing an essential component. In order to make this check easy for the reader (and also for yourself), you should make clear in your write-up where and how the assumptions are being used in your argument. In the rare case that a given assumption is not needed for the proof, mention this explicitly.

2.10
Is the proof idea clear?

As mentioned before, a proof is an argument that convinces the reader that a certain statement is true. After reading the proof, the reader may walk away with a clear and intuitive understanding of why the statement is true. Or, they may not, even if they are perfectly convinced that the statement is indeed true.

We do not want to get into a philosophical discussion about what it means to understand why something is true. But hopefully we can agree that certain proofs have more explanatory content than others.

We strongly encourage you to always write proofs with strong explanatory content! For this reason, if the motivation/intuition behind the proof is not transparent, consider adding a paragraph or two explaining the main ideas that make the proof work. This can be either added within the proof itself, or can be a separate “Proof Idea” section preceding the proof.

Summary
1. Is the basic structure right?

2. Are you starting the right way?

4. Is the purpose of every sentence clear?

5. Are you giving the right level of detail?

6. Should you break up your proof?

9. Where are you using the assumptions?

10. Is the proof idea clear?

3
An Example

Let’s now come back to the example from the beginning of Section (Guideline Points) and see how it fares. For ease of reference, we reproduce the “proof” here:

Proposed solution
1. $$F_n =$$$$2^n > n$$

2. $$F_1 =$$$$2 > 1$$$$\checkmark$$

3. $$F_n \implies F_{n+1}:$$

4. $$2^{n+1} = 2 \cdot 2^n > 2\cdot n \text{ (induction) } \geq n+1$$ because $$n \geq 1$$

5. Therefore proved.

• Is the basic structure right? It is quite easy to see that there is a gross violation of this point. The argument does not consist of sentences. In fact, there is not a single fully formed English sentence.

• Are you starting the right way? We are missing a restatement of what the given assumptions are and what needs to be derived. (Yes, this is kind of pedantic with the toy example, but still valuable in the context of more complicated examples.) Furthermore, even though the proof is supposed to be a proof by induction, we only learn about this towards the end of the proof.

• Did you read your proof out loud? If we attempt to read the argument out loud, we hear an incomprehensible sequence of words.

• Is the purpose of every sentence clear? What is the purpose of the first line? Is it a definition? Is it a claim? Is it an assumption? We can also ask these questions for lines 2 to 4.

• Are you giving the right level of detail? Implicit in the “proof” is the claim that “$$2\cdot n \geq n+1$$ because $$n\geq 1$$”. This claim is true, and would qualify as an obviously true statement. However, there is a step that is being skipped, which if included, would make the claim more immediately obvious. In particular, saying “$$2 \cdot n = n + n \geq n + 1$$, where the inequality follows because $$n \geq 1$$.” is preferable. Again, in the context of this toy example, our point may come across as too pedantic, but it is good to err on the side of caution and spell things out as much as possible. In another context/proof, a step that you skip might cause the reader a headache.

• Should you break up your proof? This proof does not require any “helper functions”. It is short and simple enough.

• Does your proof type-check? The variable $$n$$ is not introduced properly. From the problem statement, we can infer that $$n$$’s type is a natural number. However, the variable should still be declared properly. For instance, $$F_n$$ is supposed to be the statement “$$2^n > n$$”, but here $$n$$ is undefined/unquantified.

• Are your references clear? There seems to be a reference to an induction hypothesis, but it is poorly presented since there is no indication that the proof is by induction until there is an attempt to make a reference to the induction hypothesis.

• Where are you using the assumptions? The part “because $$n \geq 1$$” is a reference to an assumption that is given by the problem statement, so we expect the author to point this out explicitly. Once again, this point may seem pedantic, but in longer and more complicated proofs, these things really do make a difference.

• Is the proof idea clear? As the given problem and its proof are quite simple, a ‘proof idea’ section is not necessary.

Here is an example of how the proof can be written using good style.

Good solution

We will prove that for all integers $$n \geq 1$$, $$2^n > n$$. The proof is by induction on $$n$$.

Let’s start with the base case which corresponds to $$n=1$$. In this case, the inequality $$2^n > n$$ translates to $$2^1 > 1$$, which is indeed true.

To carry out the induction step, we want to argue that for all $$n \geq 1$$, $$2^n > n$$ implies $$2^{n+1} > n+1$$. We do so now. For an arbitrary $$n \geq 1$$, assume $$2^n > n$$. Multiplying both sides of the inequality by $$2$$, we get $$2^{n+1} > 2n$$. Note that since we are assuming $$n \geq 1$$, we have $$2n = n+n \geq n+1$$. Therefore, we can conclude that $$2^{n+1} > n + 1$$, as desired.

A remark is in order. It is widely accepted that mathematical proofs can be completely formalized in a way that can be mechanically verified, e.g. by a computer. However, writing proofs in a completely formal way is like writing computer programs using machine language. Mathematicians do not communicate proofs in that level of detail.
However, this example is in fact not made up. It was a turned-in solution in CS251 many years ago.
Avoid using long paragraphs as they hurt the clarity of the exposition.
According to Steven Pinker, a cognitive scientist, psychologist, linguist, and a popular science author, the biggest reason why many intelligent people write very poorly is “the curse of knowledge”. People have a hard time imagining what it is like for someone else to not know something that they know.


# How do I solve $32x \equiv 12 \pmod {82}$?

I am able to solve simpler linear congruences, for example $3x \equiv 2 \pmod 5$. What I would do in this case is use that $0 \equiv 10 \pmod 5$ and then utilising a theorem: $3x \equiv 12 \pmod 5$. Then I can divide by $3$ leaving me $x \equiv 4 \: \left( \mathrm{mod} \: {\frac{5}{\mathrm{GCD}(5,3)}} \right) \quad \Longleftrightarrow x \equiv 4 \pmod{5}$ which means the solution is $x = 4k + 5$ where $k \in \mathbb{Z}$.

But I cannot apply the same method to this congruence: $$32x \equiv 12 \pmod {82}$$ This is how far I got: $$8x \equiv 3 \: \left( \mathrm{mod} \: \frac{82}{\mathrm{GCD}(82, 4)} \right)$$ $$\Updownarrow$$ $$8x \equiv 3 \pmod {41}$$

What could I do next? Please provide solutions without the Euclidean algorithm.

EDIT:

What I found later is that I can say that $$0 \equiv 205 \pmod {41}$$ And then I can add it to the congruence in question and divide by $8$. So I guess my question is essentially 'How can I find a number that is a multiple of $41$ (the modulus) and which, if added to $3$ gives a number that is divisible by $8$?'

I reckon the Euclidean algorithm is something which gives an answer to these kinds of questions?!

• It's just too bad that the Euclidean algorithm is the perfect way to tackle these problems. – Lord Shark the Unknown Sep 9 '17 at 11:28
• I'd suggest turning it into algebraic equations without congruence. – user451844 Sep 9 '17 at 11:30
• @LordSharktheUnknown I am aware of that, but as I am a complete beginner in the topic, I'd firstly like to find solutions which do not involve the algorithm which I am not familiar with yet. – bertalanp99 Sep 9 '17 at 11:37
• In that case you may just observe that as $\gcd(5,41)=1$ $$8x\equiv3\pmod{41}\Leftrightarrow40x\equiv15\pmod{41}.$$ Here $40\equiv-1$, so... This amounts to replacing Extented Euclidean Algorithm with a "lucky" observation. – Jyrki Lahtonen Sep 9 '17 at 11:39
• Please see my edit – bertalanp99 Sep 9 '17 at 11:40

Hint :you can do like this $$\quad{8x \equiv 3 \pmod {41}\\ 8x \equiv 3+41 \pmod {41}\\8x \equiv 44 \pmod {41} \div4 \\ 2x \equiv 11 \pmod {41}\\2x \equiv 11+41 \pmod {41}\\2x \equiv 52 \pmod {41}\div 2\\x \equiv 26 \pmod {41}\\x=41q+26}$$

You just have to find the inverse of $8$ modulo $41$.

The general method uses the extended Euclidean algorithm, but in the particular case, it's much simpler: from $5\cdot 8=40\equiv -1\mod 41$, you get at once that $8^{-1}\equiv -5\mod 41$, so $$x\equiv -5\cdot 3=-15\equiv 26\mod 41.$$

basically you are asking how to solve $\frac 1n \mod m$ where $\gcd(m,n)=1$

where $\frac 1n \mod m$ is notation for the $x$ so that $n*x \equiv 1\mod m$.

Let $m = k + qn$ then $nx = 1 + Zm = 1 + Z(k + qn)$ implies

$x = \frac 1n + \frac {Zk}n + Zq$ where $n|Zk + 1$

Ex. $x \equiv \frac 17 \mod 67$. As $67 = 9*7 + 4$ then

$x = \frac {1+Z*4}{7} + 9$

Which means we have to find $Z \equiv -\frac 14 \mod 7$.

Oh... I guess this is Euclid's algorithm.

$7 = 3 + 4$

So $-Z = \frac{1+ 3Y}4 + 1$

Which is clearly $Y =1$ and $-Z \equiv 2\mod 7$ and $Z \equiv -2 \mod 7$

and $x = \frac {1 + 4*(-2)}7 + (-2)*9 \equiv -19 \mod 67$.

And indeed $7*(-19) \equiv -133= (-1)*67 + 1 \equiv 1 \mod 67$

... Yeah, you need Euclid's alogrithm.

