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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2answers
428 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
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1answer
1k 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 ...
2
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2answers
94 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
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1answer
93 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
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1answer
178 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
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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 ...
0
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0answers
365 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
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4answers
876 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
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1answer
90 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
201 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
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1answer
147 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
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1answer
203 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
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0answers
204 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
7k 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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2answers
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 ...
1
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1answer
156 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
129 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
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1answer
65 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
95 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 ...
4
votes
2answers
16k 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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0answers
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
77 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 ...
2
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1answer
146 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 ...
1
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1answer
85 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 ...
1
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3answers
134 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}
2
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1answer
140 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.
2
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2answers
2k views

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

So far I got $$r^2(\cos^2{\phi} - \sin^2{\phi}) -2 r\cos{\phi} = 0$$ $$r^2 \cos{(2\phi)} -2 r \cos{\phi} = 0$$
1
vote
1answer
491 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 = ...
0
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3answers
152 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) $z_{2}=\sqrt{3}+i$ ...
1
vote
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 ...
1
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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 ...
2
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0answers
798 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= ...
0
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1answer
117 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), \ \ ...
3
votes
3answers
2k 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 $\sqrt{x^2+y^2}\to 0$, ...
0
votes
2answers
165 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 ${dr\over d\theta}<0$ ...
1
vote
2answers
514 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 ...
4
votes
3answers
240 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.
0
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1answer
190 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 ...
1
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1answer
182 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. ...
3
votes
1answer
678 views

Computing gradient in cylindrical polar coordinates using metric?

I am trying to understand coordinate transformations properly (having studied some general relativity in the past). Let us consider the transformation from cartesian to cylindrical coordinates, ...
0
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1answer
657 views

What are the formulas to sum two 2D vectors?

I'm writing a PHP algorithm to calculate the sum of two 2D vectors with the same origin, given their intensity and their angle with a given axis. How to calculate the properties of the sum vector? I ...
4
votes
3answers
2k views

Writing Polar Equations In Parametric Form

For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
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vote
2answers
387 views

Changing Variables in double integral

I have these particular exercise that i cannot solve. I know i have to change the variables, but i cannot figure out if i should use polar coords or any other change. Let D be the region delimited ...
4
votes
1answer
390 views

Stokes' and Divergence Theorem Problems

I have 2 questions on stokes and divergence theorem each. I think I have done both and I just want to make sure that I did them correctly. Question 1 Let $C$ be the boundary of the surface ...
1
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1answer
44 views

Convert $r^2-5r+6=0$ to a rectangular coordinates

The factor form of that expression is: $(x-2)(x+3)$ but I could be more simpler. Thanks for a while.
2
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1answer
141 views

Explain this limit of integration for radius in polar coordinates.

Use polar coordinates to find the volume of the given solid: Inside both the cylinder $x^2 + y^2 = 4$ and the ellipsoid $4x^2 + 4y^2 + z^2 = 64$ The limit of integration for the radius goes ...
1
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0answers
1k views

Mass and center of mass of lamina in polar coordinates

I need some help with the following problem which is question number 15.5.4 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the mass and center of mass of the ...
0
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1answer
214 views

Collision detection of two circular sectors in mixed polar and Cartesian coordinate

I am trying to solve the following collision detection problem. Suppose we have two circular sectors, each described in their own polar coordinate system with four values $r_1$, $r_2$, $d_1$ and ...
1
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1answer
593 views

How to show the normal density integrates to 1?

How could you show that the normal density integrates to 1? $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1 $$
1
vote
1answer
127 views

Problem with ellipse equation

How one get from this ellipse equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ that ellipse equation $$\frac{x^2+y^2}{F(\phi)^2}=1,$$ where $$F(\phi)=\frac{ab}{\sqrt{(b\cos\phi)^2+(a\sin\phi)^2}}$$ and ...