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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1answer
245 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: $$ ...
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1answer
1k 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 ...
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3answers
538 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 ...
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1answer
219 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
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1answer
249 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 ...
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2answers
377 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? $$r=\theta$$ ...
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2answers
6k 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 ...
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1answer
4k 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 = ...
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2answers
290 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 ...
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1answer
168 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 ...
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1answer
217 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 ...
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1answer
1k 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: ...
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2answers
540 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. ...
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1answer
2k 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 ...
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2answers
99 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 ...
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1answer
96 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
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1answer
203 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 ...
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0answers
35 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 ...
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4answers
1k 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
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1answer
93 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 ...
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1answer
243 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, $$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 dr d\theta$$ ...
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1answer
153 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 ...
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1answer
219 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 ...
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0answers
228 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 ...
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2answers
10k 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.
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28 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 ...
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1answer
163 views

What happens to a line in polar coordinates when origin 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 ...
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2answers
136 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 ...
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1answer
66 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 ...
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96 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 ...
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2answers
20k views

Ellipse in polar coordinates

I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - Where $2a$ is the total height of the ellipse and $2b$ ...
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93 views

How to solve following non-linear differential equation?

Let's have an equation $$ \left(\frac{\partial f}{\partial r}\right)^{2} + \frac{1}{r^{2}}\left(\frac{\partial f}{\partial \varphi}\right)^{2} = g(r). $$ How to solve it?
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1answer
90 views

Qualitative analysis of an ordinary differential equation in polar coordinates

I want to draw the integral curves of the differential equation in polar coordinates $(\theta, \rho)$ $\frac{d\rho}{d\theta}= \rho^3-6\rho^2+8\rho$ At first I thought it would suffice to analyse ...
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1answer
148 views

Really Stuck on Partial derivatives question

Ok so im really stuck on a question. It goes: Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$. calculate $\frac{\partial u} {\partial x} (x,y)$ and ...
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1answer
97 views

Polar coordinates parameters

Sketch in the same diagram the curves with polar equations $r=2a\cos\theta$ and $2r(1+\cos\theta)=3a$ and find the polar coordinates of their points of intersection. What is the polar equation of ...
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3answers
136 views

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

So far I got here \begin{align} (r\cos\phi)^2 & + 6 r \sin\phi- 9 = 0\\ (r\cos\phi)^2 & = 9 - 6r \sin\phi \end{align}
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1answer
159 views

Converting polar to cartesian?

So far I got \begin{align} r & = 7 / (4 - 2 \cos\theta) \\ r (4 & - 2\cos\theta) = 7 \\ r (4 & - 2( x / r ) ) = 7 \end{align} I apologize in advance for the bad formatting.
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2answers
2k views

Convert $ x^2 - y^2 -2x = 0$ to polar?

So far I got $$r^2(\cos^2{\phi} - \sin^2{\phi}) -2 r\cos{\phi} = 0$$ $$r^2 \cos{(2\phi)} -2 r \cos{\phi} = 0$$
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1answer
510 views

Double integral area : how to find the curve equation

I have the following equation $$(x+y)^{4} = ax^{2}y$$ I need to find the area limited by the equation above. I know I have to transform x and y in polar coordinates: $$\begin{align*} &x = ...
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3answers
160 views

Cartesian and Polar Coordinate

I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers: a) $z_{1}=-i$ b) $z_{2}=\sqrt{3}+i$ ...
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2answers
2k views

Diff eq. transformation polar coordinates

I have $(x',y')=(x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}},x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}} )$ Now I want to use polar coordinates $(x,y)=(r\cos(t),r\sin(t))$ to get ...
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1answer
1k views

Converting to polar form

Write each of the given numbers in the polar form $re^{i\theta}$. a.) $\frac{1-i}{3}$ b.) $-8\pi (1+\sqrt 3 i)$ For a, I got: r = $\frac{\sqrt 2}{3}$ and $e^{i7\pi /2}$ since ...
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0answers
931 views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
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1answer
120 views

Find the extremal on the unit disc

I need help for finding the extremal of: $$J[u]=\int\int_D (u_x^2+u_y^2) dxdy$$ $D$ is the unit disc i.e. $x^2+y^2 \leq 1.$ My boundary condition is $$u(\cos\theta, \sin\theta)=\sin(n\theta), \ \ ...
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3answers
3k views

Calculating a limit in two variables by going to polar coordinates.

I have this limit to calculate: $$l=\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$ I solve it by going to the polar coordinates. Since $(x,y)\to 0$ means the same as $\sqrt{x^2+y^2}\to 0$, ...
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2answers
183 views

Help understanding the velocity of polar curves.

I have been studying for the AP BC Calculus exam (see this previous question) and most of the questions that deal with the first derivative in polar coordinates say that if ${dr\over d\theta}<0$ ...
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660 views

How know which direction a particle is moving on a polar curve

I have being doing problems from the released AP BC Calcululs Free-Response questions, and I have come to realize that I don't have a very good idea of explain or a deep understanding of how to tell ...
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3answers
252 views

Transform to polar the following integral $\int_0^6\int_0^y x \, dx \, dy$

I need to transform this integral $\int_0^6\int_0^y x \, dx \, dy$ to polar and then find its value. I'm stuck finding the r-limits of integration.
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1answer
214 views

equation for the region inside a circle

What equation or group of equations fill the entire or part of a region inside a circle without using inequalities? Update I don't know if this problem is already solved, I'm trying to find the ...
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1answer
208 views

Finding an argument function in a sinusoidal along a circle

I'm attempting to find a function (in polar coordinates) slightly like the one shown below --- i.e. a function which describes a sinusoidal motion along a circular path. ...