OK, without the Euclidean algorithm, you are looking for some $k$ such that $8$ divides $41k+3$, which gives you a new equation in smaller numbers (which is a Euclidean algorithm idea too, but still): $$41k+3\equiv 0 \bmod 8 \\ 41\equiv 1 \bmod 8 \\ k+3 \equiv 0 \bmod 8 \\ k\equiv 5 \bmod 8$$

So then you have your $41\times 5 = 205$ to make the divisibility.

$\, 8x = 3\!+\!41n\!\iff\!\bmod 8\!:\ 0\equiv 3\!+\!41n\equiv 3\!+\!n\!\iff\! n\equiv -3\,$ so $\,x\equiv \frac{3+41(-3))}8\equiv -15\equiv 26$

Alternatively $\bmod 41\!:\ x\equiv \dfrac{3}{8}\equiv \dfrac{15}{40}\equiv\dfrac{15}{-1}\equiv 26\$ by Gauss's algorithm

Alternatively $\bmod 41\!:\ x\equiv \dfrac{3}{8}\equiv \dfrac{44}{8}\equiv \dfrac{11}{2}\equiv\dfrac{52}{2}\equiv 26\$ by adding $\pm 41$ to simplify divisions

Alternatively $\bmod 41\!:\ x\equiv \dfrac{1}8\,\dfrac{3}1\equiv \dfrac{-40}8\ \dfrac{3}1\equiv (-5)3\equiv -15$

Remark  The first method essentially uses a single step of the (extended) Euclidean algorithm, and the second is a special case of that for prime moduli. The other methods are ad-hoc - they try to massage the fractions by adding small multiples of the modulus to make quotients exact / simpler.

Beware $\$ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus. See here for further discussion.

$$32x\equiv12\pmod{82}$$ is equivalent to $$16x\equiv6\pmod{41}$$ and since $(41,2)=1$, we can divide by $2$ $$8x\equiv3\pmod{41}$$ Then noting that $5\cdot8\equiv-1\pmod{41}$, we get that $36\cdot8\equiv1\pmod{41}$. Multiplying both sides by $36$ yields $$\bbox[5px,border:2px solid #C0A000]{x\equiv26\pmod{41}}$$ Note: When looking for the inverse of $a$ mod $m$ it is often a good idea to see if $a\mid(m-1)$ or $a\mid(m+1)$; if either of these hold, they give a quick inverse mod $m$: $a^{-1}\equiv-\frac{m-1}a$ or $a^{-1}\equiv\frac{m+1}{a}$. Above, it was noted that $8|(41-1)$ leading to $8^{-1}\equiv-\frac{41-1}8\pmod{41}$.

Alternate Method of Finding the Inverse of $\boldsymbol{8\pmod{41}}$

Since $41$ is prime and $(41,8)=1$, we know, by Fermat's Little Theorem, that $8^{39}\equiv8^{-1}\pmod{41}$. We can compute $8^{39}\pmod{41}$ using the Square and Multiply Algorithm. $39=100111_\text{two}$, therefore, \begin{align} 8^1&\equiv8&\pmod{41}\\ 8^2&\equiv23&\pmod{41}&&\text{square}\\ 8^4&\equiv37&\pmod{41}&&\text{square}\\ 8^8&\equiv16&\pmod{41}&&\text{square}\\ 8^9&\equiv5&\pmod{41}&&\text{multiply}\\ 8^{18}&\equiv25&\pmod{41}&&\text{square}\\ 8^{19}&\equiv36&\pmod{41}&&\text{multiply}\\ 8^{38}&\equiv25&\pmod{41}&&\text{square}\\ 8^{39}&\equiv36&\pmod{41}&&\text{multiply}\\ \end{align} Therefore, $8^{-1}\equiv36\pmod{41}$.

• You should elaborate on how you "note that $5\cdot 8\equiv -1$" else it amounts to pulling the inverse out of a hit - which is good for magic, but not for math. – Bill Dubuque Sep 10 '17 at 18:33
• @BillDubuque: Is it magic to realize that $5\cdot8=40$? Be careful when casting stones. This is why I added the second method. – robjohn Sep 10 '17 at 18:41
• If you don't say how you "note" that then it is magic, not math. There are of course many ways to compute such inverses, so it is strange that you want to exhibit the inverse but not say how you computed it. Helping someone improve an answer is certainly not "casting stones". In case you may not have noticed, I often give constructive feedback on answers where results are pulled out of a hat. – Bill Dubuque Sep 10 '17 at 19:17
• Certainly, in general, I would not suggest an answer by inspection, however, in this answer you seem to suggest that inspection is a valid approach. However, since I knew that was not a good general strategy, I supplied the alternate, non-Euclidean Algorithm, approach. Furthermore, I have added a description of how I came up with $5\cdot8\equiv-1\pmod{41}$, but as this doesn't work in general, I originally opted to describe the Fermat's Little Theorem approach. – robjohn Sep 10 '17 at 22:12
• Inspection (or brute-force) is much easier there being mod $3$ vs. mod $41$ (and there I say by "inspection or Euclid"). Glad to see that you added an explanation. I call this method the easy inverse case, It is equivalent to the numerator twiddling I do with fractions in my answer, i.e. try $\ 1/a\equiv (1\pm km)/a\pmod{\!m}\,$ for small $k$ to try to make the division exact (i.e. try the first few steps of a brute-force search). It often proves handy for greatly simplifying CRT calculations. – Bill Dubuque Sep 10 '17 at 22:43


Subsequences and convergence. Do all subsequences have to converge to the same limit for the sequence to be convergent?

Prove that $a_n$ converges if and only if:

$a_{2n},a_{2n+1},a_{3n}$ all converge

I thought this was an easy generic question until I read the hint which said: Note: It is not required that the three sub-sequences have the same limit. This needs to be shown

This is what is confusing me because I have found two sources stating something different:

Proposition 4.2. A sequence an converges to L ∈ R if and only if every subsequence converges to L.

and

Let $a_n$ be a real sequence. If the subsequence $a_{2n}$ converges to a real number L and the subsequence $a_{2n+1}$ converges to the same number L, then $a_n$ converges to L as well.

So my question is: for a sequence $a_n$ to converge does it's subsequences have to converge to the same limit? (I suspect not) and if the answer is no can you help me prove why?

• If a sequence converges its subsequences must converge to the same limit. This follows directly from the definition of the limit of a sequence. – John Douma Dec 1 '15 at 21:53

If a sequence has two subsequences that do not both converge to the same limit, then the sequence does not converge.

This can be proven using the $\epsilon-\delta$ definition of convergence:

• Let $a_n$ converge to $L$, and let $\{a_{n_k}\}_{k\to\infty}$ be a subsequence of $a_n$.
• Let $\epsilon > 0$.
• Then, because $a_n$ converges to $L$, there exists some $N$ such that if $n>N$, then $|a_n - L|<\epsilon$.
• Because $n_k$ is an increasing sequence of integers (by definition of a subsequence), there exists such $K$ that $n_K > N$.
• Then, if $k > K$, we have $n_k > n_K > N$, and from the previous point, we get that $|a_{n_k} - L|<\epsilon$, meaning that $a_{n_k}$ converges to $L$.

However, your case is special in that there is an overlap of sequences, for example $a_6$ is in the first and third sequence, and $a_9$ is in the second and third sequence. In fact, the third sequence alternates between the first two sequences, and you can use this to prove all three limits must be equal.

• Ok so for $a_{2n}$ and $a_{2n+1}$ I can say we are either in the odd or the even case and then $a_{3n}$ goes between the two cases. – babylon Dec 1 '15 at 22:10
• @babylon Yup. And that, forces the sequences to converge to the same number. Then you need to prove that the whole sequence also converges to the same number. – 5xum Dec 1 '15 at 22:16

I think the word "required" in the hint is confusing. Focus instead on the last sentence:

Note: It is not required that the three sub-sequences have the same limit. This needs to be shown

In other words, you need to show that as a result of the given hypotheses, the three sub-sequences do have the same limit. You need to do that precisely because it is a necessary condition for $a_n$ to converge.


# Scipy Optimization Example

This short example shows you how to use the scipy minimize function to identify model parameters. This example is set up similarly to the linear least squares example for consistency. Being able to define pretty much anything in a python function, however, gives you great power to customize this, as opposed to a specific approach like linear least squares.

First, we import all the necessary modules

%matplotlib inline
import numpy
import numpy.random
import matplotlib.pyplot as plt
import numpy.linalg
import scipy.optimize

Next create the x matrix

x = numpy.r_[-10:10:.5]

Next, define a y vector based on some model. Note: you can use any of these models or add them together. We scale the output in this example to eliminate the natural weighting of each of these functions over the given range.

