Questions on polar coordinates, a coordinate system where points are represented by their distance from the origin ($r$) and the angle the line joining the point and the origin makes with the positive horizontal axis ($\theta$).

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2
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1answer
60 views

Area of $\left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2$

I've used the modified polar coordinates: $x = 3r \cos \theta$, $y =5r \sin \theta$, which got me to $$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$ What now?
1
vote
1answer
79 views

Find polar equation from 4 polar points

Given $4$ polar coordinates $(3, -\pi/6)$, $(1, \pi/3)$, $(3, 5\pi/6)$, $(-3, 4\pi/3)$, graph and find the polar equation. I know that the general polar equation is $r = ep / 1+- e \cos (\theta)$. ...
2
votes
3answers
98 views

Finding a length of arc, what's wrong?

Find: $$ \int \sqrt{x^{2}+y^{2}}dl$$ $$L: x^{2}+y^{2}= Rx$$ (at image $p' = -R\cdot \sin(\phi)$ )
5
votes
4answers
141 views

Find the maximum value of $r$ when $r=\cos\alpha \sin2\alpha$

Find the maximum value of $r$ when $$r=\cos\alpha \sin2\alpha$$ $$\frac{\rm dr}{\rm d\alpha}=(2\cos2\alpha )(\cos\alpha)-(\sin2\alpha)(\sin\alpha)=0 \tag {at maximum}$$ How do I now find alpha? ...
8
votes
3answers
307 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
2
votes
1answer
366 views

Heat equation in polar co-ordinates

I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge $r=a$ of a circular plate is kept at temperature $f(\theta)$. The plate is insulted so that there is ...
1
vote
1answer
80 views

Sketch the polar graph $r=e^{-2\phi}$

How are you supposed to sketch this type of polar graph? Are you supposed to somehow relate this to $\cos\phi+i\sin\phi$ but can polar graphs even have an imaginary axis?! I am thinking that you ...
3
votes
3answers
763 views

Why is the formula for the area of a cardioid $ \int_a^b \frac{1}{2} r^2 d \theta$

I've seen this expression in many places :$\int_a^b \frac{1}{2} r^2 d \theta$ and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction ...
0
votes
2answers
64 views

How do you find the maximum value of $r$ in a polar function?

I have $\, r=\cos\alpha +\sin2\alpha,\quad 0\le\alpha\le\frac{\pi}{2}.$ Do you then find $\dfrac{dr}{d\alpha}$ and let that $=0$ ? I am after just a few set of instructions.
1
vote
1answer
439 views

Find the area of the shaded region between $r=e^{\theta/2}$ and $r=θ$ .

That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ...
1
vote
0answers
82 views

Curl in cylindrical coordinates

I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating: ...
0
votes
2answers
97 views

Dirac delta from polar coordinates to cartesian coordinates

I have: $$k_x = k \cos\theta\\k_y=k\sin\theta$$ I would like to rewrite in terms of $k_x$ and $k_y$: $$\exp(in\theta)\,\frac{\delta(k-\alpha)}{k}$$ I start from: ...
1
vote
2answers
93 views

$f(x,y)=\langle y- \cos y, x \sin y\rangle$

$f(x,y)=\langle y-\cos y,x\sin y\rangle$ $C$ is the circle $(x-3)^2 + (y+4)^2 = 4$ orientated clockwise. Relevant theorems: Green's theorem (this is under the Green's theorem section of our book). ...
4
votes
2answers
1k views

Dirac delta in polar coordinates

Given $$x=r\,\cos\theta\\y=r\,\sin\theta$$ and $$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$ how can I express $$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates? And the more general ...
5
votes
2answers
2k views

Plotting in the Complex Plane

I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$ What is the difference plotting this function in the complex plane or real plane?
4
votes
1answer
66 views

Polar coordinations - problem with r and $\theta$

let's take a look on Archimedean spiral. the polar equation is $r = \theta$. click here to look. but $\tan (\theta) = y/x$ and $r = \sqrt{x^2+y^2}$, so $r = \theta \rightarrow \tan(\sqrt{x^2+y^2}) ...
0
votes
1answer
34 views

Determining the correct upper bound for an integral in polar coordinates

This seems super easy. But i am just a little bit stuck here. Haven't done much calculus recently. Can someone help me out real quick? Thank you in advance!
4
votes
2answers
348 views

Integration of radial functions?

