# Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $$G=SU(N)$$.

We know there is an exact sequence: $$0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.$$

For $$G=SU(2)$$, we have:

• $$\text{Z}(SU(2)) =\mathbb Z_2$$,
• $$\text{Inn}(SU(2)) = SO(3)$$,
• $$\text{Out}(SU(2)) = 0$$,

And so $$\text{Aut}(SU(2))=SO(3)$$.

For $$N > 2$$, we have:

• $$\text{Z}(SU(N)) =\mathbb Z_{N}$$,
• $$\text{Inn}(SU(N)) = PSU(N)$$,
• $$\text{Out}(SU(N)) = \mathbb Z_2$$.

My question is:

Does $$\text{Aut}(SU(N))=PSU(N) \times \mathbb Z_2$$? If not, does this answer depend on whether $$N$$ is odd or even?

It looks to me that there is a nontrivial fibration depending on something like $$H^2(B\mathbb Z_2,PSU(N))$$ due to $$B\text{Inn}(G) \to B\text{Aut}(G) \to B\text{Out}(G) \to B^2\text{Inn}(G) \to$$ and thus $$BPSU(N) \to B\text{Aut}(G) \to B\mathbb Z_2 \to B^2PSU(N) \to$$

But I do not know how to define $$H^2(B\mathbb Z_2,PSU(N))$$, if this is a correct thing to ponder.

• this may be preliminary for answering your question: mathoverflow.net/questions/40666/… - which you may already know well. – wonderich May 1 at 3:48
• In fact this completely answers the question. – abx May 1 at 4:05
• what is the answer? I am asking the total Aut of $SU(N)$? – annie heart May 1 at 4:46
• p.s. I am not asking Inn or Out of $SU(N)$. – annie heart May 1 at 4:51
• The answer is that it is a semi-direct product $\operatorname{PSU} (N)\rtimes \mathbb{Z}/2$, with $\mathbb{Z}/2$ acting on $\operatorname{PSU}(N)$ by conjugation. – abx May 1 at 5:58

Let $$G$$ be a compact simple simply connected Lie group. Then any automorphism of $$G$$ determines an automorphism of its Lie algebra $$\mathfrak{g}$$ and visa versa. So $$\mathrm{Aut}(G)$$ is naturally isomorphic to the linear group $$\mathrm{Aut}(\mathfrak{g})$$.

The sequence $$1\to \mathrm{Inn}(\mathfrak{g})\to \mathrm{Aut}(\mathfrak{g})\to \mathrm{Out}(\mathfrak{g})\to 1$$ is split.

Moreover, $$\mathrm{Out}(G)\cong\mathrm{Out}(\mathfrak{g})\cong \mathrm{Aut}(D_\mathfrak{g})$$ where $$D_\mathfrak{g}$$ is the Dynkin diagram of $$\mathfrak{g}$$.

The upshot is:

$$\mathrm{Aut}(G)\cong \mathrm{Aut}(\mathfrak{g})\cong \mathrm{Inn}(\mathfrak{g})\rtimes \mathrm{Out}(G)\cong \mathrm{Inn}(\mathfrak{g})\rtimes \mathrm{Aut}(D_\mathfrak{g}).$$

For types $$A_1, B_n, C_n, G_2, F_4, E_7, E_8$$ there are no symmetries of the Dynkin diagram. For $$A_n$$ ($$n>1$$), $$D_n$$ ($$n\not=4$$), and $$E_6$$, we have $$\mathrm{Aut}(D_\mathfrak{g})\cong \mathbb{Z}/2\mathbb{Z}$$. And in the final case of $$D_4$$, the symmetry group is the symmetric group on three letters.

In particular, as stated in the comments: $$\mathrm{Aut}(\mathrm{SU}(n))\cong\left\{\begin{array}{ll}\mathrm{PSU}(n)\rtimes \mathbb{Z}/2\mathbb{Z},&\text{ if } n\geq 3\\ \mathrm{PU}(2),&\text{ if }n=2. \end{array}\right.$$

• PSU(2) = PU(2), yes? – annie heart May 6 at 19:27
• Does it mean that the $\mathrm{Aut}(\mathrm{PU}(n))=\mathrm{Aut}(\mathrm{PSU}(n))=\mathrm{Aut}(\mathrm{SU}(n))$, since they have the same Lie algebra: $$\mathrm{Aut}(\mathrm{PSU}(n))\cong\left\{\begin{array}{ll}\mathrm{PSU}(n)\rtimes \mathbb{Z}/2\mathbb{Z},&\text{ if } n\geq 3\\ \mathrm{PU}(2),&\text{ if }n=2. \end{array}\right. ?$$ – annie heart May 6 at 19:35
• Do we have $$\mathrm{Out}(G)\cong\mathrm{Out}(\mathfrak{g})?$$ $$\mathrm{Inn}(G)\cong\mathrm{Inn}(\mathfrak{g})?$$ $$\mathrm{Aut}(G)\cong\mathrm{Aut}(\mathfrak{g})?$$ in general? – annie heart May 6 at 19:43
• $PU(n)\cong PSU(n)$ in general. For simply-connected Lie groups, $\mathrm{Aut}(G)\cong \mathrm{Aut}(\mathfrak{g})$ and $\mathrm{Out}(G)\cong \mathrm{Out}(\mathfrak{g})$.The same argument does not work for $\mathrm{Aut}(PG)$ since $PG$ is not generally simply-connected. The inner automorphisms of the Lie algebra are the image of $G$ via the adjoint. I think you need to generally figure out the kernel. – Sean Lawton May 6 at 21:08
• I am not sure. I think it is true for simple, simply-connected compact Lie groups. "Proof:" Since $G$ is simple, the Lie algebra of $\mathrm{Inn}(\mathfrak{g})$ is $\mathfrak{g}$ itself. Thus, since $G$ is simply-connected and compact, the abstract Lie group structure on $\mathrm{Inn}(\mathfrak{g})$ is a finite central quotient of $G$. Since the center is in the kernel, it must be $PG$. And $\mathrm{Inn}(G)\cong PG$ always. $\Box$ – Sean Lawton May 6 at 21:17


# Basis for eigenspace of Identity Matrix

Let $A=\begin{pmatrix} 1&0\\ 0&1\\ \end{pmatrix}$. Find the bases for the eigenspaces of the matrix $A$. I know the bases for the eigenspace corresponding to each eigenvector is a vector (or system) that can scale to give any other vector contained in that said eigenspace. Thus, we see that the identity matrix has only one distinct eigenvalue $\lambda=1$. Thus the eigenvector satisfies the equation $(A-\lambda I)\vec{x}=\vec{0}$. Then we must solve $\begin{pmatrix} 0&0&0\\ 0&0&0\\ \end{pmatrix}$. I say that any vector satisfies this equation however the key says that the basis is $\lbrace (1,0),(0,1) \rbrace$. How do they come up with this solution. Thanks in advance!

It is true that any vector satisfies the equation. So, our eigenspace is the whole space $\mathbb{R}^2$. Since $\{(1,0),(0,1)\}$ is a basis for $\mathbb{R}^2$, it is also a basis for this eigenspace.

• Yes. This is precisely the explanation I needed. Thank you! – coreyman317 Apr 5 '18 at 8:50


# Solve the recurrence relation $T(n)=4T(n/3) + n$, $T(3) = 1$, $n = 3^k$ then determine upper and lower bounds

I would like to solve the following recurrence relation. After which I must find it's upper and lower bounds.

$T(n)=4T(\frac{n}{3})+n, T(3)=1, n = 3^k$

I attempted to solve with substitution:

k = 1: $T(n)=4T(\frac{n}{3})+n$
k = 2: $T(n)=4^2T(\frac{n}{3^2})+\frac{4n}{3}+n$
k = 3: $T(n)=4^3T(\frac{n}{3^3})+\frac{4^2n}{3^2}+\frac{4n}{3}+n$

General: $T(n)=4^kT(\frac{n}{3^k})+\sum_{i=1}^k(\frac{4^{i-1}}{3^{i-1}})n=4^kT(\frac{3^k}{3^k})+...=4^kT(1)+...$

Q1: Here is where my first problem arrived; I was not given a relation for T(1) so I can not finish with substitution, right? Perhaps my professor made a typo and meant to put T(1) = 1?

Assuming that he didn't make a typo, I decided to try and solve the problem using Master Theorem:

Master Theorem: $T(n)=aT(\frac{n}{b})+n^d$ where $a,b > 0$ and $d\ge0$ then

$T(n)=\Theta(n^d)$ if $d>\log_b(a)$
$T(n)=\Theta(n^d\log(n))$ if $d=\log_b(a)$
$T(n)=\Theta(n^{\log_b(a)})$ if $d<\log_b(a)$

Using this method I get $T(n)=\Theta(n^{1.262})$

Q2: I read that $\Theta(g(n))$ is both the upper bound $O(g(n))$ and the lower bound $\Omega(g(n))$ so does that mean that the three answers to my question would be...

Solution to recurrence relation: $T(n)=\Theta(n^{1.262})$
Upper bound: $O(n^{1.262})$
Lower bound: $\Omega(n^{1.262})$

• Actually, you have $T(3) = 1$ so you can just use that. – Qudit Oct 11 '17 at 7:00
• The case in when $T(3) = 1$ has the overall term of $4^{k-1}T(\frac{n}{3^{k-1}})$. When this occurs, and after the substitution $T(3) = 1$, the term becomes $4^{k-1}$. – user316270 Oct 11 '17 at 7:08

Q1: Here is where my first problem arrived; I was not given a relation for T(1) so I can not finish with substitution, right? Perhaps my professor made a typo and meant to put T(1) = 1?

No, your professor was correct in providing $T(3) = 1$, given $n = 3^{k}$. As you may have noticed when using substitution method, there is generally a relationship in the power of the coefficient - in this case, $4$ - and the power of the denominator in $T()$. Realizing this, you can often quickly generalize and jump to the base condition. To explicitly illustrate:

\begin{align} T(n) &= 4^{1}T(\frac{n}{3^{1}}) + n \\ &= 4^{2}T(\frac{n}{3^{2}}) + 4n + n \\ &= 4^{3}T(\frac{n}{3^{3}}) + 4^{2}n + 4n + n \\ & = \ ... \end{align}

At this point, look to the value within $T()$ in order to determine the remaining number of additional substitutions need to take place. For this problem, the base case is $T(3) = 1$, and,

$$3 = \frac{n}{3^{k-1}}$$

therefore, we get

\begin{align} T(n) &= 4^{k-1}T(\frac{n}{3^{k-1}}) + 4^{k-2}n + ... + 4^{k - (k - 1)}n + n \\ & = 4^{k-1} + 4^{k-2}n + ... + 4^{k - (k - 1)}n + n \end{align}

Now, these terms need to be summed. Re-expressing $4 = 2^{2}$ and factoring out $n$,

\begin{align} T(n) & = 2^{2(k-1)} + 2^{2(k-2)}n + ... + 2^{2(k - [k - 1])}n + n \\ & = 2^{2(k-1)} + n(2^{2(k-2)} + ... + 2^{2(k - [k - 1])} + 1) \end{align}

Recognizing the parenthesized terms collectively represent a geometric series, we can easily sum the terms to get

\begin{align} T(n) &= 2^{2(k-1)} + n\sum_{i=1}^{k-2} \ (2^{2})^{i} \\ &= 2^{2(k-1)} + n(2^{2(k-2)+1} - 1)\end{align}

Lastly, back substitute $k$ to get

\begin{align} T(n) &= 4^{(log_{3}(n)-1)} + n(4^{(log_{3}(n)-2)+1} - 1) \\ &= 4^{log_{3}(n)-1}(n+1)-n \end{align}

Q2: I read that Θ(g(n)) is both the upper bound O(g(n)) and the lower bound Ω(g(n)) so does that mean that the three answers to my question would be...