#y = x
y = x**2
#y = x**3
#y = numpy.sin(x)
y /= y.max()

Add some noise to y:

rand = numpy.random.randn(*y.shape)/10
y_rand = y + rand

plt.plot(x,y)
plt.plot(x,y_rand,'o')

[<matplotlib.lines.Line2D at 0x7f860d882a30>]

Create an A matrix consisting of several different models

A = numpy.array([(x),(x)**2,(x)**3,numpy.sin(x)]).T

Now create a function that outputs the sum of squared error of each model applied to the given x. In this case we are solving for the weighting coefficients, $k$, as in the linear least squares example:

def myfunc(k):
# make sure our coefficients are in the form of a numpy array
k = numpy.array(k)
# generate y* = Ak^T
y_model = A.dot(k.T)
# sum the square of the error of our model against the input data, y_rand
error = ((y_model-y_rand)**2).sum()
#return the error
return error

Create an initial guess for each of the weights. In this case we just give each coefficient the value of 1 as an initial guess

ini = [1]*A.shape[1]
ini

[1, 1, 1, 1]

Now, call the minimize function. The first value should be the function you are trying to minimize, and the For more information see the optimization function page

sol = scipy.optimize.minimize(myfunc,ini)
sol

      fun: 0.29336548348717617
hess_inv: array([[ 2.35078351e-03, -1.23710160e-05, -3.31813423e-05,
4.86251949e-04],
[-1.23710160e-05,  6.46844761e-06,  3.00084328e-07,
-3.66529338e-05],
[-3.31813423e-05,  3.00084328e-07,  5.62458836e-07,
-1.91576150e-05],
[ 4.86251949e-04, -3.66529338e-05, -1.91576150e-05,
2.78470176e-02]])
jac: array([3.7252903e-09, 3.7252903e-09, 3.7252903e-09, 0.0000000e+00])
message: 'Optimization terminated successfully.'
nfev: 65
nit: 9
njev: 13
status: 0
success: True
x: array([-1.13654781e-03,  1.09398078e-02, -2.22292008e-05,  1.93916411e-02])

sol.x contains the solution for $k$

k_optimum = sol.x
k_optimum

array([-1.13654781e-03,  1.09398078e-02, -2.22292008e-05,  1.93916411e-02])

xx = numpy.r_[:4]
labels = '$x$','$x^2$','$x^3$','$\sin(x)$'
f = plt.figure()
ax.bar(xx,k_optimum)
ax.set_xticks(xx)
ax.set_xticklabels(labels)

[Text(0, 0, '$x$'),
Text(1, 0, '$x^2$'),
Text(2, 0, '$x^3$'),
Text(3, 0, '$\\sin(x)$')]

Now generate $y^*$

y_model = A.dot(k_optimum.T)

Plot the model against the input data

fig = plt.figure()
a = ax.plot(x,y_rand,'.')
b = ax.plot(x,y_model)
ax.legend(a+b,['data','model'])

<matplotlib.legend.Legend at 0x7f860ce494f0>

And plot the residual as well

plt.figure()
plt.plot(x,y_model-y_rand)

[<matplotlib.lines.Line2D at 0x7f860cdb2d00>]

Now try other models, higher resolution data, and different domains


# Algorithm to find a basis of a quotient space $R^n/R^m$.

I have a set of $$m$$ vectors $$\{x_i\}$$, $$x_i \in R^n$$. How can I obtain a basis for $$R^n/span(\{x_i\})$$?

Find a basis of $$\text{span(\{x_i\})}=:W$$, say $$\{x_1,x_2,... ,x_k\}$$, and a basis of $$\mathbb{R}^n$$ of the form $$\{x_1,x_2,...,x_k\}\cup\{y_1,y_2,...,y_{n-k}\}$$. Then the classes $$y_j+W, 1\leq j\leq n-k$$, are a basis of $$\mathbb{R}^n/W$$.
• Yes, but how do we obtain the $\{y_i\}$?. In 3D (n=3), if m=2, we can take the cross product. If m=1, and $x_0=[a,b,c]$ then take $y_0=[−b,a,0]$ and $y_1=x_0 \times y_0$ (with some linear independence assumptions). Does this algorithm generalize? – Scott Oct 19 '18 at 8:56