Let $f(|x|)$ be a integrable radial function in $\mathbb{R}^n$ ($|\cdot|$ denotes the euclidean norm as in convention). The following identity is used to simplify computations ...
1
vote
1answer
131 views

How did theta become equal to 3pi/4 here?

How did theta become equal to 3π/4 in this particular example? Find a set of polar coordinates (r,θ) of the cartesian point (-4,4) such that -2π ≤ θ ≤ 2π and a. r > 0 and θ > 0 b. ...
1
vote
1answer
149 views

Inaccuracy in numerical calculation of arclength of part of an ellipse

I am trying to numerically calculate the arclength of part of an ellipse according to: $$ L = \int_0^{\phi_s}\sqrt{r^2+\left(\frac{dr}{d\phi}\right)^2} d\phi $$ where $r$ is defined as: $$ ...
1
vote
1answer
623 views

Finding area between two polar curves using double integrals

I have a homework question that is asking me to find the area that lies: Inside the curve $r=2+cos(2\theta)$ But outside the curve $r=2+sin(\theta)$ I think I'm supposed to be using a double ...
4
votes
3answers
349 views

Trying to understand the meaning of symmetry

The picture below is the solution to the following problem as presented in my book: Find the area of the region that lies inside both curves $$r = 8 + \cos \theta \\r = 8 − \cos θ$$ According to ...
1
vote
1answer
162 views

Moment of inertia of a circle

A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$ Thank you
4
votes
1answer
203 views

Mexican Hat wavelet in polar coordinates

I'm interested in wavelet framework for polar coordinates. In the paper of Hou&Qin (2012) was proposed a general method for definition of MH wavelets on a certain manifold. In short, first we ...
2
votes
2answers
171 views

Find Cartesian equation of $r=\theta$

I solved this problem, but I'm not sure my answer is correct as it seems very complex (compared to the polar equation). Did I make some mistake along the way or is it the right solution? $$r=\theta$$ ...
0
votes
2answers
2k views

Don't understand how to use jacobian for transformation of coordinates

Hello. I fail to understand why the Jacobian matrix is used to transform Cartesian coordinates to polar coordinates. If I'm not misunderstanding, it is assumed that the matrix ...
2
votes
1answer
2k views

Find a Cartesian equation of $r = 4\cos\theta$

I was able to figure the substitutions inside the equation, but I'm stuck with the equation's manipulation that will give me the solution. What would be my next step? $$r = 4\cos\theta$$ $$r^2 = ...
4
votes
2answers
201 views

Parametrization of a curve in polar coordinates

I'm trying to change this parametrics equations to polar coordinates $$ X(t) = 2\cos(t) - \sin(2t) \\ Y(t) = 2\sin(t) - \cos(2t) $$ What i tryed to do was raise the two equations squared, sum ...
2
votes
1answer
101 views

Line integral of $F = r \times k$ on hemisphere

Exam revision - Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere $r=c$, with $z \geq 0$, where $F = r \times k$ and $k$ is the unit vector ...
1
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1answer
164 views

Polar Coordinates: Dividing by the variable “r.”

Evaluate the iterated integral by converting to polar coordinates: $\large \int^2_0 \int^{\sqrt{2x-x^2}}_0 xy~dy~dx$ I successfully completed most of the problem; however, I had difficulty ...
0
votes
1answer
431 views

How does one interpolate between polar coordinates?

I'm finding that when I try to use the standard methods of interpolation in polar space, the result is not what I would expect. For example, when interpolating between the following polar coordinates: ...
0
votes
2answers
292 views

Finding the centroid of a polar curve

The curve is $r = e^{-b\theta}$ where $b > 0$ and $θ \in [0, \infty)$. I got that the arc length is $\frac{\sqrt{b^2 + 1}}{b}$ (is this correct?), but computing the centroid $(x, y)$ looks awful. ...
3
votes
1answer
821 views

How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?