Using case 1 of the Master Theorem as described by here, $T(n) \in \Theta(n^{1.262})$. Using theta notation alone will suffice for describing the bounds for this function, as it's implied in its definition. Consider the following graph:


# Convergence of the series $\sum_{n=1}^{\infty}\frac{3\ln(n)}{n^7}$; I say it converges, WebWork tells me this is incorrect.

I'm almost certain that $$\sum_{n=1}^{\infty}\frac{3\ln(n)}{n^7}$$ converges. However, WebWork tells me that this is incorrect. I have been in that situation before but I obviously can't assume that I'm right and the computer is wrong based on that. I don't know how the system work and I'm not sure whether what I do is correct anymore.

By the $p$-test, I know $$3\sum_{n=1}^{\infty}\frac{1}{n^7}$$ converges. I also know that $\ln(n)$ grow very slowly so I use the comparison $\ln(n)\le Cn$ which I think will not change that fact that the sum converges such that $$\sum_{n=1}^\infty\frac{3\ln(n)}{n^7}\le \sum_{n=1}^\infty\frac{Cn}{n^7}= C\sum_{n=1}^\infty\frac 1{n^6}\lt\infty$$ which makes me quite certain that $\sum_{n=1}^{\infty}\frac{3\ln(n)}{n^7}$ is convergent.

I also checked that \begin{align} \lim_{\epsilon \to \infty}\int_1^\epsilon\frac{3\ln(x)}{x^7}\,dx & = 3\lim_{\epsilon \to \infty}\left(-\frac{\log (x)}{6 x^6}\bigg|_{1}^{\epsilon} +\frac{1}{6}\int_1^\epsilon \frac {dx}{x^7}\right)\\ & = 3\lim_{\epsilon \to \infty}\left( -\frac{\log (x)}{6 x^6}-\frac{1}{36 x^6}\right)\bigg|_{1}^{\epsilon}\\ & = \frac 1{12}\\ \end{align}

Can someone please shine some light on this for me?

• Besides mixing $n$'s and $x$'s in your comparison, you are correct. The series converges. – carmichael561 Apr 4 '17 at 0:53
• WolframAlpha agrees with you that this converges. – Mark Apr 4 '17 at 0:53
• @carmichael561 Thank you I will edit that, I put that on the count of not being properly caffeinated... – user409521 Apr 4 '17 at 0:54

The series is convergent by p test and we may evaluate the “closed form”. The Riemann zeta function is defined as$$\zeta\left(s\right)=\sum_{n\geq1}\frac{1}{n^{s}},\,\textrm{Re}\left(s\right)>1.$$It is absolute convergent in the region $\textrm{Re}\left(s\right)>1$ so $$\zeta'\left(s\right)=\frac{d}{ds}\left(\sum_{n\geq1}\frac{1}{n^{s}}\right)=\sum_{n\geq1}\frac{d}{ds}\left(\frac{1}{n^{s}}\right)=-\sum_{n\geq1}\frac{\log\left(n\right)}{n^{s}}$$ hence $$S=3\sum_{n\geq1}\frac{\log\left(n\right)}{n^{7}}=\color{red}{-3\zeta'\left(7\right)}\approx0.0181.$$

• Thanks for sparking my interest for Riemann's zeta function and giving me some insight. – user409521 Apr 4 '17 at 17:21
• @InfiniteMonkey You're welcome. – Marco Cantarini Apr 4 '17 at 18:17

I would try the "Integral Test for Convergence" which says:

Consider an integer $N$ and a non-negative, continuous function $f$ defined on the unbounded interval $[N, ∞)$, on which it is monotone decreasing. Then the infinite series

$$\sum_N^{\infty} f(n)$$

converges to a real number if and only if the improper integral

$$\int_N^{\infty} f(x) dx$$

is finite.

Look at

$$\sum_{n=1}^{\infty}\frac{3\ln(n)}{n^7}$$

and the improper integral

$$\int _1^{\infty }\frac{3 \ln (n)}{n^7} = \frac{1}{12},$$ which is finite.

Thus, by the Integral Test for Convergence, you can say that the infinite series $\sum_{n=1}^{\infty}\frac{3\ln(n)}{n^7}$ converges.

• The p test is more than enough. You don't need to solve that integral to know that the series converges, it is as immediate as $7 > 1$ – Ant Apr 4 '17 at 8:49

It really, really converges. The only common convergence tests I can think of that don't help are the root test and the alternating series test (the latter for the trivial reason that the summand is always positive).

Ratio test: $$\lim_{n \to \infty} \frac{\log(n+1)}{n \log(n)} = 0$$ which you can do by just "limit of a product is the product of the limits" on $\frac{\log(n+1)}{\log n} \times \frac{1}{n}$.

Comparison with $\frac{1}{n^2}$: $$\frac{\log n}{n^7} \leq \frac{1}{n^2}$$ if and only if $n^5 \geq \log n$, which is true for all positive $n$.

Integral test someone else has covered.

Cauchy condensation: it converges if and only if $$\sum_{n=0}^{\infty} \frac{n \log(2)}{2^{7n-n}}$$ does, but that is absurdly quickly convergent.

• I was not aware of Cauchy condensation I'm reading about it now thanks! – user409521 Apr 4 '17 at 17:27


# Determinant with Very Hard Algebra

• August 7th 2007, 02:36 AM
binarybob0001
Determinant with Very Hard Algebra
The problem is
$
\det \left [ \begin{matrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{matrix} \right ]
$

Sorry I'm new. That's the best I could write the problem. It's easy to find the determinant with expansion by minors but to get the simplified solution involves some sort of clever trick that I don't know of. Here's what I can get to.
$
abc(bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b)
$

There are many ways to factor out terms from this.
$
abc[bc(c - b) + ac(a - c) + ab(b - a)] =
$

$
abc[b(c + a)(c - a) + c(a + b)(a - b) + a(b + c)(b - c)] =
$

$
abc[c^2(b - a) + b^2(a - c) + a^2(c - b)]
$

There are even more ways of course. None of these ways lead me to the simple solution:
$
abc(b - a)(c - a)(c - b)
$

Can someone explain how to factor this? Thanks.
• August 7th 2007, 03:43 AM
JakeD
Quote:

Originally Posted by binarybob0001
The problem is
$
\det \left [ \begin{matrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{matrix} \right ]
$

Sorry I'm new. That's the best I could write the problem. It's easy to find the determinant with expansion by minors but to get the simplified solution involves some sort of clever trick that I don't know of. Here's what I can get to.
$
abc(bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b)
$

There are many ways to factor out terms from this.
$
abc[bc(c - b) + ac(a - c) + ab(b - a)] =
$

$
abc[b(c + a)(c - a) + c(a + b)(a - b) + a(b + c)(b - c)] =
$

$
abc[c^2(b - a) + b^2(a - c) + a^2(c - b)]
$

There are even more ways of course. None of these ways lead me to the simple solution:
$
abc(b - a)(c - a)(c - b)
$

Can someone explain how to factor this? Thanks.

Multiply out

\begin{aligned}(b - a)(c - a)(c - b) &= (bc -ac -ab +a^2)(c-b) \\
&= bc^2 - ac^2 -abc + a^2c -b^2c +abc + ab^2 - a^2b \\
&= bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b \end{aligned}

as you wanted.
• August 7th 2007, 07:28 AM
binarybob0001
You can only do that if you already know the solution. What if you didn't know the solution? I would have just stopped thinking I was done. Is there a systematic way of arriving at that result?
• August 7th 2007, 09:06 AM
Soroban
Hello, binarybob0001!

Quote:

The problem is: . $
\begin{bmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{bmatrix}$

Sorry I'm new. . . . . Welcome aboard!
That's the best I could write the problem. . . . . Good job!

It's easy to find the determinant with expansion by minors
but to get the simplified solution involves some sort of clever trick that I don't know of.

Here's what I can get to:
. . $abc(bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b)$

There are many ways to factor out terms from this.
None of these ways lead me to the simple solution:
. . $abc(b - a)(c - a)(c - b)$
Can someone explain how to factor this?

Here's an approach I "discovered" years ago . . .

Collect terms with respect to one of the variables.

Let's use $c$
. . Collect terms with $c^2$, and with $c$, and 'constants' (no $c$'s).

We have: . $abc\bigg[bc^2 - ac^2 - b^2c + a^2c + ab^2 - a^2b\bigg]$

Factor "by grouping": . $abc\bigg[c^2(b-a) - c(b^2-a^2) + ab(b-a)\bigg]$
. . $= \;abc\bigg[c^2(b-a) - c(b-a)(b+a) + ab(b-a)\bigg]$

Factor out $(b-a)\!:\;\;abc(b-a)\bigg[c^2 - c(b+a) + ab\bigg]$
. . $= \;abc(b-a)\bigg[c^2 - bc - ac + ab\bigg]$

Factor "by grouping": . $abc(b-a)\bigg[c(c-b) - a(c-b)\bigg]$

Factor out $(c-b)\!:\;\;abc(b-a)(c-b)(c-a)$ . . . . ta-DAA!

• August 8th 2007, 01:35 AM
binarybob0001
ahah, so I just need to chose a letter, stupid me. Thank you very much. Hey, are there more problems this hard to practice on?


# Generating Perfect Model Data

Let’s generate some model x1 and y1 data. First we will create an independent variable x1 that is a column vector ranging from 0 to 20 in steps of 2 and then we can create y1 which is a polynomial with respect to x1. We will store this in a table called data:

x1=[0:2:20]'; y1=0.1*x1.^2+1.2*x1+5; data=table(x1,y1); clear x1; clear y1;

For this quadratic polynomial y1=0.1*x1.^2+1.2*x1+5 of the form y1=p2(1)*x1.^2+p2(2)*x1+p2(3) the three coefficients are:

p2(1)=0.1
p2(2)=1.2
p2(3)=5

p2(1) is the x.^2 coefficient p2(2) is the x coefficient and p2(3) is the constant.