## Polar Coordinates Pdf

Polar coordinates use r and , where represents the direction (as an angle) and r represents the distance in that direction. 2 (No Test this week) 10. To use polar coordinates to specify a point, enter a distance and an angle separated by an angle bracket (<). 10 (Intro to Polar packet): 1-12 all. The location of a point is expressed according to its distance from the pole and its angle from the polar axis. To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. 2 , 53 o) to rectangular coordinates to three decimal places. My data set is defined in (R, theta) coordinates. Examples Convert ( 6;2) to polar coordinates Solution: r = p ( 6)2 +22 = p 40 ˇ6:325 tan = 1 3, so we ﬁnd tan 1 1 3 ˇ 18:4 , but is in the second quadrant, so ˇ161:6 Convert r = 10, = 276 to Cartesian coordinates. Two points are specified using polar coordinates. The spherical polar coordinate system is like the polar coordinate system, except an additional angle variable is used, frequently labeled as phi (φ). In polar coordinates, the position of the point of contact of the ball at times t and t = 0 respectively are (r,θ) and (r 0, θ 0). 6 Velocity and Acceleration in Polar Coordinates 2 Note. All four types are used in CNC applications, for different machines and different kinds of work. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. Definition of Polar Coordinates. Polar coordinates with polar axes. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees. edu is a platform for academics to share research papers. They plot and label points and identify alternative coordinate pairs for given points. There are other possibilities, considered degenerate. You will then need something like the Free Printable Polar Coordinate Graph Paper. To plot the coordinate, draw a circle centered on point O with that radius. Department of Mathematics - University of Houston. Defining Polar Coordinates. ? $\endgroup$ – Will Jun 10 '15 at 20:41. The coordinate system in such a case becomes a polar coordinate system. If a curve is given in polar coordinates , an integral for the length of the curve can be derived using the arc length formula for a parametric curve. By printing out this quiz and taking it with pen and paper creates for a good variation to only playing it online. 7 7, 6 ⎛⎞π ⎜⎟ ⎝⎠ 2. We convert from polar coordinates to rectangular coordinates and from rectangular coordinates to polar coordinates. Use rectangular coordinates when the number is given in rectangular form and polar coordinates when polar form is used. The polar coordinate system provides an alternative method of mapping points to ordered pairs. When you drag the red point, you change the polar coordinates $(r,\theta)$, and the blue point moves to the corresponding position $(x,y)$ in Cartesian coordinates. Two diﬀerent polar coordinates, say (r 1,θ 1) and (r 2,θ 2), can map to the same point. In this section we will introduce polar coordinates an alternative coordinate system to the 'normal' Cartesian/Rectangular coordinate system. For clockwise rotation, it decreases. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Polar Coordinates x is a function of r and ; y is a function of r and. Polar coordinates use an angle measurement from a polar axis, which is usually positioned as horizontal and pointing to the right. Introduction to polar coordinates. r = secθcscθ ⇒ 24. Polar coord unit vectors and normal. Examples on Converting Polar and Rectangular Coordinates Example 1 Convert the polar coordinates (5 , 2. Until now, we have worked in one coordinate system, the Cartesian coordinate system. The area of a region in polar coordinates defined by the equation $$r=f(θ)$$ with $$α≤θ≤β$$ is given by the integral $$A=\dfrac{1}{2}\int ^β_α[f(θ)]^2dθ$$. Thus, to nd. Then a number of important problems involving polar coordinates are solved. Diﬀerentiatingur anduθ with respectto time t(and indicatingderivatives with respect to time with dots, as physicists do), the Chain Rule gives. txt) or read online for free. Our sailing "Polar" is a diagram showing boatspeed across a range of wind angles and wind speeds, displayed in polar coordinates. Graphing in Polar Coordinates Jiwen He 1 Polar Coordinates 1. Deﬁnition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ), with r > 0 and θ ∈ [0,2π). 42008 S3 Q3 The point P(acos ;bsin ), where a>b>0, lies on the ellipse x2 a2 + y2 b2 = 1: The point S( ea;0), where b2 = a2(1 e2), is a focus of the ellipse. 3) Instead of using (x;y), we describe a point by (r; ) in the polar coordinates where ris its dis-tance from the origin and is the angle it makes with the positive x axis. 2 Calculus In The Polar Coordinate System Contemporary Calculus 4 Area in Rectangular Coordinates (Fig. Polar Coordinates (r,θ) Polar Coordinates (r,θ) in the plane are described by r = distance from the origin and θ ∈ [0,2π) is the counter-clockwise angle. To convert polar coordinates to rectangular coordinates use the formulas: x = r cos y = r sin To convert rectangular coordinates to polar coordinates use the following formulas: r = √x2 + y2 θ = tan-1 (when x > 0) θ = tan-1 + π (when x < 0) y x y x (OR + 180o if it's in degrees). So depending upon the flow geometry it is better to choose an appropriate system. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. 5 Test Review Polar. 2_practice_solutions. r=8sin(θ) Example: The graph of 2 /3 is shown below. To find the coordinates of a point in the polar coordinate system, consider Figure 7. Using di erent names for the radial coordinate, on the other hand, causes few problems. For example, the coordinates of [2, π] do not satisfy the equation. The card10id is a special kind of limaçon. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. as a function of. For example in Lecture 15 we met spherical polar and cylindrical polar coordinates. In polar coordinates rectangles are clumsy to work with, and it is better to divide the region into wedges by using rays. polar coordinates project - Free download as Word Doc (. This system divides the earth into latitude lines, which indicate how far north or south of the equator a location is, and longitude lines, which indicate how far east or west of the prime meridian a location is. 4 The Reference 21 4. Displacements in Curvilinear Coordinates. r is the radius, and θ is the angle formed between the polar axis (think of it as what used to be the positive x-axis) and the segment connecting the point to the pole (what used to be the origin). Unique cylindrical coordinates. Normally, angle x is. In the Menu Bar, choose Layer > Merge Layers. 6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O;the rotating ray or half line from O with unit tick. With the polar grid paper, you can locate someone's exact location. 4 1 x y a The magnitudeof a determines the spread of the parabola: for j a very small, the curve is narrow, and as j a gets large, the parabola broadens. theta: variable to map angle to (x or y) start: Offset of starting point from 12 o'clock in radians. So far, we have described plane curves by giving: y. What is the. Introduction of Polar Coordinates. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. Let r1 denote a unit vector in the direction of the position vector r , and let θ1 denote a unit vector perpendicular to r, and in the direction of increasing θ, see Fig. The coordinates are displayed in the form (r, ). There are some aspects of polar coordinates that are tricky. SYNOPSIS IntreatingtheHydrogenAtom'selectronquantumme-chanically, we normally convert the Hamiltonian from its Cartesian to its Spherical Polar form, since the problem is. Please update your bookmarks accordingly. Polar Coordinates The system was introduced by Newton to more easily describe curves in the rectangular coordinate system and consequently perform Calculus with those curves. Graphs of Polar Equations. Rectangular form to polar form Change x2 + y2 – 2y = 0 to polar form. Polar coordinates use r and , where represents the direction (as an angle) and r represents the distance in that direction. The use of r for the spherical radial coordinate can be confused with the radial coordinate in polar or cylindrical coordinates, but computations requiring both at the same time are rare. Example Sketch the curve described by the polar equation. (If r was negative, then we would head in the opposite direction. We ﬁnd from the above equations that dur dθ = −(sinθ)i +(cosθ)j = uθ duθ dθ = −(cosθ)i−(sinθ)j = −ur. Cartesian coordinates need two lines within an orthogonal system. To view the value of θ. A consensus was reached that planetocentric coordinates should be used and that the selected Lunar Coordinate System should be compatible with the one used within the PDS for Clementine data. The polar axis is usually horizontal and directed toward the right. 5 Polar Coordinates. 2 The naddplot Command: Coordinate Input. See figure -1. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees. Some properties of polar coordinates. Notice that this solution can be transformed back into rectangular coordinates but it would be a mess. Polar Form of an Ellipse—C. This can happen in the following ways: (a) It can happen if r 2 = r 1 and θ 2 = θ 1 ± 2πn for any. Then each point in the plane can be assigned polar coordinates as follows. This is a subtle point but you need to keep that in mind. My data set is defined in (R, theta) coordinates. 5 Test Review Polar. Polar Coordinates x is a function of r and ; y is a function of r and. Navy are to declare the ability to operate and deploy the F-35 in 2016 and 2018 respectively, and full-rate production of the aircraft is to begincapability”) in 2016 and 2018 respectively, and full-rate production decision of the program is planned for 2019. A line through the pole, making angle 0 with the polar axis, has an equation. To use polar coordinates to specify a point, enter a distance and an angle separated by an angle bracket (<). r = tanθ ⇒ 10. The origin is the vertex of the parabola. PHYS 419: Classical Mechanics Lecture Notes POLAR COORDINATES A vector in two dimensions can be written in Cartesian coordinates as r = xx^ +yy^ (1) where x^ and y^ are unit vectors in the direction of Cartesian axes and x and y are the components of the vector, see also the ﬂgure. Consider this exam question to be reminded how well this system works for circular motion:. Solution This time we find x and y from the polar coordinates. But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. The location of P in the plane can also be described using polar coordinates. (5, 960°) SOLUTION: Let P(r, θ) = (5, 960°). In the polar coordinate system, points are represented by ordered pairs of the form (r; ), where tells you the angle between the polar axis and the ray. First try to convert to x and y coordinates, by multiplying by r if necessary and/or a suitable trig substitution. 10 (Intro to Polar packet): 1-12 all. Thus, in this coordinate system, the position of a point will be given by the ordered. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. Examples on Converting Polar and Rectangular Coordinates Example 1 Convert the polar coordinates (5 , 2. If f: [a;b]! Rbe a continuous function and f(x) ‚ 0 then the area of the region between the graph of f and the x-axis is. We also know. • θis measured from an arbitrary reference axis • e r and eθ are unit vectors along +r & +θdirns. Cylindrical Coordinates. HPC - Polar Coordinates Unit Test Sample Open Response Answer Key - Page 2. 21 Locating a point in polar coordinates Let’s look at a specific example. , the z coordinate is constant), then only the first two equations are used (as shown below). 5) i Real Imaginary 6) (cos isin ) Convert numbers in rectangular form to polar form and polar form to rectangular form. There are a total of thirteen orthogonal coordinate systems in which Laplace’s equation is separable, and knowledge of their existence (see Morse and Feshbackl) can be useful for solving problems in potential theory. The fixed point is called the pole and the fixed line is called the polar axis. This article is about Spherical Polar coordinates and is aimed for First-year physics students and also for those appearing for exams like JAM/GATE etc. Note that this definition provides a logical extension of the usual polar coordinates notation, with remaining the angle in the – plane and becoming the angle out of that plane. This allows you to fully utilize the paper size that you have on hand. In polar coordinates, angles are labeled in either degrees or radians (or both). The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. Use Page 2. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. Spherical coordinates system (or Spherical polar coordinates) are very convenient in those problems of physics where there no preferred direction and the force in the problem is spherically symmetrical for example Coulomb's Law due to point. Example contributed by Armin Moser. Thus, in this coordinate system, the position of a point will be given by the ordered. Professional Publications, Inc. De nition (polar coordinate system). State three other pairs of polar coordinates for each point where —2m < 9 < 2m. A point P in the plane, has polar coordinates (r; ), where r is the distance of the point from the origin and is the angle that the ray jOPjmakes with the positive x-axis. The area of a region in polar coordinates defined by the equation $$r=f(θ)$$ with $$α≤θ≤β$$ is given by the integral $$A=\dfrac{1}{2}\int ^β_α[f(θ)]^2dθ$$. b) In the rotated system of Cartesian coordinates (X r, Y r) the X r-axis is parallel to the direction of vector c, defined by initial position r 0 = Xi +Yj and velocity V 0 = Ui. Recall from trigonometry that if x, y, r are real numbers and r 2 = x 2 + y 2, then there is a unique number θ with 0 ≤ θ < 2π such that. Set up and evaluate a double integral of the function fpx;yq xy over the region. Polar coordinates with polar axes. y x x r y θ. 30 Coordinate Systems and Transformation azimuthal angle, is measured from the x-axis in the xy-plane; and z is the same as in the Cartesian system. Find a different pair of polar coordinates for each point such that 0 ≤ ≤ 180° or 0 ≤ ≤ π. Instead, we design P-RSDet which is an anchor-free detector modeled in polar coor-dinates. Math 215 Examples Double Integrals in Polar Coordinates. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates. Stewart Calculus 7e Solutions Chapter 10 Parametric Equations and Polar Coordinates Exercise 10. In polar coordinates, the position of the point of contact of the ball at times t and t = 0 respectively are (r,θ) and (r 0, θ 0). You can select different variables to customize these graphing worksheets for your needs. txt) or read online for free. 6) continued… If the particle is constrained to move only in the r – plane (i. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. Polar coordinates use a different kind of graph instead, based on circles: The center point of the graph (or "origin" in a rectangular grid) is the pole. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. Polar coordinate lines. 1 Þ Locate each of the following points on the polar coordinate system. Thus, to nd. Then we count out a distance of three units along the. In the equation = 5ˇ 4, ris free, so we plot all of the points with polar representation r;5ˇ 4. 3D surface with polar coordinates¶ Demonstrates plotting a surface defined in polar coordinates. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. What is the distance between polar coordinates #(-2, 300^circ)# and #(2, 10^circ)#? What's the difference in finding the distance between two polar coordinates and two rectangular See all questions in Finding Distance Between Polar Coordinates. Double integrals in polar coordinates (Sect. Complete the back of Graphing Roses Revisited and also p. A general system of coordinates uses a set of parameters to deﬁne a vector. You should pay attention to the following: 1. Before we can start working with polar coordinates, we must define what we will be talking about. Two points are specified using polar coordinates. A consensus was reached that planetocentric coordinates should be used and that the selected Lunar Coordinate System should be compatible with the one used within the PDS for Clementine data. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. Plot each point in the complex plane. The divergence We want to discuss a vector ﬂeld f deﬂned on an open subset of Rn. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). 2 We can describe a point, P, in three different ways. In spherical polar coordinates we describe a point (x;y;z) by giving the distance r from the origin, the angle anticlockwise from the xz plane, and the angle ˚from the z-axis. Until now, we have worked in one coordinate system, the Cartesian coordinate system. Polar coordinates When you were ﬁrst introduced to coordinate systems you will have used cartesian coordinates. Example contributed by Armin Moser. This Precalculus video tutorial provides a basic introduction into polar coordinates. Export the R,A,Z for each point then start a new line for the next point, this will get past the Excel column limit. Polar Coordinates Graphs of Polar Equations An equation expressed in terms of polar coordinates is called a polar equation. 2 (No Test this week) 10. Find a formula for. The old vvvv nodes Polar and Cartesian in 3d are similar to the geographic coordinates with the exception that the angular direction of the longitude is inverted. The value of r can be positive, negative, or zero. We will now look at graphing polar equations. 1] can lie on a curve given by a polar equation although the coordinates. 2) Convert the following to polar coordinates: :4,150° ; (‐6, 2) 3) Typical Polar Graphs: Make sure you watch the Application Walk Through Video to see how you should graph these. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). Polar Coordinates Identify the curve by ﬁnding a Cartesian equation for the curve. New Music Updates in your inbox! Enter your email address:. 4 Polar Equations Polar coordinate system is a plane with point O, the pole and a ray from O, the polar axis. I Calculating areas in polar coordinates. In its basic form, Newton's Second Law states that the sum of the forces on a body will be equal to mass of that body times the rate of. Before we can start working with polar coordinates, we must define what we will be talking about. Draw a horizontal line to the right to set up the polar axis. µ = tan¡1 ‡y x ·, if x > 0, 3. Unique cylindrical coordinates. The X and Y relative coordinates are signed numbers. This is the result of the conversion to polar coordinates in form. It is a two-dimensional coordinate system in which each point is at a definite distance from the reference point. r (x ;y)=( rcos( ) sin( )) =ˇ 6 =ˇ 3 Polar coordinates are related to ordinary (cartesian) coordinates by the formulae x = r cos( ) y = r sin( ) r = p x 2+ y = arctan(y=x):. Until now, we have worked in one coordinate system, the Cartesian coordinate system. In Lemma we have seen that the vector r(t) × r˙(t) = C is a constant. Exploring Space Through Math. Polar Curves Curves in Polar Coordinate systems are called Polar Curves, which can be written as r = f(µ) or, equivalently, as F(r;µ) = 0. Different microphones have different recording patterns depending on their purpose. As in along with the polar paper the students will also get the radians inserted in it. to describe using polar coordinates. Then each point in the plane can be assigned polar coordinates as follows. The activity is designed as a puzzle sort and match. com, a free online dictionary with pronunciation, synonyms and translation. In this case, the path is only a function of F r = ma. Allows students to discover what polar coordinates are and how math and art can work together. This substitution would result in the Jacobian being multiplied by 1. Formula Sheet Parametric Equations: x= f(t); y= g(t); t Slope of a tangent line: dy dx = dy dt dx dt = g0(t) f0(t) Area: Z g(t)f0(t)dt Arclength: Z p (f0(t))2 + (g0(t))2dt Surface area: Z p 2ˇg(t) (f0(t))2 + (g0(t))2dt Polar Equations: r= f( ); Polar coordinates to cartesian: x= rcos( ); y= rsin( ) Cartesian coordinates to polar: r= p x2 + y2. Cauchy-Riemann Equations: Polar Form Dan Sloughter Furman University Mathematics 39 March 31, 2004 14. Show the angle θ between two lines with slopes m 1 and m 2 is given by the equation tanθ = m 2 −m 1 1−m 2m 1 I’ve added some more information to the diagram, based on the hint to include the angle the lines make with the x-axis. Arc length and surface area of parametric equations. A polar equation is an equation that tells about the details of the relation between the origin and the coordinates. The 2-D polar coordinates #P ( r, theta)#, r = #sqrt (x^2 + y^2 ) >= 0#. Until now, we have worked in one coordinate system, the Cartesian coordinate system. This creates a visual bias that does not portray actual data. TrigCheatSheet. (Angles may be in degrees or radians) 8. In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. You can use absolute or relative polar coordinates (distance and angle) to locate points when creating objects. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there. We would like to be able to compute slopes and areas for these curves using polar coordinates. An Introduction to Polar Coordinates Polar coordinates are used in many, many fields even at an introductory level. Let (r,θ) denote the polar coordinates describing the position of a particle. Infinitely many angles, and r can also be negative. Then a number of important problems involving polar coordinates are solved. Convert the following equation to polar coordinates: y = − 4 3 x 6. State three other pairs of polar coordinates for each point where —2m < 9 < 2m. By default, angles increase in the counterclockwise direction and decrease in the clockwise direction. 31) Polar coordinates can be calculated from Cartesian coordinates like. If you are looking for basic graph paper, then the Graph Paper Template is the resource you need. com, a free online dictionary with pronunciation, synonyms and translation. Math 2300 Practice with polar coordinates (c) r= 3sin2 0 1 2 3 0 ˇ=2 ˇ 3ˇ=2 Solution: The graph hits the origin at = ˇ 2 and = ˇ, = 3ˇ 2, and = 2ˇ. The method of setting, water coordinates in the AutoCAD by. as a function of. 11) ( , ), ( , ) 12) ( , ), ( , ) Critical thinking question: 13) An air traffic controller's radar display uses polar coordinates. Figure 3: Relationship between coordinate plane and polar plane determine this is shown in gure 3. However, we can use other coordinates to determine the location of a point. Is the point that coordinates are just labels to keep track of where all the points on the manifold are, so within a given patch we are free to choose any coordinate system we like (although in practice we would choose one that suited the problem at hand), not just Cartesian or spherical polar etc. Preview Activity 11. Polar Coordinates Polar coordinates of a point consist of an ordered pair, r θ( , ), where r is the distance from the point to the origin, and θ is the angle measured in standard position. r is a directed distance from the pole to P. Also, you have a DeltaMath assignment that is due Thursday morning. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. doc), PDF File (. In this handout we will ﬁnd the solution of this equation in spherical polar coordinates. Since the axis of the parabola is vertical, the form of the equation is Now, substituting the values of the given coordinates into this equation, we obtain Solving this system, we have Therefore, y 5 or 5x2 14x 3y 9 0. Angles are measured relative to the wind, and shown as "true wind angle" or TWA. theta# determines the direction. Polar coordinates are in the form r, , where is the independent variable. The origin is the vertex of the parabola. k = 5 Since k is odd, we need to replace r with -r to obtain the correct polar coordinates. 11, page 636. 4 The Reference 21 4. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations. 4 5, 4 S SS S SS · ¸ rr ¹ · ¸ r ¹ Yes, there are infinitely many polar coordinates for a given pair of rectangular coordinates. do not satisfy the equation. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). Rectangular form to polar form Change x2 + y2 - 2y = 0 to polar form Solution : Use: r2 = x2 + y2 and y = r sin(θ). Once we've moved into polar coordinates $$dA \ne dr\,d\theta$$ and so we're going to need to determine just what $$dA$$ is under polar coordinates. Polar coordinates When you were ﬁrst introduced to coordinate systems you will have used cartesian coordinates. Preview Activity 11. • Polar–Rectangular conversions where coordinates of points in polar coordinates, say bearings and distances, are converted to rectangular coordinates. 1 Polar form of the Cauchy-Riemann Equations Theorem 14. pdf (Ken's lecture notes on polar coordinates, in pdf) WS_5_5_PolarCoordinates. the standard n-dimensional polar coordinates. In polar coordinates, lines occur in two species. r = sin(3θ) ⇒ 22. coordinates. For example, think of a circle of radius centred on the point. Polar Rectangular Regions of Integration. Representing Polar Coordinates Well, as you already know, a point in the Rectangular or Cartesian Plane is represented by an ordered pair of numbers called coordinates (x,y). A system of coordinates in which the location of a point is determined by its distance from a fixed point at the center of the coordinate space. Example (FEIM): A 2500 kg truck skids with a deceleration of 5 m/s2. Test multiples of 180. the given equation in polar coordinates. There are approximately 20 problems on this. I Calculating areas in polar coordinates. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Points are. Polar Coordinates-measures the distances (and direction) from the origin (radius)& the circle •• (r, f), (radius): •• ndusionf Rectangular Coordinates deal with horizontal & vertical distances, whereas polar coordinates deal with diagonal & circular distances. the part of the solution depending on spatial coordinates, F(~r), satisﬁes Helmholtz’s equation ∇2F +k2F = 0, (2) where k2 is a separation constant. Convert the following equation to polar coordinates: y = − 4 3 x 6. com, a free online dictionary with pronunciation, synonyms and translation. 2_practice_solutions. (If r was negative, then we would head in the opposite direction. r=−2sinθ Identify the polar graph (line, circle, cardioid, limacon, rose): If a circle, name the center (in polar coordinates) and the radius. polar coordinate system synonyms, polar coordinate system pronunciation, polar coordinate system translation, English dictionary definition of polar coordinate system. This point will be labeled with rectangular coordinates instead of polar coordinates. All four types are used in CNC applications, for different machines and different kinds of work. many polar representations in addition to the standard one in the picture above where r >0 and 02≤θ< π. To convert the point (x, y, z) from rectangular to cylindrical coordinates we use: 222 y. b) In the rotated system of Cartesian coordinates (X r, Y r) the X r-axis is parallel to the direction of vector c, defined by initial position r 0 = Xi +Yj and velocity V 0 = Ui. Look it up now!. The point N is the foot of the perpendicular from the origin, O, to the tangent to the ellipse at. When we defined the double integral for a continuous function in rectangular coordinates—say, $$g$$ over a region $$R$$ in the $$xy$$-plane—we divided $$R$$ into subrectangles with sides parallel to the coordinate axes. ) for polar coordinates are shown. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. The point with rectangular coordinates (-1,0) has polar coordinates (1,pi) whereas the point with rectangular coordinates (3,-4) has polar coordinates (5,-0. Polar coordinates use an angle measurement from a polar axis, which is usually positioned as horizontal and pointing to the right. Therefore, in rectangular coordinates, r=sin( ) is written as p x2 + y2=y/ p x2 + y2. ) The graph of = , where is a constant, is the line of inclination. 1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. tan y x θ = y r = sinθ 2 2 2 r x y = + Example 1: Convert the polar coordinate 2 2, 3 π to rectangular form. Press b and choose Trace⎮ Trace Settings to adjust the trace step. In the polar coordinate system, a circle centered at the origin with a radius a units has equation r = a 7KH dartboard has a radius of 225 mm, so its boundary equation is r = 225. The method of setting, water coordinates in the AutoCAD by. 1 r =4secθ r =4secθ ⇒ r secθ =4 ⇒ 4cos(θ) ⇒ x =4 Thus,theCartesianequationisx =4. 4, - 14 A point in polar coordinates is given. To convert polar coordinates to rectangular coordinates use the formulas: x = r cos y = r sin To convert rectangular coordinates to polar coordinates use the following formulas: r = √x2 + y2 θ = tan-1 (when x > 0) θ = tan-1 + π (when x < 0) y x y x (OR + 180o if it's in degrees). I Formula for the area or regions in polar coordinates. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). Math 126 Worksheet 5 Polar Coordinates Graphing Polar Curves The aim of this worksheet is to help you familiarize with the polar coordinate system. The spherical polar coordinate system is like the polar coordinate system, except an additional angle variable is used, frequently labeled as phi (φ). In fact, we will look at how to calculate the area given one polar function, as well as when we need to find the area between two polar curves. 2 , 53 o) to rectangular coordinates to three decimal places. Polar Coordinates (r-θ)Ans: -0. So far, we have described plane curves by giving: y. In the polar coordinate system, the ordered pair will now be (r, θ). In this fun Polar Coordinates, No Prep, Interactive Activities for PreCalculus and Trigonometry your students practice both graphing polar coordinates and also finding equivalent forms of polar coordinates. Complete the unit circle with each angles' coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. In polar coordinates, if ais a constant, then r= arepresents a circle. Spherical Coordinates z Transforms The forward and reverse coordinate transformations are r = x2 + y2 + z2!= arctan" x2 + y2,z # $% &= arctan(y,x) x = rsin!cos" y =rsin!sin" z= rcos! where we formally take advantage of the two argument arctan function to eliminate quadrant confusion. To convert from Polar coordinates to Cartesian coordinates, draw a triangle from the horizontal axis to the point. The coordinate systems allow the geometrical problems to be converted into a numerica. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. For example, we will review trigonometric concepts, such as trigonometric identities and real valued functions with points on the coordinate plane, when learning the polar coordinate system. 3 1 x y a Figure 11. The magnetic turbulence is confined near the auroral zone and is similar to that seen at higher altitudes by HEOS-2 in the polar cusp. 1 POLAR COORDINATES Polar coordinate system: a pole (fixed point) and a polar axis (directed ray with endpoint at pole). [email protected] Polar Coordinates T NOTES MATH NSPIRED ©2015 Texas Instruments Incorporated education. Here we provide you with free printable graph paper pdf. 3: Double Integrals in Polar Coordinates We usually use Cartesian (or rectangular) coordinates (x;y) to represent a point P in the plane. Student information Link. NCT program example to show how G81 drilling cycle can be used to drill in a circle using G15 G16 Polar Coordinate Commands and G81 Drilling Cycle. It has been accepted for inclusion in Chemistry Education Materials by an authorized administrator of [email protected] In the ﬁrst two cases,. 3) Instead of using (x;y), we describe a point by (r; ) in the polar coordinates where ris its dis-tance from the origin and is the angle it makes with the positive x axis. Unique cylindrical coordinates. I Double integrals in disk sections. Double Integrals in Polar Coordinates 1. Applications [ edit ] Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. Find a formula for. Its graph is the circle of radius k, centered at the pole. 4 2D Elastostatic Problems in Polar Coordinates Many problems are most conveniently cast in terms of polar coordinates. Convert the following rectangular coordinates to polar form. The equations are easily deduced from the standard polar triangle. 