How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
1
vote
2answers
81 views

Integration, polar coordinates

My question is general rather than specific.If a problem requires to find the area of a figure bounded by a curve given in polar coordinates,how do we find the limits of integration analytically ...
3
votes
1answer
76 views

polar coordinates ..question about the answer from the solution manual

Im trying to figure out but for some reason I dont know how to...could someone please tell me how did they get this answer from the solution manual....they skipped steps so I have no idea
1
vote
1answer
117 views

Express in Rectangular Form

a) $(-1+i)^{-i}$ so I know that the answer is $9.92-3.58i$. My track getting there is off. I know that $x=-1$ and $y=1$, so $r = \sqrt{2}$, also that $\displaystyle \theta=-\frac{pi}{4}$. I've ...
2
votes
0answers
33 views

Pure differential equation whose solution is a siluroid?

I am trying to find a differential equation for the siluroid that DOES NOT contain explicitly $\theta$, $\sin\theta$, or $\cos\theta$, but only $\rho$, $\dot\rho$, $\ddot\rho$. The siluroid equation ...
0
votes
0answers
204 views

Polar Fourier transform in Matlab

I have a 2D signal: sg=sin(x+y). To represent it in 2D I use meshgrid: [xx,yy]=meshgrid(x,y) and I plotted it with ...
2
votes
4answers
436 views

Converting x^2 + 6y - 9 = 0 to polar

Hi I'm trying to solve this problem but am having difficulty removing the remaining r. I have tried http://i.imgur.com/iJk9b2g.jpg but cannot get an answer Any help is appreciated
1
vote
1answer
79 views

Polar coordinate

Let $f(x,y)$ be a differntiable function in $\mathbb{R}^2$ so that $f_x(x,y)y=f_y(x,y)x$ for all $(x,y)\in\mathbb{R}^2$. Find $g(r)$ so that $g(\sqrt{x^2+y^2})=f(x,y)$ and $g$ is differentiable in ...
2
votes
1answer
122 views

evaluation of double order integral using polar co-ordinates

When evaluating double integral using polar co-ordinates, does the order of $dr ~ d\theta$ make any difference? Suppose, $$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 dr d\theta$$ ...
1
vote
1answer
113 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
1
vote
1answer
161 views

triple integral - ecliptic coordinate

I need to find the $V$ by triple integral. the limits from up is (1) - ecliptic cone. and from dwon - (2) - elepsoide. $$(1) \ \ \ \ z=-\sqrt{3x^2+5y^2}$$ $$(2) \ \ \ \ {3 \over 10}x^2+5y^2+{z^2 ...
1
vote
0answers
170 views

gradient of an axis symmetric vector field in cylindical coordiantes

I am trying to calculate with a general approach the gradient of an axis symmetric vector field in cylindrical coordinates and then express it in cartesian coordinates. First I write my vector ...
1
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2answers
4k views

Find the area of the Rose's petal.

If a Rose leaf is described by the equation $r = \sin 3\theta$, find the area of one petal.
0
votes
2answers
24 views

Polar coordinates that uses $\frac { 1 }{ Z_1 }$

I am doing polar coordinates, and I am stuck when my book asks to do $\frac { 1 }{ Z_1 }$. I have no problems with $\frac { Z_1 }{ Z_2 }$ and $Z_1Z_2$. Here is the values for $Z_1$ I'm not so much ...
1
vote
1answer
135 views

What happens to a line in polar coordinates when orgin is moved and rotated in cartesian coordinates?

Let's say we have an Archimedean spiral in Cartesian coordinates. This corresponds to a line in polar system (i.e. $r=a\theta+b$). Now if I move the origin of the Cartesian coordinates system to ...
1
vote
2answers
118 views

Coordinate system conversion: what it is called what I'm doing?

I want to do a simple coordinate transformation and would like to know what is the rigorous way to describe this mathematically in order to be able to search for algorithms for more complex ...
0
votes
1answer
61 views

Polar form $\frac{dy}{dx}$

Trying to find the derivative $\dfrac{dy}{dx}$ in polar form, where: $$x=r\cos\theta \,\text{ and } \, y=r\sin\theta$$ Seems like the common approach (on Wikipedia and other sites) is to assume that ...
0
votes
2answers
93 views

What is the inverse $z^{-1}(z)$ of $z(\varphi)=e^{i\varphi}$ with $\varphi\in\Bbb N_0$?

Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me). How ...