# Scatter Plot

figure(1); scatter(data.x1,data.y1,100,'fill',... 'MarkerEdgeColor',[112/255,48/255,160/255],... 'MarkerFaceColor',[0/255,176/255,80/255],... 'LineWidth',1,...
'MarkerFaceAlpha',0.5,...
'MarkerEdgeAlpha',0.5)
set(gca,'FontSize',20)
xlabel('x')
ylabel('y')
title('Polynomial')
grid('minor')
legend('y1',... 'Location','NorthEast')

# Using polyfit to Find the Polynomial Coefficients

We can use polyfit to evaluate the polynomial coefficients. Polyfit has the form

[pn]=polyfit(x,y,n)

Where the input arguments are the x data, y data and n, the order of polynomial. The output arguments is a row vector pn of the polynomial coefficients. In most cases we would try fitting from a first order polynomial upwards however as we know our data is a second order polynomial, as it is created from a second order polynomial we can try fitting to a second order polynomial first.

[p2]=polyfit(data.x1,data.y1,2)

$\displaystyle \text{p2=}\left[ {\begin{array}{*{20}{c}} {0.100000} & {1.200000} & {5.000000} \end{array}} \right]$

These are the three polynomial coefficients for the quadratic equation:

$\displaystyle 0.100000{{x}^{2}}+1.200000x+5.000000$

The original polynomial was y1=0.1*x1.^2+1.2*x1+5 and the three coefficients returned match y1=p2(1)*x1.^2+p2(2)*x1+p2(3) as expected.

Which as expected is a perfect match to our original equation.

# Using polyval to Evaluate the Polynomial

We can use polyval to evaluate the polynomial at specified x values. Let’s first create a new variable x2 and assign it as a column in a new table called datafit.

x2=[1:2:19]'; datafit=table(x2); clear x2;

We can use polyval to evaluate the polynomial coefficients. polyval has the form

[newy]=polyval(pn,newx);

We can use the function

[datafit.y2]=polyval(p2,datafit.x2);

To find the value of the polynomial for each x2 value.

# Plotting the Fitted Data with Respect to the Raw Data

We can plot this data as a scatter plot alongside the original data:

figure(1); scatter(data.x1,data.y1,100,'fill',... 'MarkerEdgeColor',[112/255,48/255,160/255],... 'MarkerFaceColor',[0/255,176/255,80/255],... 'LineWidth',1,...
'MarkerFaceAlpha',0.5,...
'MarkerEdgeAlpha',0.5)
hold('on'); scatter(datafit.x2,datafit.y2,100,'fill',... 'MarkerEdgeColor',[255/255,0/255,0/255],... 'MarkerFaceColor',[0/255,112/255,92/255],... 'LineWidth',1,... 'MarkerFaceAlpha',0.5,... 'MarkerEdgeAlpha',0.5); hold('off'); set(gca,'FontSize',20)
xlabel('x')
ylabel('y')
title('Polynomial')
grid('minor')
legend('y1','y2',... 'Location','NorthEast')

It may be more common to plot the fitted curve as a line:

figure(1); scatter(data.x1,data.y1,100,'fill',... 'MarkerEdgeColor',[112/255,48/255,160/255],... 'MarkerFaceColor',[0/255,176/255,80/255],... 'LineWidth',1,...
'MarkerFaceAlpha',0.5,...
'MarkerEdgeAlpha',0.5)
hold('on'); plot(datafit.x2,datafit.y2,... 'color','r',... 'LineWidth',3); hold('off'); set(gca,'FontSize',20)
xlabel('x')
ylabel('y')
title('Polynomial')
grid('minor')
legend('y1','y2',... 'Location','NorthEast')

If we want to print the equation to the chart we can use:

equation=sprintf('y=(%f)x^2+(%f)x+(%f)',p2(1),p2(2),p2(3)); legend('data',equation,'location','northwest','FontSize',8);

The function sprintf has the same form as fprintf but prints to a string variable opposed to outputting in the Command Window.

# Underfitting – Fitting a 1st Order Polynomial to 2nd Order Polynomial Data

The code above can be modified to fit to a 1st order polynomial (i.e. a straight line). Here it is obvious by eye that it is an unsuitable fit:

The reader is recommended to try to use polyfit and polyval to try and fit the above curve to a 1st order polynomial (straight line).

x2=[1:2:19]'; [p1]=polyfit(data.x1,data.y1,1) datafit=table(x2); clear x2; [datafit.y2]=polyval(p1,datafit.x2);

The user is recommended to plot the graph above, with the updated equation in the legend.

figure(1); scatter(data.x1,data.y1,100,'fill',... 'MarkerEdgeColor',[112/255,48/255,160/255],... 'MarkerFaceColor',[0/255,176/255,80/255],... 'LineWidth',1,...
'MarkerFaceAlpha',0.5,...
'MarkerEdgeAlpha',0.5)
hold('on'); plot(datafit.x2,datafit.y2,... 'color','r',... 'LineWidth',3); hold('off'); set(gca,'FontSize',20)
xlabel('x')
ylabel('y')
title('Polynomial')
grid('minor') equation=sprintf('y=(%f)x+(%f)',p2(1),p2(2)); legend('data',equation,'location','northwest','FontSize',8);

# Overfitting – Fitting a 3rd Order Polynomial to 2nd Order Polynomial Data

Here from the values of the coefficients i.e. the cubic term being extremely small 1.6509e-17 it is obvious that the polynomial is overfitted. This term can be rounded to 0 and we return with the quadratic fit. This is perfect model data so the fitting is relatively easy in real life it may be harder to distinguish data which has been over-fitted.

The reader is recommended to try to use polyfit and polyval to try and fit the above curve to a 3rd order polynomial.

x2=[1:2:19]'; [p3]=polyfit(data.x1,data.y1,3) datafit=table(x2); clear x2; [datafit.y2]=polyval(p3,datafit.x2);

The user is recommended to plot the graph above, with the updated equation in the legend.

figure(1); scatter(data.x1,data.y1,100,'fill',... 'MarkerEdgeColor',[112/255,48/255,160/255],... 'MarkerFaceColor',[0/255,176/255,80/255],... 'LineWidth',1,...
'MarkerFaceAlpha',0.5,...
'MarkerEdgeAlpha',0.5)
hold('on'); plot(datafit.x2,datafit.y2,... 'color','r',... 'LineWidth',3); hold('off'); set(gca,'FontSize',20)
xlabel('x')
ylabel('y')
title('Polynomial')
grid('minor') equation=sprintf('y=(%f)x^3+(%f)x^2+(%f)x+(%f)',p3(1),p3(2),p3(3),p3(4)); legend('data',equation,'location','northwest','FontSize',8);

# Generating Model Data with Noise

rng('default'); x1=[0:2:20]'; [m,n]=size(x1); noise=rand(m,n); y1=0.1*x1.^2+1.2*x1+5+noise; data=table(x1,y1); clear x1; clear y1; clear noise; clear m; clear n;

We see a deviation from the starting polynomial with:

p2(1)=0.1
p2(2)=1.2
p2(3)=5

The deviation influences the constant term the most, followed by the x term.

If we times the noise by 5, we see a larger deviation:

If we times the noise by 10 we see more deviation:

And if we times the noise by 25 we see an even larger deviation again:

If we times the noise by 250, the original function is not recognisable:


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# Which of the following is an integer?

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Which of the following is an integer?  [#permalink]

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Updated on: 07 Oct 2019, 01:36
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Which of the following is an integer?

I. $$\frac{12!}{6!}$$

II. $$\frac{12!}{8!}$$

III. $$\frac{12!}{7!5!}$$

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

Originally posted by NickTW on 18 Aug 2009, 19:23.
Last edited by Bunuel on 07 Oct 2019, 01:36, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: Which of the following is an integer?  [#permalink]

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08 May 2015, 15:47
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Hi All,

Beyond the arithmetic that sdrandom1 pointed out, these calculations fall into some patterns that you are likely to see (in some form) on Test Day.

12!/6!

When dealing with INDIVIDUAL factorials, if the factorial in the numerator is GREATER than the factorial in the denominator, then you will end up with an integer. Here, since 12 > 6, 12!/6! will simplify to an integer.

12!/7!5!

You might not have studied the Combination Formula yet, but this calculation is what you would end up with when answering the question "how many different combinations of 7 people can you form from a group of 12 people?" In these types of "Combination" calculations, the number of groups will always be an integer, so 12!/7!5! will simplify to an integer.

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Re: Which of the following is an integer?  [#permalink]

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18 Aug 2009, 19:30
2
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NickTW wrote:
Did a search, could not find anything on this one (the question stem is pretty vague). Would appreciate some enlightenment on the shortcut I'm missing!

Which of the following is an integer?

I. 12! / 6!
II. 12! / 8!
III. 12! / 7!5!

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

Source: GMAT Prep Exam. Solution : E

To me the question looks straight forward.
we know 12! is multiple of all numbers from 1 to 12, so dividing it by 6! or 8! is pretty much sure an integer.

now coming to the 3rd options 12! by 5! leaves 12*11*10*9*8*7*6/1*2*3*4*5*6*7 which results in integer, so the answe E is right
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Re: Which of the following is an integer?  [#permalink]

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18 Aug 2009, 19:45
1
1
There is no shortcut persay...

Clear E.

12!/6! = 12*11*10*9*8*7*6!/6! = Integer
12!/8! = 12*11*10*9*8!/8! = Integer
12!/7!*5! = 12*11*10*9*8/5*4*3*2*1 = integer

So all 3 are correct.
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Re: Which of the following is an integer?  [#permalink]

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19 Aug 2009, 08:32
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Every factorial is divisible by every smaller factorial, so 12!/6! and 12!/8! are integers. 12!/(7!*5!) is the number of ways to choose 7 things from a group of 12 if order is irrelevant, so it must also be an integer.
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Re: Which of the following is an integer?  [#permalink]

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03 May 2015, 01:42
case 1 and case 2 are simple but case 3 is harder. for case 3, we have to write down all the factors.
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Re: Which of the following is an integer?  [#permalink]

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03 Dec 2017, 17:39
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NickTW wrote:
Which of the following is an integer?

I. 12! / 6!
II. 12! / 8!
III. 12! / 7!5!

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

Before actually solving this problem, let's review how factorials can be expanded and expressed. As as example, we can use 5!.

5! could be expressed as:

5!

5 x 4!

5 x 4 x 3!

5 x 4 x 3 x 2!

5 x 4 x 3 x 2 x 1!

Understanding how this factorial expansion works will help us work our way through each answer choice, especially answer choices 1 and 2.

I. 12!/6!

Since we know that factorials can be expanded, we now know that:

12! = 12 x 11 x 10 x 9 x 8 x 7 x 6!