4) I Review: Polar coordinates. b) In the rotated system of Cartesian coordinates (X r, Y r) the X r-axis is parallel to the direction of vector c, defined by initial position r 0 = Xi +Yj and velocity V 0 = Ui. Conversion: Rectangular to Polar/ Polar to Rectangular 2011 Rev by James, Apr 2011 1. To this end, first the governing differential equations discussed in Chapter 1 are expressed in terms of polar coordinates. Mon Nov 11 - I retaught graphing roses and then we began converting from polar form to rectangular and rectangular to polar. The 50 -point section has a radius of 6. Coordinates in AutoCAD. I Calculating areas in polar coordinates. Corrective Assignment. The axial coordinate or height z is the signed distance from the chosen plane to the point P. 7) Partition the domain x of the rectangular coordinate function into small pieces ∆x. The fact that a single point has many pairs of polar coordinates can cause complications. (As a teacher, one of my favorite questions on homework or exams will be to ask what happens when $$r$$ is negative. In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Introduction to the Polar Coordinate System A polar coordinate system consists of a fixed point (called the pole or origin) and a ray from the origin (called the polar axis). We will look at polar coordinates for points in the xy-plane, using the origin (0;0) and the positive x-axis for reference. In this polar coordinates worksheets, students change ordered pairs from rectangular form to polar form. [2] Polar Coordinate System, Summary article about the polar coordinate system. Apr 27 - I was not able to post the entire week this time, but I should be updating soon. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. Click on the tags below to find other worksheets in the same. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. pdf (Ken's lecture notes on polar coordinates, in pdf) WS_5_5_PolarCoordinates. The polar coordinates (r,θ) are related to the usual rectangular coordinates (x,y) by by x = r cos θ, y = r sin θ The ﬁgure below shows the standard polar triangle relating x, y, r and θ. See Large Polar Graph Paper. the report the polar coordinates of each hit(you can get the polar values at that time), to export the data into a CSV file it is probably easier to create your own utility to do that. Integration in polar coordinates Polar Coordinates Polar coordinates are a diﬀerent way of describing points in the plane. Coordinates in AutoCAD. In Polar Coordinate System, the references are a fixed point and a fixed line. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. Polar Form of an Ellipse—C. Cylindrical and spherical coordinates Recall that in the plane one can use polar coordinates rather than Cartesian coordinates. Consider this exam question to be reminded how well this system works for circular motion:. In fact, we will look at how to calculate the area given one polar function, as well as when we need to find the area between two polar curves. pdf (Worksheet practicing this material, in pdf) WS_Soln_5. There are a total of thirteen orthogonal coordinate systems in which Laplace’s equation is separable, and knowledge of their existence (see Morse and Feshbackl) can be useful for solving problems in potential theory. ;) 21) ( , ), ( , ) 22) ( , ). Concentric Circles: 17 vs 13 Polar Radians. Frame of Reference In the polar coordinate system, the frame of reference is a point O that we call the pole and a ray that. To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. We ﬁnd from the above equations that dur dθ = −(sinθ)i +(cosθ)j = uθ duθ dθ = −(cosθ)i−(sinθ)j = −ur. The area of a region in polar coordinates defined by the equation with is given by the integral ; To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas. An angle is considered positive if measured in the counterclockwise direction from the polar axis, and negative if measured in the. 7) Partition the domain x of the rectangular coordinate function into small pieces ∆x. 2) Convert the following to polar coordinates: :4,150° ; (‐6, 2) 3) Typical Polar Graphs: Make sure you watch the Application Walk Through Video to see how you should graph these. Using standard trigonometry we can find conversions from Cartesian to polar coordinates and from polar to Cartesian coordinates Example. Let r1 denote a unit vector in the direction of the position vector r , and let θ1 denote a unit vector perpendicular to r, and in the direction of increasing θ, see Fig. Polar Coordinates. WaterproofPaper. 31) Polar coordinates can be calculated from Cartesian coordinates like. 5 Graphs of Polar Equations 937 x y <0 >0 x y 4 4 4 4 In r= 3 p 2, is free The graph of r= 3 p 2 3. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. L-01 (Cartesian and Polar coordinates ). The polar coordinate system will be useful for many problems you encounter at MIT, such as those involving circular motion or radial forces. 3 mm, so its boundary equation is r = 6. Watch today's lesson and complete pp. My data set is defined in (R, theta) coordinates. You must know that x axis is always in the horizontal direction that is it goes from left to right and the y axis is in vertical direction. coordinates. Such definitions are called polar coordinates. 1 Polar Coordinates - PRACTICE. 2 Slopes in r pola tes coordina When we describe a curve using polar coordinates, it is still a curve in the x-y plane. Preview Activity 11. Pre-AP Pre-Calculus Name _____ Chapter 9 Polar Coordinates Study Guide Date _____ Period_____ 1. 1 DEFINITION OF CYLINDRICAL COORDINATES A location in 3-space can be defined with (r, θ, z) where (r, θ) is a location in the xy plane defined in polar coordinates and z is the height in units over the location (r, θ)in the xy plane Example Exercise 11. Complete the unit circle with each angles’ coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. In Lemma we have seen that the vector r(t) × r˙(t) = C is a constant. (As a teacher, one of my favorite questions on homework or exams will be to ask what happens when $$r$$ is negative. 4 (Circular motion). In polar coordinates, every point is located around a central point, called the pole, and is named (r,nθ). Use double integrals in polar coordinates to calculate areas and volumes. You should pay attention to the following: 1. png 488 × 468; 81 KB. There are approximately 20 problems on this. Precalculus Examples. Apr 11, 2014 - Explore brittanykaye911's board "polar coordinates", followed by 154 people on Pinterest. Consider Figure 13. 4 Polar Coordinate System Blank; 6. 5 Systems of Linear Inequalities; 7. Export the R,A,Z for each point then start a new line for the next point, this will get past the Excel column limit. a polar equation is the set of all points in the plane that can be described using polar coordinates that satisfy the equation. Cartesian coordinates need two lines within an orthogonal system. Find polar coordinates for the point with rectangular coordinates 00,. The arc length of a polar curve defined by the equation with is given by the integral. The polar coordinate system provides an alternative method of mapping points to ordered pairs. units away from the last point entered. Suppose f is deﬁned on an neighborhood U of a point z 0 = r 0eiθ 0, f(reiθ) = u(r,θ)+iv(r,θ), and u r, u θ, v r, and v θ exist on U and are continuous at (r 0,θ 0). The polar coordinate system (r, θ) and the Cartesian system (x, y) are related by the following expressions: With reference to the two-dimensional equ ations or stress transformation. In this section we will look at converting integrals (including dA) in Cartesian coordinates into Polar coordinates. Note that this definition provides a logical extension of the usual polar coordinates notation, with remaining the angle in the – plane and becoming the angle out of that plane. In Polar Coordinate System, the references are a fixed point and a fixed line. I Double integrals in disk sections. rectangular coordinates ⇒ polar coordinates polar coordinates ⇒ rectangular coordinates N=√ T2+ 2 U, 𝜃= P T= N K O𝜃 U= N O𝑖𝜃 The angle, θ, is measured from the polar axis to a line that passes through the point and the pole. Equilibrium equations in polar coordinates Hooke’s Law in polar coordinates √ Miner’s rule Crack Propagation √ √ @ A @ A Strain displacement Equations in Polar Coordinates Airy stress function in polar Coordinates Fracture mechanics √ von Mises effective stress: for 2-D √ Maximum Distortion Energy Theory:. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). 3 WS Polar Coordinates (Answers). Pre-Calculus Worksheet Name: _____ Section 10. A region R in the xy-plane is bounded below by the x-axis and above by the polar curve defined by 4 1 sin r T for 0 ddTS. Polar Graph Paper – free line graph paper polar in petech pregenerated files usually i put the most useful outputs here i still did that but i also tried odd things what happens when you divide a circle by 365 25 and also 12 5 free printable polar graph paper in pdf polar graph paper radians this is an advanced form of paper that will be available by us as in along with the polar paper the. 1 The Axis-Environments. Area of regions in polar coordinates (Sect. Spherical-polar coordinates. Department of Mathematics - University of Houston. Input the Cartesian coordinates of P (1, 1), x first. We recall that the Dirichlet problem for for circular disk can be written in polar coordinates with 0 r R, ˇ ˇ as u= u rr+ 1 r u r+ 1 r2 u = 0 u(R; ) = f( ): 6. 2 (pdf) S&Z 11. So depending upon the flow geometry it is better to choose an appropriate system. We need to show that ∇2u = 0. Cartesian coordinate system: start with xand yaxes. Watch today's lesson and complete pp. Polar Graph Paper – free line graph paper polar in petech pregenerated files usually i put the most useful outputs here i still did that but i also tried odd things what happens when you divide a circle by 365 25 and also 12 5 free printable polar graph paper in pdf polar graph paper radians this is an advanced form of paper that will be available by us as in along with the polar paper the. The 2-D polar coordinates #P ( r, theta)#, r = #sqrt (x^2 + y^2 ) >= 0#. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. So let us first set us a diagram that will help us understand what we are talking about. Polar coordinates. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. The use of r for the spherical radial coordinate can be confused with the radial coordinate in polar or cylindrical coordinates, but computations requiring both at the same time are rare. 1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. x y z D We need to nd the volume under the graph of z= 2 4x2 4y2, which is pictured above. pdf from MATH 111 at American Public University. This is one application of polar coordinates, represented as (r, θ). Professional Publications, Inc. [2] Polar Coordinate System, Summary article about the polar coordinate system. I also presume length judgements in polar coordinates are more difficult. Example of finding the polar coordinates of a point Give the four basic polar coordinates of points A, B, C, and D shown in the figure. 5) i Real Imaginary 6) (cos isin ) Convert numbers in rectangular form to polar form and polar form to rectangular form. The Laplacian in Spherical Polar Coordinates C. Cylindrical coordinates are defined with respect to a set of Cartesian coordinates, and can be converted to and from these coordinates using the atan2 function as follows. based and Cartesian coordinates modeling. Complete the unit circle with each angles' coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. We can thus regard f as a function from Rn to Rn, and as such it has a derivative. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. Coordinates were specified by the distance from the pole and the angle from the polar axis. For example the graph of the equation x2 + y2 = a we know to be a circle, if a > 0. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. To use polar coordinates to specify a point, enter a distance and an angle separated by an angle bracket (<). Notice that this solution can be transformed back into rectangular coordinates but it would be a mess. Solution: The function that we need to use in this example is G, which converts the pair of rectangular coordinates (x,y) into the polar coordinates (r,!). The coordinates of a point determine its location. 1 Equilibrium equations in Polar Coordinates One way of expressing the equations of equilibrium in polar coordinates is to apply a change of coordinates directly to the 2D Cartesian version, Eqns. In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. L-01 (Cartesian and Polar coordinates ). 3 mm, so its boundary equation is r = 6. Polar Coordinates-measures the distances (and direction) from the origin (radius)& the circle •• (r, f), (radius): •• ndusionf Rectangular Coordinates deal with horizontal & vertical distances, whereas polar coordinates deal with diagonal & circular distances. By printing out this quiz and taking it with pen and paper creates for a good variation to only playing it online. If we restrict rto be nonnegative, then = describes the. ) for polar coordinates are shown. The distance of these lines passing throw the origin or pole is called radians. Plane Curvilinear Motion Polar Coordinates (r -θ) The particle is located by the radial distance r from a fixed point and by an angular measurement θto the radial line. 11, page 636. 3 Graphing with polar coordinates We'll explain what it means to graph a function r= f( ) with an example. 1] can lie on a curve given by a polar equation although the coordinates. Homework 2: Spherical Polar Coordinates Due Monday, January 27 Problem 1: Spherical Polar Coordinates Cartesian coordinates (x,y,z) and spherical polar coordinates (r,θ,ϕ) are related by x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ. 4 2D Elastostatic Problems in Polar Coordinates Many problems are most conveniently cast in terms of polar coordinates. The divergence We want to discuss a vector ﬂeld f deﬂned on an open subset of Rn. 1 Cylindrical coordinates If P is a point in 3-space with Cartesian coordinates (x;y;z) and (r; ) are the polar coordinates of (x;y), then (r; ;z) are the cylindrical coordinates of P. In this note, I would like to derive Laplace’s equation in the polar coordinate system in details. But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Example contributed by Armin Moser. Double integrals in polar coordinates (Sect. Unit Vectors The unit vectors in the cylindrical coordinate system are functions of position. The polar coordinate system provides an alternative method of mapping points to ordered pairs. Tangents of polar curves. In certain problems, like those involving circles, it is easier to define the location of a point in terms of a distance and an angle. Applications [ edit ] Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. NCT program example to show how G81 drilling cycle can be used to drill in a circle using G15 G16 Polar Coordinate Commands and G81 Drilling Cycle. A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. We are used to using rectangular coordinates, or xy-coordinates. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to. Polar Coordinates. Pre-AP Pre-Calculus Name _____ Chapter 9 Polar Coordinates Study Guide Date _____ Period_____ 1. 180 Spoke Radians. pdf), Text File (. In Lemma we have seen that the vector r(t) × r˙(t) = C is a constant. x2 24y 96 0 x2 4 6 y 4 x h 2 4p y k 25. Coordinates in AutoCAD. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. 12/9- Polar and rectangular coordinates VECTORS Re-TEST THIS WEEK 12/10- converting polar to rectangular equations 12/11- exploration of special polar equations 12/12- Group Project - finish special polar graphs 12/13 - review 12/16 - Test day PROJECT DUE 12/19 5/13 - review TEST 12/16 PROJECT DUE: 12/19. On questions 7-10, you should write your answers in degrees. This can happen in the following ways: (a) It can happen if r 2 = r 1 and θ 2 = θ 1 ± 2πn for any. TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES 3 Notice how easy it is to nd the area of an annulus using integration in polar coordinates: Area = Z 2ˇ 0 Z 2 1 rdrd = 2ˇ[1 2 r 2]r=2 r=1 = 3ˇ: [We are nding an area, so the function we are integrating is f= 1. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. a) , Š ‹%$ 1 b) , Œ % # \$ 1 c) ,Œ % & % 1 d) , Œ " (' 1. 686 CHAPTER 9 POLAR COORDINATES AND PLANE CURVES The simplest equation in polar coordinates has the form r= k, where kis a positive constant.
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NumPy: Eigenvalues & Eigenvectors