Plugging this in for answer choice 1, we have:

(12 x 11 x 10 x 9 x 8 x 7 x 6!)/6! = 12 x 11 x 10 x 9 x 8 x 7, which is an integer.

II. 12!/8!

Once again, since we know that factorials can be expanded, we now know that:

12! = 12 x 11 x 10 x 9 x 8!

Plugging this in for answer choice 2, we have:

(12 x 11 x 10 x 9 x 8!)/8! = 12 x 11 x 10 x 9, which is an integer.

III. 12!/(7!5!)

Once again, since we know that factorials can be expanded, we now know that:

12! = 12 x 11 x 10 x 9 x 8 x 7!

Plugging this in for answer choice 3 gives us:

(12 x 11 x 10 x 9 x 8 x 7!)/(7!5!)

(12 x 11 x 10 x 9 x 8)/(5 x 4 x 3 x 2 x 1)

(12 x 11 x 10 x 9 x 8)/(12 x 10 x 1)

11 x 9 x 8, which is an integer.

We see that the quantities in Roman numerals I, II and III are all integers.

Alternate solution:

For any positive integers m, n and p,

1) If m > n, then m!/n! is always an integer.

2) If m = n + p, then m!/(n!p!) is always an integer (which is in fact mCp).

From the above two facts, we see that all three quotients in the Roman numerals must be integers.

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Re: Which of the following is an integer?  [#permalink]

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20 Jan 2019, 08:00
NickTW wrote:
Which of the following is an integer?

I. 12! / 6!
II. 12! / 8!
III. 12! / 7!5!

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

For III, I went through and crossed out everything that's reduced to visualize...
12*11*102*93*8*7*6*5*4*3*2*1 / 7*6*5*4*3*2*1 * 5*4*3*2*1

So you're left with 12*11*2*3 (from taking 5 and 3 from 10 and 9) = 792. Makes sense.

ScottTargetTestPrep
As a rule, you said m = n + p, then m!/(n!p!). When does the factorial in the denominator make the result not an integer? Am I right to think this has to do with the number of prime factors?
Because 12!/8!5! and 12!/9!5! are also integers, while 12!/10!5! isn't. If we had 11*2^3*3^2 left with 7! on the bottom then 8! would be 2^3 and 9! would be 3^2 both of which we can still reduce... while 10! gives us a factor of 5 which we don't have.
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Re: Which of the following is an integer?  [#permalink]

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22 Jan 2019, 17:19
energetics wrote:
NickTW wrote:
Which of the following is an integer?

I. 12! / 6!
II. 12! / 8!
III. 12! / 7!5!

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

For III, I went through and crossed out everything that's reduced to visualize...
12*11*102*93*8*7*6*5*4*3*2*1 / 7*6*5*4*3*2*1 * 5*4*3*2*1

So you're left with 12*11*2*3 (from taking 5 and 3 from 10 and 9) = 792. Makes sense.

ScottTargetTestPrep
As a rule, you said m = n + p, then m!/(n!p!). When does the factorial in the denominator make the result not an integer? Am I right to think this has to do with the number of prime factors?
Because 12!/8!5! and 12!/9!5! are also integers, while 12!/10!5! isn't. If we had 11*2^3*3^2 left with 7! on the bottom then 8! would be 2^3 and 9! would be 3^2 both of which we can still reduce... while 10! gives us a factor of 5 which we don't have.

So you ask a really great qestion. So, as you've shown above, when m < n + p, there is no hard and fast rule to determien when m!/(n!p!) is an integer and when it is not.
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Re: Which is integer?   [#permalink] 07 Oct 2019, 01:36


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# The figure above represents a semicircular archway over a flat street.

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The figure above represents a semicircular archway over a flat street.  [#permalink]

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The figure above represents a semicircular archway over a flat street. The semicircle has a center at O and a radius of 6 feet. What is the height h, in feet, of the archway 2 feet from its center?

A. √2

B. 2

C. 3

D. 4√2

E. 6

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The figure above represents a semicircular archway over a flat street.  [#permalink]

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22 Jun 2018, 01:41
Let's disturb Pythagoras once again:

$$h^2+2^2=6^2$$
h=$$\sqrt{32}$$

In My Opinion
Ans: D
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Re: The figure above represents a semicircular archway over a flat street.  [#permalink]

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22 Jun 2018, 05:24
2

Solution

Given:
• The figure depicts a semi-circular archway over a flat street
• The semi-circle has a center at O and a radius of 6 feet

To find:
• The length of the height h, in feet, of the archway 2 feet from its center

Approach and Working:

If we add the point O and A, we get a right-angled triangle OAB, right-angled at point B and hypotenuse AO

As the radius is 6 feet, we can say
• AO = 6 feet

Applying Pythagoras Theorem in triangle OAB, we can write:
• $$AO^2 = AB^2 + BO^2$$
Or, $$6^2 = h^2 + 2^2$$
Or, $$h^2 = 36 – 4 = 32$$
Or, $$h = 4\sqrt{2}$$

Hence, the correct answer is option D.

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Re: The figure above represents a semicircular archway over a flat street.  [#permalink]

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23 Jun 2018, 09:52
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Bunuel wrote:

The figure above represents a semicircular archway over a flat street. The semicircle has a center at O and a radius of 6 feet. What is the height h, in feet, of the archway 2 feet from its center?

A. √2

B. 2

C. 3

D. 4√2

E. 6

NEW question from GMAT® Official Guide 2019

(PS05957)

Attachment:
PS05957_f009.jpg

Let's add the radius of 6 feet to the diagram to get:

From here, we can see the right triangle hiding within the diagram, which means we can apply the Pythagorean Theorem.
Se can write: h² + 2² = 6²
Simplify: h² + 4 = 36
So, we get: h² = 32
This means h = √32
Check the answer choices. . . √32 is not among them.
Looks like we need to simplify √32

We'll use the fact that √(xy) = (√x)(√y)
So, √32 = √[(16)(2)] = (√16)(√2) = (4)(√2) = 4√2
Check the answer choices. . . D

RELATED VIDEO (simplifying roots)

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Re: The figure above represents a semicircular archway over a flat street.  [#permalink]

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25 Jun 2018, 12:06
1
Bunuel wrote:

The figure above represents a semicircular archway over a flat street. The semicircle has a center at O and a radius of 6 feet. What is the height h, in feet, of the archway 2 feet from its center?

A. √2

B. 2

C. 3

D. 4√2

E. 6

Since the radius is 6 feet, we can create a right triangle with hypotenuse of 6, leg of 2, and leg of h; thus, we have:

2^2 + h^2 = 6^2

4 + h^2 = 36

h^2 = 32

h = √16 x √2 = 4√2

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Re: The figure above represents a semicircular archway over a flat street.  [#permalink]

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02 Aug 2018, 13:30
if the radius is 6 max height of the arch can be 6 from the center of semi-circle. So height asked in question which is two feet away from center is definitely less than 6 . Also point 2 feet away from center to end of semicircle the dist is 4 so height is little more than 4 . So the only answer choice that matches it is D
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Re: The figure above represents a semicircular archway over a flat street.   [#permalink] 02 Aug 2018, 13:30
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# Polar Coordinates Pdf

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In polar coordinates the position of an object $$R$$ distance from the origin as represented in the diagram above is modelled $$\mathbf{r} = R \hat{r}$$ The velocity and acceleration in polar coordinates is derived by differentiating the position vector. Some useful properties about line integrals: 1. Michael VanValkenburgh To make it easier to type and easier to read, this handout will focus on the computational aspects of integration in polar coordinates. If you were to add a true position characteristic it would look like this. However, we can use other coordinates to determine the location of a point. However, we can still rotate around the system by any angle we want and so the coordinates of the origin/pole are (0,θ). Download the pdf file and print. Please read through this supplement before going to quiz section for the polar worksheet on Thursday. 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. Show Instructions. 2) Equal amounts of the Primaries produce white. r = 2 and θ= 30°, so P is located 2 units from the origin in the positive direction on a ray making a 30°angle with the polar axis. Unit Six Precalculus Practice Test Vectors & Polar Graphs Page 3 of 6 14. 5 3, 3 For problems 17-20, convert the rectangular coordinates to polar coordinates with r 0 and 0 2. Cartiesian Coordinate System. Home Decorating Style 2020 for Polar Coordinate System Pdf, you can see Polar Coordinate System Pdf and more pictures for Home Interior Designing 2020 76667 at Manuals Library. ) Abbreviated podcast notes on lecture 5. Use a double integral in polar coordinates to calculate the area of the region which is common to both circles r= 3sin and r= p 3cos. Solution; The Cartesian coordinate of a point are $$\left( { - 8,1} \right)$$. 5 Polar Coordinates and Graphs-1. 4 Many problems are more easily stated and solved using a coordinate system other than rectangular coordinates, for example polar coordinates. A 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. x t y t9cos , 9sin 16. You can copy that worksheet to your home. 2 We can describe a point, P, in three different ways. 686 CHAPTER 9 POLAR COORDINATES AND PLANE CURVES The simplest equation in polar coordinates has the form r= k, where kis a positive constant. Home Decorating Style 2020 for Polar Coordinate System Pdf, you can see Polar Coordinate System Pdf and more pictures for Home Interior Designing 2020 76667 at Manuals Library. This is the xy-plane. Find the distance between the points. a) r=3secθ b) r=−3sinθ c) rcsc 1θ= 5) Convert the rectangular equation to polar form. 9) ( , ) 10) ( , ) Two points are specified using polar coordinates. Different radials and degree over them are known as the polar paper. 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. And polar coordinates, it can be specified as r is equal to 5, and theta is 53. …Polar-coordinates are entered…in a magnitude direction format. Getting Started To assist you, there is a worksheet associated with this lab that contains examples and even solutions to some of the exercises. The answer is: (r,θ) Polar = (p x2 +y2, arctan y x) Polar Meanwhile, for a point given by Polar coordinates, (r,θ) Polar, we need to specify the coordinates in Cartesian form in terms of the Polar data r and θ. 3) Rectangular coordinates of point P are given. Customize Polar Axes. On the ﬁrst region we would have −2 6 x 6 2 and √ 4−x2 6 y 6 √ 9−x2, on the second region −3 6 x 6 2. First, fix an origin (called the pole) and an initial ray from O. My knows are (R, r, theta, Phi) My unknowns are (Phi1, R1) If you guys are up for it, could you assist me in establishing a formula for R1 and Phi1 in terms of R, r, theta, and Phi?. A polar coordinate graph paper that’s perfect for when you need to compare two graphs that have minor differences. In addition to, It converts complex number into polar form and vice versa. 5,13) into polar coordinates. Graphing Worksheets for Practice. But what about r f(T)? At first you might think dr dT is the slope of the tangent line to the curve but consider r = constant e. The unit tangent vector to the curve is then Tˆ = ˙xˆı+ ˙y ˆ (2) where we have used a dot to denote derivatives with respect to s. The initial line may be identified with the x-axis of rectangular Cartesian coordinates, as. 5 , 0 1 2 ≤ ≤θ π. 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. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, Calculating a limit in two variables by going to polar coordinates. Use a double integral in polar coordinates to calculate the area of the region which is common to both circles r= 3sin and r= p 3cos. 5 3, 3 For problems 17-20, convert the rectangular coordinates to polar coordinates with r 0 and 0 2. The rectangular coordinates for P (5,20°) are P (4. Check out our many other free graph/grid paper styles. pdf (392 KB) Equella is a shared content repository that organizations can use to easily track and reuse content. (i) Plot each point. Defining Polar Coordinates. Large single polar graph--Thirty scale marks for r in increments of five degrees. It can be found by the "gradient in polar coordinates" googling. TrigCheatSheet. Polar coordinates with polar axes. 841d f0, 2pd u3 5 cos21s2y3d 5 0. Review: Polar coordinates Deﬁnition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ) deﬁned by the picture. We see this general pattern in the circle of gure 2. This easy-to-use packet is full of stimulating activities that will give your students a solid introduction to polar coordinates and trigonometric form!. their direction does not change with the point r. Ciencia y Tecnología, 32(2): 1-24, 2016 - ISSN: 0378-0524 3 II. POLAR COORDINATES (OL]DEHWK :RRG DEFINITION OF POLAR COORDINATES. The angular dependence of the solutions will be described by spherical harmonics. The divergence We want to discuss a vector ﬂeld f deﬂned on an open subset of Rn. However, in polar coordinates we have u(r,θ) = r sinθ r2 = sinθ r so that u r = − sinθ r2, u. If we restrict rto be nonnegative, then = describes the. Laplace’s equation in polar coordinates, cont. In case n = 3, the polar coordinates (r,θ,φ) are called spherical coordinates, and we have y = x1, x = x2, z = x3, r2 = x2 + y2 + z2, x = rsinφsinθ, y = rsinφcosθ, and x = rcosφ, so we can take r3 = r, φ2 = θ. 6) Area in Polar Coordinates (Fig. In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. 1 Background on Polar Coordinates. 7 - Polar Coordinates ° 3,225 3,0. The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). The small change r in rgives us two concentric circles and the small change in gives us an angular wedge. To convert from Cartesian co-ordinates to polar use the transformation $y=r\sin { \theta }$ and [m. In this system coordinates for a point P are and , which are indicated in Fig. See Large Polar Graph Paper. Mechanics 1: Polar Coordinates Polar Coordinates, and a Rotating Coordinate System. 5 MM Graph Paper. Viewed 11k times 3. Trigonometry - Trigonometry - Polar coordinates: For problems involving directions from a fixed origin (or pole) O, it is often convenient to specify a point P by its polar coordinates (r, θ), in which r is the distance OP and θ is the angle that the direction of r makes with a given initial line. The ordered pairs, called polar coordinates, are in the form $$\left( {r,\theta } \right)$$, with $$r$$ being the number of units from the origin or pole (if $$r>0$$), like a radius of a circle, and $$\theta$$ being the angle (in degrees or radians) formed by the ray on the positive $$x$$ – axis (polar axis), going counter-clockwise. The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system. Cartesian/Polar Coordinates Junior high school The connection between Cartesian coordinates and Polar coordinates is established by basic trigonometry. The transformation from Cartesian coordinates to spherical coordinates is. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. This article explains how to convert between polar and cartesian coordinates and also encourages the creation of some attractive curves from some relatively easy equations. The point P has. For any point P consider the two distances:. In the polar coordinate system, the ordered pair will now be (r, ). 4 Many problems are more easily stated and solved using a coordinate system other than rectangular coordinates, for example polar coordinates. These charts print on a standard sheet of 8 1/2 x 11 paper. Home Decorating Style 2020 for Polar Coordinate System In Autocad Pdf, you can see Polar Coordinate System In Autocad Pdf and more pictures for Home Interior Designing 2020 4680 at Manuals Library. ﬁnd the x and y coordinates of a point (r, θ)), we use the following formulas: x = r cos θ, y = r sin θ. Polar Coordinates (r-θ)Ans: -0. First, let's get some preliminaries out of the way. Free Polar Graph Paper Template. We make the convention (−r,θ) = (r,θ +π). Determine a set of polar coordinates for the point. The orientation of a plane curve can be represented by arrows drawn along the curve. (− −5, 5) 9. (iii) Find the Cartesian coordinates of the point. Spherical Coordinates and the Angular Momentum Operators. Students will plot points in the polar coordinate system, convert coordinates and convert equations from rectangular to polar form and vice versa. b) Set up an expression with two or more integrals to find the area common to both curves. 1 New Optional Features pgfplots has been written with backwards compatibility in mind: old TEX les should compile without modi cations and without changes in the appearance. 6 Complex Polar Coordinates (slides, 4-to-1). If we restrict rto be nonnegative, then = describes the. For coordinate conversions: Example 2: Find the rectangular coordinate for the point whose polar coordinates are (a ) 4 5, 3 (b ) 5 4, 6 Example 3: Convert the following rectangular coordinate into four different, equivalent polar coordinates. to describe using polar coordinates. 442, C 5 s21, 0. The initial line may be identified with the x-axis of rectangular Cartesian coordinates, as. 197t 12 16) x = — I pairs of polar coordinates that describe the same point as the provided polar (-q 1-71/2) Convert each pair of polar coordinates to rectangular coordinates. pdf (condensed podcast notes, 4 slides to a page) Presentations (slides without audio) on lecture 5. 3 Double Integrals in Polar Coordinates In Chapter 10, we explored polar coordinates and saw that in certain situations they simplify problems considerably. R Pr( converts a rectangular form to r in polar coordinates. But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. Lecture #6: The Jacobian for Polar Coordinates Recall. Polar coordinates (Introduction and conversion) Sketching polar curves. For problems 5 and 6 convert the given equation into an equation in terms of polar coordinates. The second point lies on the positive 'y' axis, so the angle in polar coordinates is. The fact that a single point has many pairs of polar coordinates can cause complications. The polar coordinates (r,θ) are deﬁned by r2 = x2 + y2, (2) x = rcosθ and y = rsinθ, so we can take r2 = r and φ2 = θ. The conversion from polar to rectangular coordinates is the same idea as converting rectangular form to polar form in complex numbers. Because Dis a circular disk, we will set up the integral in polar coordinates. Source: Wikipedia - Polar Coordinate System. Cartesian coordinate system: start with xand yaxes. To Convert from Cartesian to Polar. Please read through this supplement before going to quiz section for the polar worksheet on Thursday. (iii) Find the Cartesian coordinates of the point. If there are three variables, the graph is 3D. Polar coordinates part 4 This is a continuation of the polar coordinates part 3. The azimuthal angle, now designated as ϕ, specifies the rotational orientation. 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,!). 2 Calculus In The Polar Coordinate System Contemporary Calculus 4 Area in Rectangular Coordinates (Fig. 3 WS Polar Coordinates (Answers). Convert the following equation of a circle to polar coordinates: 4x2 + 3 2 x +4y2 +1y. 1 Polar Coordinates and Rectangular Coordinates In astronomical calculations, polar coordinate systems are usually used. There are three types of polar graph that are Large Single Polar Graph which has thirty marks for r in increment of five degrees, Smaller (Double) Polar Graph which has two polar graphs on one page, each with twenty scale marks for r increment of 5 degrees and Combined Cartesian and Polar has three pages here, One is a large cartesian grid, one a large polar grid and the third one has one. I still did that, but I also tried odd things. You should be familiar with the Cartesian Coordinate System, also called rectangular coor- dinates, and with the deﬁnitions of sin and cos. Polar coordinates with polar axes. in polar from f(r,θ ) = 1 r D E T 3 2 1 cos(T) r 1 r r ´ µ. The Cartesian coordinate of a point are $$\left( {2, - 6} \right)$$. Precalculus. The polar coordinates of a point are given. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. The transformation from spherical coordinates to Cartesian coordinate is. 