In this tutorial, we will explore NumPy's numpy.linalg.eig() function to deduce the eigenvalues and normalized eigenvectors of a square matrix.

Let $A$ be a square matrix.

In Linear Algebra, a scalar $\lambda$ is called an eigenvalue of matrix $A$ if there exists a column vector $v$ such that

$$Av = \lambda v$$

and $v$ is non-zero. Any vector satisfying the above relation is known as eigenvector of the matrix $A$ corresponding to the eigen value $\lambda$.

We take an example matrix from a Schaum's Outline Series book Linear Algebra (4th Ed.) by Seymour Lipschutz and Marc Lipson1.

Given the matrix

$$A = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix},$$

the column vector

$$v = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$$

is its eigenvector corresponding to the eigenvalue $\lambda = 1$ as

$$Av = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ -2 \end{bmatrix} = \begin{bmatrix} 1 \\ -2 \end{bmatrix} = 1v$$

Also,

$$u = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$

is its another eigenvector corresponding to the eigenvalue $\lambda = 4$ as

$$Au = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 4 \end{bmatrix} = 4 \begin{bmatrix} 1 \\ 1 \end{bmatrix} = 4u$$

NumPy has the numpy.linalg.eig() function to deduce the eigenvalues and normalized eigenvectors of a given square matrix. And since the returned eigenvectors are normalized, if you take the norm of the returned column vector, its norm will be 1.

So, take the cue from here. Since the returned eigenvectors are NORMALIZED, they may not always be the same eigenvectors as in the texts you are referring.