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. In these notes, we want to extend this notion of different coordinate systems to consider arbitrary coordinate systems. The graph of an equation in polar coordinates is the set of points which satisfy the equation. edu is a platform for academics to share research papers. The small change r in rgives us two concentric circles and the small change in gives us an angular wedge. In this system coordinates for a point P are and , which are indicated in Fig. 5 3, 3 For problems 17-20, convert the rectangular coordinates to polar coordinates with r 0 and 0 2. If we restrict rto be nonnegative, then = describes the. 23 17_2_polar_coordinates. You can modify certain aspects of polar axes in order to make the chart more readable. Selection File type icon File name Description Size Revision Time User; Ċ: D32. Complete the unit circle with each angles’ coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. a) Find the polar coordinates of the points of intersection between the two curves. This de nition is worded as such in order to take into account that each point in the plane can have in nitely many representations in polar coordinates. Spherical polar coordinates. This is the xy-plane. Polar Coordinates (r-θ)Ans: -0. Therefore r˙(t) = (˙rcosθ − rθ˙sinθ)i + (˙rsinθ + rθ˙cosθ)j. But most commercial motion control cards do not support the polar coordinate, so this paper presents a program module based on polar coordinate system, which can be integrated into computer numeric control (CNC) controller based on motion control cards. Shade the. (b) Compute the Christoﬀel symbols of S in polar coordinates. 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. 5: Polar Coordinates Polar coordinate system, introduced by Isaac Newton, is often more convenient in some applications than the more traditional Cartesian, or rectangular, coordinate system. And you'll get to the exact same point. G15 and G16 G-Codes [Polar Coordinates and CNC Bolt Circles] CNCCookbook's G-Code Training What are Polar and Cartesian Coordinates? Until this point, we've strictly been using Cartesian Coordinates where X, Y, and Z represent distances from part zero (absolute coordinates) or from the current position (relative coordinates). Consider the top which is bounded above by z= p 4 x2 y2 and bounded below by z= p x2 + y2, as shown below. Choose a point in the plane that is called the pole (origin) and labeled O. 6 Graphs of Rational Functions. Polar coordinates part 4 This is a continuation of the polar coordinates part 3. In this section, we discuss how to graph pdfedit ubuntu package equations in polar coordinates on the. If we wish to relate polar coordinates back to rectangular coordinates (i. The graph above shows symmetry with respect to the y-axis. 4 2D Elastostatic Problems in Polar Coordinates Many problems are most conveniently cast in terms of polar coordinates. 8, as outlined in the. 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. SPHERICAL POLAR COORDINATES. It is sometimes convenient to refer to a point by name, especially when this point occurs in multiple \draw commands. Double Integrals in Polar Coordinates “Integrating Functions over circular regions” Suppose we want to integrate the function f(x,y) = x2 over the fol-lowing region: 2 3 We would have to break the region up into three pieces. Mon Nov 11 - I retaught graphing roses and then we began converting from polar form to rectangular and rectangular to polar. Michael VanValkenburgh To make it easier to type and easier to read, this handout will focus on the computational aspects of integration in polar coordinates. 3 WS Polar Coordinates (Answers). Conversion: Rectangular to Polar/ Polar to Rectangular 2011 Rev by James, Apr 2011 1. [See how to convert rectangular and polar forms in the complex numbers chapter. 841 cos us d 5 2y3, 3 cos sud 5 2, 1 2 3 cos sud 5 21, s2r, u 1 pd fsud 5 2gsu 1 pd, p, sr, ud. Convert the following equation of a circle to polar coordinates: 2x2 +3x+2y2 + −5y = 7 7. We know sine starts at zero, and then grows until the function reaches a height of one at ˇ=2. SCHROEDINGER'S EQUATION IN SPHERICAL POLAR COORDINATES The magnitude of a central force on an object depends on only the distance of that object. a) Find the polar coordinates of the points of intersection between the two curves. Most of the things we've done can also be done in the polar, cylindrical, and spherical coordinate as well. Describe the graph. However, we can use other coordinates to determine the location of a point. 21 Locating a point in polar coordinates Let’s look at a specific example. ) The graph of = , where is a constant, is the line of inclination. Thus its area will be Z 2π 0 R2 2 dϑ = R2 2 x 2π 0. 54 Example 1 – Using Polar Coordinates to Describe a Region Use polar coordinates to describe each region shown in. This is an advantage of using the polar form. Polar Rectangular Regions of Integration. Convert each pair of rectangular coordinates to polar coordinates where r and. We basically use a 2D formation having two coordinates x and y, if you are wishing to create graph points on a coordinate plane then below we are providing instructions of doing that. θ π 4 is the straight line through the origin pole making an angle of π 4 University of Calgary MATH 267 - Summer 2019 Math267-Polar-Coordinates-Double-Integrals. We shall show how easy it becomes using polar coordinates instead. 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. Example 3 2Given the ellipse with equation 9 x2 + 25 y = 225, find the major and minor axes, eccentricity, foci and vertices. The initial line may be identified with the x-axis of rectangular Cartesian coordinates, as. Viewed 11k times 3. If we restrict rto be nonnegative, then = describes the. Symmetry with. 25 and also 12? How about 17 and 13? Good times. View Notes - Polar-Coordinate-System. 5, 30°), (-1. • θis measured from an arbitrary reference axis • e r and eθ are unit vectors along +r & +θdirns. I'm always amazed by what my students come up with on this one - I've even had a student who designed a penguin using polar equations!. Extensions Graphing the polar equations will help the students to make the connection when they are learning to change polar coordinates to rectangular coordinates and back The students can visually see the points on the polar axis and compare the point on the rectangular axis. This discussion is critical for you to understand in order to correctly determine the polar coordinates. 1 Illustrating polar coordinates. The graphing worksheets are randomly created and will never repeat so you have an endless supply of quality graphing worksheets to use in the classroom or at home. d) ˜˝3, ˝Π 6 ˚. Then a number of important problems involving polar coordinates are solved. Convert the following equation to polar coordinates: y = − 4 3 x 6. Find the distance between the points. pdf View Download. In particular, how the angle increases counter-clockwise and how the radius rincreases going away from the origin. Convert the equation of the circle r= 2sinto rectangular coordinates and nd the center and radius of the circle. 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. Polar coordinates are not unique. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). 3 convert the equation from polar to rectangular and draw the graph (a) r = 2secθ (b) θ = π/3 (c* graph only) r = θ 4 find the slope of the tangent line at the given value of θ (a) r = 3, θ = π/6 (b) r = 1+3sinθ, θ = π/4 5* What is the distance formula for two points in polar coordinates, (r1, θ1) & (r2, θ2) ?. This is the result of the conversion to polar coordinates in form. Arc length of polar curves. r = 2 and θ= 30°, so P is located 2 units from the origin in the positive direction on a ray making a 30°angle with the polar axis. z axis up, θ is sometimes called the zenith angle and φ the azimuth angle. We begin with a brief review of polar coordinates. Polar sun path chart program This program creates sun path charts using polar coordinate for dates spaced about 30 days apart, from one solstice to the next. Here, the two-dimensional Cartesian relations of Chapter 1 are re-cast in polar coordinates. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations. The rst coordinate is the distance of the point from the origin (0;0), and the second coordinate is the angle, in standard. Creating Constellations on a Coordinate Plane Grade Level/Subject: Science or math (Could be simplified to 3rd grade level by making the coordinates all positive. If we restrict rto be nonnegative, then = describes the. We see this general pattern in the circle of gure 2. We interpret r as the distance from the sun and θ as the planet's angular bearing, or its direction from a fixed point on the sun. $\begingroup$ See answer here with diagram showing the directions and magnitudes of the changes in velocity (in polar form). This is the xy-plane. If C is a circle of radius R, then its polar equation is f(ϑ) = R where 0 6 ϑ 6 2π. 8 Polar Equations of Conics We have seen that geometrically the conic sections are related since they are all created by intersecting a plane with a right circular cone. 442, C 5 s21, 0. Home Decorating Style 2020 for Polar Coordinate System Pdf, you can see Polar Coordinate System Pdf and more pictures for Home Interior Designing 2020 76667 at Manuals Library. A point P is located at (r,θ) in a polar coordinate system if the distance from P. Showing top 8 worksheets in the category - Number Planes. Therefore r˙(t) = (˙rcosθ − rθ˙sinθ)i + (˙rsinθ + rθ˙cosθ)j. However, doing the math is the tricky part. We shall show how easy it becomes using polar coordinates instead. Polar - Rectangular Coordinate Conversion Calculator. We basically use a 2D formation having two coordinates x and y, if you are wishing to create graph points on a coordinate plane then below we are providing instructions of doing that. However, the Coriolis acceleration we are discussing here is a real acceleration and which is present when rand both change with time. (5, 960°) SOLUTION: Let P(r, θ) = (5, 960°). 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. This is an advantage of using the polar form. Then we will use these formulas to convert Cartesian equations to polar coordinates, and vice versa. Changing the solvent has little. 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. polar coordinates pl (plural only) ( mathematics ) The coordinates of a point in a plane, measured as its Cartesian distance from the origin and the angle measured anticlockwise / counterclockwise from the x -axis to a line joining the point to the origin. Graphing in Polar Coordinates Jiwen He 1 Polar Coordinates 1. We need to subtract 960 by 180k, such that the result is between 0 and 180. Eliminate the parameter and identify the graph of the parametric curve. The equations of the 10 - and 20 - radius circles are r = 10 and r = 20, respectively. A point in polar coordinates requires an angle a, in degrees, and distance from the origin, r. It’s easy to convert rectangular coordinates to polar coordinates when the angle of the polar coordinate is 0°, 30°, 45°, 60°, or 90°. Spherical polar coordinates provide the most convenient description for problems involving exact or approximate spherical symmetry. Lesson 6: Polar, Cylindrical, and Spherical coordinates 1. How does this compare with the computation in the usual coordinate system ψ(x,y) = (x,y,0)? Answer: In order to compute the Christoﬀel symbols, ﬁrst we need to compute the partials of E, Fand G: E ρ = 0 E θ = 0 F ρ = 0 F θ = 0 G ρ = 2ρ G θ = 0. Symmetry with. 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):. Spherical-polar coordinates. Search this site. Superposition of separated solutions: u = A0=2 + X1 n=1 rn[An cos(n ) + Bn sin(n )]: Satisfy boundary condition at r = a,. 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. To find the game you're looking for, use the filter below. For example, the behavior of the drum surface when you hit it by a stick would be best described by the solution of the wave equation in the polar coordinate system. pdf), Text File (. Elasticity equations in polar coordinates (See section 3. d is the perpendicular distance from the line to the origin. In this section, we explore the question of how to quantize a system in curvilinear coordinates, using plane polar coordinates as an example. The r represents the distance you move away from the origin and θ represents an angle in standard position. Lecture 36: Polar Coordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point Oand a half line or ray starting at the point O. Review: Polar coordinates Deﬁnition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ) deﬁned by the picture. b) Show that the area of R is 1 (9 3 2) 16 − π. We will look at polar coordinates for points in the xy-plane, using the origin (0;0) and the positive x-axis for reference. And if we talking about polar paper for maths so this is a type graph paper which is used in many projects and also. A location is defined by its distance in x,y from the origin point. POLAR COORDINATE GRAPH PAPER Author: Frances Elizabeth Blount Created Date: 10/26/2007 9:37:15 AM. The easiest kind of region R to work with is a rectangle. 5: Polar Coordinates Polar coordinate system, introduced by Isaac Newton, is often more convenient in some applications than the more traditional Cartesian, or rectangular, coordinate system. Convert Rectangular to Polar Coordinates Polar Axis If a point P has rectangular coordinates (x, y) then the polar coordinates (r, e) of P are given by and tan-I y —, when x > O tan tan — or + 1800, when x < O. Complete the unit circle with each angles’ coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. Polar Coordinates - Solution Question 1 Plot the points with Cartesian coordinates A 8 p 3;8 and B 5 4;5 p 3 4 and then convert them to polar coordinates. Apr 9 - Today I handed back and we went over the unit 5 and Unit 6 tests. Write an equation for this curve in rectangular coordinates. The polar coordinate system,(r, ), is convenient if we want to consider radial distance from a ﬁxed point (origin, or pole) and bearing (direction). The orientation of a plane curve can be represented by arrows drawn along the curve. Selection File type icon File name Description Size Revision Time User; Ċ: D32. The photochemistry of 4-haloanilines and 4-halo-N,N-dimethylanilines has been studied in apolar, polar aprotic, and protic solvents. In polar coordinates the position of an object $$R$$ distance from the origin as represented in the diagram above is modelled $$\mathbf{r} = R \hat{r}$$ The velocity and acceleration in polar coordinates is derived by differentiating the position vector. The following steps can be used for graphing polar curves: 1. Using relative coordinate, points entered by typing @x,y [Enter] Polar Coordinates Polar coordinates used when you need to draw the next points at specify angle. Polar coordinate lines. Polar, Cylindrical, and Spherical Coordinates 1. 64 Spoke Degrees. The polar coordinate system is a two-dimensional coordinate system using a polar grid: The r and θ coordinates of a point P measure respectively the distance from P to the origin O and the angle between the line OP and the polar axis. The solution is such that the stress components are in the form of a Fourier series in θ {\displaystyle \theta \,}. Polar Coordinates. Find the distance between the points. The area element in polar coordinates In polar coordinates the area element is given by dA = r dr dθ. 1 Exponential Equations Blank. Graphing in Polar Coordinates Jiwen He 1 Polar Coordinates 1. Find a different pair of polar coordinates for each point such that 0 ≤ ≤ 180° or 0 ≤ ≤ π. 2 Calculus In The Polar Coordinate System Contemporary Calculus 4 Area in Rectangular Coordinates (Fig. 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. 3 WS Polar Coordinates (Answers). The general idea behind graphing a function in polar coordinates is the same as graphing a function in rectangular coordinates. Pre-Calculus Worksheet Name: _____ Section 10. Polar-coordinate equations for lines A polar coordinate system in the plane is determined by a Pole Pand a half-line called the polar axis, extending from Pto the right in Figure 1. However, it still is a useful tool to give you an introduction to the concepts pertaining to polar coordinates. 04 Double Integrals in Polar Coordinates. Find the polar coo dinate. 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. 54 Example 1 – Using Polar Coordinates to Describe a Region Use polar coordinates to describe each region shown in. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. Media in category "Polar coordinate system" The following 124 files are in this category, out of 124 total. It only takes a minute to sign up. The use of polar graph paper or circular graph paper uses, in schools and colleges math teachers, are also still making assignments that require students to make a graph and draw my own by hands. Coordinate Graph Paper PDF. 7 is self explanatory. • θis measured from an arbitrary reference axis • e r and eθ are unit vectors along +r & +θdirns. 0 International License. Polar Coordinates. This coordinate system is convenient to use when the distance and direction of a particle are measured relative to a fixed point or when a particle is fixed on or moves along a rotating arm. The polar coordinate system is an adaptation of the two-dimensional coordinate system invented in 1637 by French mathematician Ren é Descartes (1596 – 1650). But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. As previously noted, the Cartesian coordinate (a,b) refers to the point a cen-timeters in the x-direction and b centimeters in the y-direction. We need to subtract 960 by 180k, such that the result is between 0 and 180. In particular, how the angle increases counter-clockwise and how the radius rincreases going away from the origin. pdf), Text File (. 5) 1,150 6) 1, 240 7) Plot 3, 4 A on the polar grid and find three additional pairs of polar coordinates that name the point if 22. Double Integrals in Polar Coordinates. A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. Polar coordinates and Defining the Polar Coordinate Axes Any right triangle can be defined on a circle with radius r. This introduction to polar coordinates describes what is an effective way to specify position. 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). (ii) Find two other pairs of polar coordinates for each point, one with r ˜ 0 and one with r ˚ 0. The first equation looks easy but there is a hidden assumption that you need to be aware of. 2 Calculus In The Polar Coordinate System Contemporary Calculus 4 Area in Rectangular Coordinates (Fig. Write the inequalities for. Polar coordinates are a set of values that quantify the location of a point based on 1) the distance between the point and a fixed origin and 2) the angle between. 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 emanates from it that we call the polar axis. Since the axis of the parabola is vertical, the form of the equation is Now, substituting the val-ues of the given coordinates into this equation, we obtain. We must also know how to convert from rectangular to polar coordinates and from polar coordinates to. Apr 9 - Today I handed back and we went over the unit 5 and Unit 6 tests. 0 Unported by Lantonov. The initial line may be identified with the x-axis of rectangular Cartesian coordinates, as. 841d f0, 2pd u3 5 cos21s2y3d 5 0. Use absolute polar coordinates when you know the precise distance and angle coordinates of the point. To specify a clockwise direction, enter a negative value for the angle. Math 126 Worksheet 5 Polar Coordinates Graphing Polar Curves The aim of this worksheet is to help you familiarize with the polar coordinate system. 10) It is often convenient to work with variables other than the Cartesian coordinates x i ( = x, y, z). Pre-Calculus Notes Name: _____ Section 10. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. Graph the point P, (r; ) = 3;ˇ 3. There are some aspects of polar coordinates that are tricky. org are unblocked. 11) ( , ), ( , ) 12) ( , ), ( , ) Critical thinking question: 13) An air traffic controller's radar display uses polar coordinates. Polar Coordinates. 5 3, 3 For problems 17-20, convert the rectangular coordinates to polar coordinates with r 0 and 0 2. The equations of the 10 - and 20 - radius circles are r = 10 and r = 20, respectively. location than conventional Cartesian coordinates. 362 Chapter 10 Conics, Parametric Equations, and Polar Coordinates 21. The conversion from polar to rectangular coordinates is the same idea as converting rectangular form to polar form in complex numbers. Review: Polar coordinates Deﬁnition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ) deﬁned by the picture. First measure a circle feature. We graph some of the basic functions in polar coordinates using LiveMath and a graphing calculator. Polar coordinates An alternative to using rectangular coordinates (x and y) to specify points in the plane is to specify how far the point is from the origin and the direction it lies in. The ordered pair specifies a point's location based on the value of r and the angle, θ, from the polar axis. 5355 0 -10] x = 1×4 5. 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. Ciencia y Tecnología, 32(2): 1-24, 2016 - ISSN: 0378-0524 3 II. (b) Find the velocity of the particle in polar coordinates. The graph above shows symmetry with respect to the y-axis. More Graphing Polar Equations. Enter this lesson and corresponding worksheet covering the basics of the polar coordinate system. What happens when you divide a circle by 365. d) ˜˝3, ˝Π 6 ˚. r is a directed distance from the pole to P. As the function approaches ˇ, the value reduces back to zero. 2 : Apr 12, 2018, 11:37 AM. (it can be positive, negative, or zero. (iii) Find the Cartesian coordinates of the point. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. a) r=3secθ b) r=−3sinθ c) rcsc 1θ= 5) Convert the rectangular equation to polar form. Beautifull!! #2 Andre, December 19, 2009 at 12:22 p. Graph the point P, (r; ) = 3;ˇ 3. SCHROEDINGER'S EQUATION IN SPHERICAL POLAR COORDINATES The magnitude of a central force on an object depends on only the distance of that object. In other words, we need to rewrite the equation so that the denominator begins with 1. These charts print on a standard sheet of 8 1/2 x 11 paper. Example: What is (12,5) in Polar Coordinates?. Home Decorating Style 2020 for Polar Coordinate System Pdf, you can see Polar Coordinate System Pdf and more pictures for Home Interior Designing 2020 76667 at Manuals Library. I Computing volumes using double integrals. 686 CHAPTER 9 POLAR COORDINATES AND PLANE CURVES The simplest equation in polar coordinates has the form r= k, where kis a positive constant. k = 5 Since k is odd, we need to replace r with -r to obtain the correct polar coordinates. I Changing Cartesian integrals into polar integrals. 0 Unported by Lantonov. The area inside the polar curve r = 3 + 2cos q is-4 -2 2 4-4-2 2 4 (A) 9. Cylindrical Coordinates Transforms The forward and reverse coordinate transformations are != x2+y2 "=arctan y,x ( ) z=z x =!cos" y =!sin" z=z where we formally take advantage of the two argument arctan function to eliminate quadrant confusion. Polar coordinates An alternative to using rectangular coordinates (x and y) to specify points in the plane is to specify how far the point is from the origin and the direction it lies in. Polar coordinates and applications Let’s suppose that either the integrand or the region of integration comes out simpler in polar coordinates (x= rcos and y= rsin ). (ii) Find two other pairs of polar coordinates for each point, one with r ˜ 0 and one with r ˚ 0. Precalculus: Polar Coordinates Practice Problems 3. Let (r,θ) denote the polar coordinates describing the position of a particle. Cartesian/Polar Coordinates Junior high school The connection between Cartesian coordinates and Polar coordinates is established by basic trigonometry. Polar Graph Paper. A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. Ask Question Asked 7 years, 3 months ago. Algebra of complex numbers You should use the same rules of algebra as for real numbers,. POLAR COORDINATES CONTINUED Writing Equations in Polar Form x y r r r, , cos , sin oo T T T I can convert an equation from Rectangular Form to Polar Form. Any geometric object in the plane is a set (collection) of points, so we can describe it by a set of coordinate pairs. (i) Plot each point. To evaluate ZZ R f(x,y)dxdy proceed as follows: • work out the limits of integration if they are not. Show Step-by-step Solutions. The ranges of the variables are 0 < p < °° 0 < < 27T-00 < Z < 00 A vector A in cylindrical coordinates can be written as (2. 23 17_2_polar_coordinates. 7 is self explanatory. The fact that a single point has many pairs of polar coordinates can cause complications. Example Sketch the curve described by the polar equation. This allows you to fully utilize the paper size that you have on hand. 1 Polar Coordinates and Polar Equations OBJECTIVE 1: Plotting Points Using Polar Coordinates In this section we begin our study of the polar coordinate system. You can see this by just drawing unit vectors at each point on, say, a circle: (draw). The transformation from spherical coordinates to Cartesian coordinate is. (5, 960°) SOLUTION: Let P(r, θ) = (5, 960°). A CNC program module based on polar coordinate system Article (PDF Available) in International Journal of Advanced Manufacturing Technology 68(5-8) · September 2013 with 2,790 Reads. No oﬃce hours Tuesday 2/19. y x y x P(x,y) 0 y x y x P(x,y) 0 r a) b. You should be familiar with the Cartesian Coordinate System, also called rectangular coor- dinates, and with the deﬁnitions of sin and cos. A point P is located at (r,θ) in a polar coordinate system if the distance from P. A 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. This measurement will display in the Cartesian coordinates. The polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine: = , = . 21 Locating a point in polar coordinates Let’s look at a specific example. Basic polar coordinates are those coordinates with angles not lower than -360° and not higher than +360°. The location of P in the plane can also be described using polar coordinates. Polar Coordinates This file contains one interactive page that your students could use to practice plotting polar coordinates. First, fix an origin (called the pole) and an initial ray from O. Consider the top which is bounded above by z= p 4 x2 y2 and bounded below by z= p x2 + y2, as shown below. How does this compare with the computation in the usual coordinate system ψ(x,y) = (x,y,0)? Answer: In order to compute the Christoﬀel symbols, ﬁrst we need to compute the partials of E, Fand G: E ρ = 0 E θ = 0 F ρ = 0 F θ = 0 G ρ = 2ρ G θ = 0. Polar coordinates are in the form r, , where is the independent variable. We want to nd another way to get to the point (x;y). The Polar Coordinate System is a different way to express points in a plane. The polar form of (a,b) is illustrated in Figure 1. Number Planes. Polar coordinates and applications Let's suppose that either the integrand or the region of integration comes out simpler in polar coordinates (x= rcos and y= rsin ). Plane polar coordinates pdf Plane polar coordinates pdf Plane polar coordinates pdf DOWNLOAD! DIRECT DOWNLOAD! Plane polar coordinates pdf Polar Coordinates r, θ in the plane are described by r distance from the origin and θ 0, 2π is the counter-clockwise angle. It is sometimes convenient to refer to a point by name, especially when this point occurs in multiple \draw commands. generally start to learn to think in terms of polar coordinates. Consider the top which is bounded above by z= p 4 x2 y2 and bounded below by z= p x2 + y2, as shown below. 2 We can describe a point, P, in three different ways. Extensions Graphing the polar equations will help the students to make the connection when they are learning to change polar coordinates to rectangular coordinates and back The students can visually see the points on the polar axis and compare the point on the rectangular axis. Double Integrals in Polar Coordinates 1. Finally, the Coriolis acceleration 2r Ö. 5355 0 -10] x = 1×4 5. Stirling's Web Site. Compass Labels on Polar Axes.