Note the two variables w and v assigned to the output of numpy.linalg.eig(). The first variable w is assigned an array of computed eigenvalues and the second variable v is assigned the matrix whose columns are the normalized eigenvectors corresponding to the eigenvalues in that order.

import numpy as np

a = np.array([[3, 1], [2, 2]])
w, v = np.linalg.eig(a)

print(w)
print(v)

Executing the above Python script, the output is as follows:

Here we will explain the output. The first printed array is w, which constitutes the eigenvalues. The second printed matrix below it is v, whose columns are the eigenvectors corresponding to the eigenvalues in w. Meaning, to the w[i] eigenvalue, the corresponding eigenvector is the v[:,i] column in matrix v.

In NumPy, the ith column vector of a matrix v is extracted as

v[:,i]

So, the eigenvalue

• w[0] goes with v[:,0]
• w[1] goes with v[:,1]

We will now check if the condition

$$Av = \lambda v$$

holds here.

The LHS of the above equation $Av$ here is

np.dot(a,v[:,0])

and the RHS part $\lambda v$ is

np.dot(w[0],v[:,0])

So if the returned eigenvalues and eigenvectors are correct, the following line of script should return True

print(np.allclose(np.dot(a,v[:,0]),np.dot(w[0],v[:,0])))

Also, just to see if the returned eigenvectors are normalized, use the numpy.linalg.norm() function to cross-check them. The below script should return 1.0 in both the print() statements.

print(np.linalg.norm(v[:,0]))
print(np.linalg.norm(v[:,1]))

Notes

• 1) Seymour Lipschutz and Marc Lipson, Linear Algebra. McGraw-Hill Companies, Inc, 2009. Chapter 9: Diagonalization: Eigenvalues and Eigenvectors, p. 297, Ex. 9.5.
• The eigenvectors returned by the numpy.linalg.eig() function are normalized. So, you may not find the values in the returned matrix as per the text you are referring.


# Thread: *sigh*... need to prove somthing...

1. ## *sigh*... need to prove somthing...

I'm told that gcd(a,n)=1 and gcd(b,n)=1, a,b,n are natural.
I need to prove that gcd(ab,n)=1.
I don't succeed :-\ Tried millions of things!

2. Originally Posted by aurora
I'm told that gcd(a,n)=1 and gcd(b,n)=1, a,b,n are natural.
I need to prove that gcd(ab,n)=1.
I don't succeed :-\ Tried millions of things!
Say $\gcd(ab,n)=d > 1$. Then $d|ab$ and $d|n$. Note $\gcd(d,a)=1$ since $d|n$ and that would imply $\gcd(a,n) \geq d > 1$ a contradiction, so $\gcd(d,a)=1$. But then $d|ab \implies d|b$. And so $\gcd(b,n)\geq d > 1$ a contradiction. Thus, $d=1$.

3. Originally Posted by aurora
I'm told that gcd(a,n)=1 and gcd(b,n)=1, a,b,n are natural.
I need to prove that gcd(ab,n)=1.
I don't succeed :-\ Tried millions of things!
so we have $ra+sn=kb+\ell n=1,$ for some integers $r,s,k,\ell.$ thus $(kr)ab + (ksb + \ell)n=1. \ \ \square$

4. Thank you very much!
However, NonCommAlg, correct me if I'm mistaken - but the fact that you find this sort of linear presentation of a and n that equals 1 doesn't mean that 1 is their gcd...

5. Originally Posted by aurora
Thank you very much!
However, NonCommAlg, correct me if I'm mistaken - but the fact that you find this sort of linear presentation of a and n that equals 1 doesn't mean that 1 is their gcd...
You are mistaken..

If gcd(n, ab) = d > 0 then d|n and d|ab => d| (kr)ab + (ksb + l)n => d|1 => d = 1

However note that NonCommAlg's trick only works for gcd of 1

6. Originally Posted by Isomorphism
You are mistaken..

If gcd(n, ab) = d > 0 then d|n and d|ab => d| (kr)ab + (ksb + l)n => d|1 => d = 1

However note that NonCommAlg's trick only works for gcd of 1
I agree. However, in my algebra class, my professor would require that you show the extra step (what you just did), so i see where aurora is coming from. i would have done it the way NonCommAlg did it. but i like the contradiction method, it seems elegant to me

7. Thank you guys

8. Originally Posted by Jhevon
I agree. However, in my algebra class, my professor would require that you show the extra step (what you just did), so i see where aurora is coming from. i would have done it the way NonCommAlg did it. but i like the contradiction method, it seems elegant to me
I like the contradiction method too. But I was telling aurora about the idea because its useful in an exam. I think its more mechanical.Contradiction may not hit at the right time

9. Originally Posted by Isomorphism
I like the contradiction method too. But I was telling aurora about the idea because its useful in an exam. I think its more mechanical.Contradiction may not hit at the right time
I agree. that's one of the reasons i like it. it is not so obvious at the moment you see the problem. NonCommAlg's way is something you would do on impulse to see how it works out, which is the charm of that method. good for tests when you can't think clearly or be fancy, but are racing against the clock and just need to get the job done


# Prove the fractional field of an integral domain is the smallest field containing the integral domain

I have two questions about the fractional field of an integral domain.

Given an integral domain $D$:

1. Is there a difference between saying "the fractional field of $D$ is the smallest field containing $D$" or "the fractional field of $D$ is the smallest field containing an embedding of $D$"?

2. How do you prove that the fractional field is the smallest field containing $D$ (or an embedding of $D$, if there is a difference...)? Specifically, I want to show that if $F$ is any field containing $D$, then $F$ must contain the fractional field of $D$.

• What is your definition of "fractional field of integer domain"? For me, it is precisely the minimal field containing the domain. – Timbuc Oct 11 '14 at 18:52
• @timbuc *integral domain. My definition is based on the construction. First, given an integral domain $D$, we define the set of ordered pairs where the second coordinate is non-zero. Then we define an equivalence relation where $(a,b) \sim (c,d)$ if $ad = bc$. Then we call the equivalence class containing $(a,b)$ as $\frac{a}{b}$. Then we define addition and multiplication on this set of equivalence classes in the same way as it is defined in $\mathbb{Q}$. This forms a field, what we call the fractional field. – layman Oct 11 '14 at 18:55
• Ok, but then it is trivial, isn't it? I mean, any field containing $\;D\;$ must contain all the multiplicative inverses of non-zero elements $\;d\in D\;$ and thus their product by any element in $\;D\;$ , and this means (by the definition!) that $\;d_1\cdot\frac1{d_2}:=\frac{d_1}{d_2}\;$ is in the field, for any $d_1,d_2\in D\;,\;\;d_2\neq 0\;$ , which means any such field contains the fractions field of $\;D\;$ . – Timbuc Oct 11 '14 at 18:57
• @Timbuc Well now I need to know the answer to my first question. Is there a difference between saying a field contains $D$ and saying it contains an embedding of $D$? – layman Oct 11 '14 at 19:00
• Well, formally there is, yet for most usual cases one doesn't usually pay attention to that slight difference. – Timbuc Oct 11 '14 at 19:01

Let $F'$ be a smallest field containing an embedding of $D$ ($f:D\to F'$), and $F$ a field of fraction of $D$.

We can extend $f$ to morphism of field $\tilde f:F\to F'$ by $\tilde f(a/b)=f(a)/f(b)$.

Now we have that $\tilde f(F)\subseteq F'$ and $\tilde f(F)$ containing an embedding of $D$ , by smallest property we have $\tilde f(F)=F'$.

So the tow fields $F$ and $F'$ are isomorphic.

Edit: If $F$ is any field containing $D$. And denote $K$ the field of fraction of $D$.

Let $a/b\in K$, $a\in D$ and $0\neq b\in D$, hence $a,b\in F$, it follow that $a$ and $1/b$ are in $F$ so $a. (1/b)=a/b\in F$. Thus $K\subseteq F$.

• Is your extension of $f$ well defined? – layman Oct 11 '14 at 19:26
• I guess it is since $f$ is a homomorphism, and if $a/b = c/d$, then $ad = bc$, so $f(a)f(d) = f(b)f(c)$ which implies $f(a)/f(b) = f(c)/f(d)$. – layman Oct 11 '14 at 19:28
• Yes $b\neq 0$ implies $f(b)\neq 0$ because $f(0)=0\neq f(b)$ ($f$ is injective). – Hamou Oct 11 '14 at 19:28
• I guess I would need to do a little bit more work because I would need to show the image of the field $F$ under the extension of $f$ is a field. – layman Oct 11 '14 at 19:29
• When $\frac ab\neq 0$, $\tilde f(a/b)\tilde f(b/a)=1$, hence $\tilde f(a/b)$ is invertible is in $\tilde f(F)$. – Hamou Oct 11 '14 at 19:32

The right (i.e. categorical) way to say this (without the ambiguities of words like "smallest", "containing", etc.) ought to be that the inclusion $\iota: D\to Q(D)$ has the following universal property:

If $K$ is a field, and $f: D\to K$ is any morphism of rings, then there is a unique morphism of fields $g : Q(D) \to K$ such that $f = g \circ \iota$.

(In particular, $Q(D)$ embeds into any field that $D$ embeds into.)

This property uniquely determines (up to isomorphism) not only $Q(D)$, but $\iota$ as well.

And it's easily proved, since $g(1/b)g(b)=g(1)$ forces $g(a/b) = f(a)/f(b)$, so this amounts to checking that $a/b \mapsto f(a)/f(b)$ is actually a homomorphism.