1. Dec 12, 2013

### Appleton

1. The problem statement, all variables and given/known data

$\int \frac{1}{\sqrt{1-x^{2}}} dx$

I seem to be getting two incompatible answers when I substitute x with $sin \theta$ and $cos \theta$. Could someone please help me with where I'm going wrong.

2. Relevant equations

3. The attempt at a solution

first attempt (the correct answer I believe):

$let\ x = sin \theta$

$dx = cos \theta\ d\theta$

$\int \frac{cos \theta}{\sqrt{1-sin^{2}\theta}} d\theta$

$= \int \frac{cos \theta}{\sqrt{cos^{2}\theta}} d\theta$

$= \int \frac{cos \theta}{{cos\theta}} d\theta$

$= \int 1\ d\theta$

$= \theta$

$x = sin \theta$

$\theta = arcsin x$

second attempt (the wrong answer I believe)

$let\ x = cos \theta$

$dx = -sin \theta\ d\theta$

$\int \frac{-sin \theta}{\sqrt{1-cos^{2}\theta}} d\theta$

$= \int \frac{-sin \theta}{\sqrt{sin^{2}\theta}} d\theta$

$= \int \frac{-sin \theta}{{sin\theta}} d\theta$

$= \int -1\ d\theta$

$= -\theta$

$x = cos \theta$

$\theta = -arccos x$

2. Dec 12, 2013

### sandyp

Its simple...we know that d/dx arc sinx = (1-x^2)^-1/2
d/dx arc cos x = -(1-x^2)^-1/2 (notice the minus sign)
as integral is antiderivative of a function you are getting the same values
yes integral arc sinx=(minus)integral arc cos x..Hope your doubt is clarified

3. Dec 12, 2013

### ShayanJ

In fact $\int \frac{1}{\sqrt{1-x^2}}dx=\sin^{-1}{x}+C_1 \ and \ \int \frac{1}{\sqrt{1-x^2}}dx=-\cos^{-1}{x}+C_2$
If you take a look at the plots of the functions $\sin^{-1}{x}$ and $\cos^{-1}{x}$,you can see that you can get one of them by multiplying the other by a minus sign and then adding a constant(or vice versa!)...and you can see that you have the negative sign and also the constant in the results of the integral!

4. Dec 12, 2013

### Appleton

Thanks for clearing that up for me.

5. Dec 12, 2013

### Staff: Mentor

No one has mentioned this. There's a trig identity that can shed some light here.

$sin^{-1}(x) + cos^{-1}(x) = \pi/2$
The two functions differ by a constant. If an indefinite integral produces two different antiderivatives, those functions can differ by at most a constant.


Integral - substitution method problem

Yankel

Active member
Hello all

I am working on this integral

$\int \frac{x^{2}+1}{x^{4}+1}dx$

Now, I have tried this way:

$u=x^{2}+1$

after I did:

$\int \frac{x^{2}+1}{\left ( x^{2}+1 \right )\left ( x^{2}-1 \right )}dx$

But I got stuck, I got:

$\frac{1}{2}\cdot \int \frac{1}{u\sqrt{u-1}}dx$

I thought of making another substitution, but I tried and failed. Help required

Prove It

Well-known member
MHB Math Helper
Hello all

I am working on this integral

$\int \frac{x^{2}+1}{x^{4}+1}dx$

Now, I have tried this way:

$u=x^{2}+1$

after I did:

$\int \frac{x^{2}+1}{\left ( x^{2}+1 \right )\left ( x^{2}-1 \right )}dx$

But I got stuck, I got:

$\frac{1}{2}\cdot \int \frac{1}{u\sqrt{u-1}}dx$

I thought of making another substitution, but I tried and failed. Help required
Well first of all, \displaystyle \displaystyle \begin{align*} x^4 + 1 \end{align*} is NOT equal to \displaystyle \displaystyle \begin{align*} \left( x^2 + 1 \right) \left( x^2 - 1 \right) \end{align*}.

I would try

\displaystyle \displaystyle \begin{align*} x^4 + 1 &= x^4 + 2x^2 + 1 - 2x^2 \\ &= \left( x^2 + 1 \right) ^2 - \left( \sqrt{2} \, x \right) ^2 \\ &= \left( x^2 - \sqrt{2}\, x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) \end{align*}

and so

\displaystyle \displaystyle \begin{align*} \int{ \frac{x^2 + 1}{x^4 + 1}\,dx} &= \int{ \frac{x^2 - \sqrt{2}\,x + 1 + \sqrt{2}\,x}{ \left( x^2 - \sqrt{2}\,x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) } \, dx} \\ &= \int{ \frac{1}{x^2 + \sqrt{2}\,x + 1} \,dx} + \sqrt{2} \int{ \frac{x}{ \left( x^2 - \sqrt{2}\,x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) } \, dx} \\ &= \int{ \frac{1}{x^2 + \sqrt{2}\,x + \left( \frac{\sqrt{2}}{2} \right) ^2 - \left( \frac{\sqrt{2}}{2} \right) ^2 + 1 } \, dx} + \sqrt{2} \int{ \frac{x}{x^4 + 1} \, dx} \\ &= \int{ \frac{1}{ \left( x + \frac{\sqrt{2}}{2} \right) ^2 + \frac{1}{2} } \, dx} + \frac{\sqrt{2}}{2} \int{ \frac{2x}{ \left( x^2 \right) ^2 + 1 }\, dx} \end{align*}

Now in the first integral, let \displaystyle \displaystyle \begin{align*} x + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}\tan{(\theta)} \implies dx = \frac{\sqrt{2}}{2}\sec^2{(\theta)}\,d\theta \end{align*} and in the second integral let \displaystyle \displaystyle \begin{align*} x^2 = \tan{(\phi)} \implies 2x\,dx = \sec^2{(\phi)}\,d\phi \end{align*} and the integrals become

\displaystyle \displaystyle \begin{align*} \int{ \frac{1}{\left( x + \frac{\sqrt{2}}{2} \right) ^2 + \frac{1}{2} } \, dx} + \frac{\sqrt{2}}{2} \int{ \frac{2x}{ \left( x^2 \right) ^2 + 1 } \, dx} &= \int{ \frac{1}{ \left[ \frac{\sqrt{2}}{2} \tan{(\theta)} \right] ^2 + \frac{1}{2} } \cdot \frac{\sqrt{2}}{2}\sec^2{(\theta)}\,d\theta } + \frac{\sqrt{2}}{2} \int{ \frac{\sec^2{(\phi)}}{ \tan^2{(\phi)} + 1 } \, d\phi } \\ &= \frac{\sqrt{2}}{2} \int{ \frac{\sec^2{(\theta)}}{ \frac{1}{2}\tan^2{(\theta)} + \frac{1}{2} } \, d\theta} + \frac{\sqrt{2}}{2} \int{ \frac{\sec^2{(\phi)}}{\sec^2{(\phi)}}\,d\phi} \\ &= \sqrt{2} \int{ \frac{\sec^2{(\theta)}}{ \tan^2{(\theta)} + 1 } \, d\theta} + \frac{\sqrt{2}}{2} \int{ 1 \, d\phi } \\ &= \sqrt{2} \int{ \frac{\sec^2{(\theta)}}{\sec^2{(\theta)}}\,d\theta} + \frac{\sqrt{2}}{2} \phi + C_1 \\ &= \sqrt{2} \int{ 1\,d\theta } + \frac{\sqrt{2}}{2} \arctan{ \left( x^2 \right) } + C_1 \\ &= \sqrt{2} \, \theta + C_2 + \frac{\sqrt{2}}{2}\arctan{ \left( x^2 \right) } + C_1 \\ &= \sqrt{2} \arctan{ \left( \sqrt{2}\, x + 1 \right) } + \frac{\sqrt{2}}{2} \arctan{ \left( x^2 \right) } + C \textrm{ where } C = C_1 + C_2 \end{align*}

Yankel

Active member
sorry, I thought it was minus when I tried, of course x^4+1 can't go the way I wrote it.

I have a basic question, how did you got to trigonometry ? did you move to polar system or something ? I don't get it. I mean, how did you know you need trigonometry in the first place ?

Thanks !

paulmdrdo

Active member
I would try this method but this is open for any correction

$\displaystyle \int\frac{x^2+1}{x^4+1}dx$

by letting

$\displaystyle u\,=\,x^2$
$\displaystyle du\,=\,2xdx$
$\displaystyle dx\,=\,\frac{du}{2x}$
$\displaystyle x\,=\,u^{\frac{1}{2}}$

can you continue..?

ZaidAlyafey

Well-known member
MHB Math Helper
For practice try

$$\displaystyle \int \frac{1}{1+x^4}\,dx$$

$$\displaystyle \int \frac{1}{\sqrt[4]{1+x^4}}\, dx$$

$$\displaystyle \int \frac{x^2}{1+x^4}\, dx$$

They are a little bit tougher .

[*] By the way I have a way to solve the integral using complex numbers , the answer is nasty .

MarkFL

Staff member
Using partial fractions, we may write:

$$\displaystyle \int\frac{x^2+1}{x^4+1}\,dx=\frac{1}{2}\int\frac{1}{x^2+\sqrt{2}x+1}+\frac{1}{x^2-\sqrt{2}x+1}\,dx$$

Completing the squares...

$$\displaystyle \int\frac{x^2+1}{x^4+1}\,dx=\frac{1}{2}\int\frac{1}{\left(x+\frac{1}{\sqrt{2}} \right)^2+\frac{1}{2}}+\frac{1}{\left(x-\frac{1}{\sqrt{2}} \right)^2+\frac{1}{2}}\,dx$$

Now, we may simply apply the formula:

$$\displaystyle \int\frac{du}{u^2+a^2}=\frac{1}{a}\tan^{-1}\left(\frac{u}{a} \right)+C$$

to get (after simplification):

$$\displaystyle \int\frac{x^2+1}{x^4+1}\,dx=\frac{1}{\sqrt{2}} \left(\tan^{-1}\left(\sqrt{2}x+1 \right)+ \tan^{-1}\left(\sqrt{2}x-1 \right) \right)+C$$

Prove It

Well-known member
MHB Math Helper
sorry, I thought it was minus when I tried, of course x^4+1 can't go the way I wrote it.

I have a basic question, how did you got to trigonometry ? did you move to polar system or something ? I don't get it. I mean, how did you know you need trigonometry in the first place ?

Thanks !
It's a method known as trigonometric substitution. I suggest you read what Mark has written in post #8 of this thread.