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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384 views

Harmonic functions and polar differential forms

Given a harmonic function $u$, its differential and conjugate differential are $$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy,\qquad ^{*}du = -\frac{\partial u}{\partial y}dx ...
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2answers
24 views

Area between two polar curves method

The question is not too hard. I sketched them and they were correct which was not too bad. I then did the second part by finding the intersection points between the two curves which are $\frac{\pi}{...
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0answers
20 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
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2answers
32 views

Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
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1answer
137 views

Laplace Challenge in One Examples, Is there any help?

this question is taken from 2014 exam on CE Entrance Exam, Question $32$ on the end of page $6$. Consider the Laplace equation of following polar coordination, $$\frac{1}{r}\frac{\partial}{\partial ...
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24 views

trying to figure out how to set up polar area given two equations

$R=3$ and $R=4-2sin(\theta)$ Find the area I got the second part of the equation right ($1/2$ the integration of $\pi/6$ to $5\pi/6$ of $(4-2sin(x))^2)$ but he first part confuses me. The answer ...
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1answer
111 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system $(r, \theta, \phi)$, $\theta$ can range from $0$ to $2\pi$, but $\phi$ only varies from $0$ to $\pi$. Why is that?
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Calculus 3 Integrals

Find the volume of the solid bounded by $z=4x^2+4y^2, z=0, x^2+y^2=1$ and $x^2+y^2=2$. What I know: I know that when I draw the graph I will get two paraboloids giving me a radius of $1$ and $2$,$\...
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2answers
93 views

Plotting complicated polar curve without calculator

I'm wondering if anyone can give me tips or guidance on how to plot complicated polar curves without the use of a calculator. Most notably, I am trying to plot: Based on a graphing calculator, I ...
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1answer
2k views

Find the area of the region inside $r=a$ and outside $r=a(1-cos (\theta))$.

I found the intersecting point to be at $\pi/2$ and ${3\pi}/2$ $a=a(1-\cos(\theta) )$ $\cos (\theta)=0$ $\theta= \pi/2$, $3\pi/2$ I'm confused as to what angles to use for integration. As for $r=...
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2answers
36 views

Triple integral using Cylindrical Coordinates

Evaluate the iterated integral $x= 0$ to $x=1$, $y= -\sqrt{1-x^2}$ to $y=\sqrt{1-x^2}$ and $z= 0$ to $z=2-x^2-y^2$ and $f(x,y,z)= \sqrt{x^2+y^2}$ What I know: I know that I have to use cylindrical ...
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4answers
85 views

Triple integral of $e^{{(x^2+y^2+z^2)}^\frac32}$

I'm trying to find the result of $$ \iiint_{R} e^{(x^2+y^2+z^2)^{3/2}} \,dz\,dy\,dx $$ with $$ R = \left\lbrace (x,y,z) \in \mathbb{R}^3 \mid -1 \le x \le 1, 0\le y \le \sqrt{1-x^2}, 0\le z \...
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0answers
54 views

Difference of Entropy of two-dimensional Gaussians

I encountered a putative contradiction. Assume we have two 2-dim. Gaussian variables $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ with all components being independent, normal distributed variables: $x_1,...
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1answer
22 views

Why Circle is traced counterclockwise and ellipse is traced clock wise?

In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise. "Which was this one here. And first we noted that this does parameterize, as we ...
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0answers
39 views

Area of the graph $z=\sqrt{x^2+y^2}$ over the ring $r^2 <(x^2+y^2)<4$ with $0<r<2$

Could someone please help me calculate the surface area of the graph $z=\sqrt{x^2+y^2}$ over the ring $r^2 <(x^2+y^2)<4$ with $0<r<2$ Thanks!
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1answer
13 views

Help writing a parametric equation from this complex polar one

A particle is moving along the curve $r=4-2\sin(\theta)$ at the moment when $\theta = t^2$. I need to write a x(t) and y(t) function that will model the particle behavior with its x position and y ...
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1answer
21 views

Is there any way that a polar coordinate function can be shifted inward or outward without changing the derivative where it exists?

question Let f(x) be a polar coordinate function defined from 0 to pi Is there a translation that pulls f outward (like in the sense f(x) + c does) without altering the polar coordinate derivative? ...
3
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1answer
23 views

Is there a way to shift a portion of a polar coordinate graph without disrupting its shape?

A polar coordinate function f(x) can be rotated around the axis by h with the shift f(x - h). However, this rotates the graph. Is there a way to shift a polar coordinate function by x and y ...
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2answers
33 views

Calculate the area bounded by the polar curve $ r(\theta)=2+\sin(\theta)$

How can I calculate the area bounded by the following curve: $$ r(\theta)=2+\sin(\theta) $$
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0answers
22 views

Converting a higher order operator to polar coordinates?

The operator: $$\frac{\partial^4}{\partial x^4}+2\frac{\partial^4}{\partial x^2 \, \partial y^2}+\frac{\partial^4}{\partial y^4}$$ ... appears in the equation of motion of an oscillating rigid ...
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1answer
21 views

polar cordinates, finding teta

the integral,and a picture of the graph of D in the left corner you have the integral and below it the range D. the right side has the ranges i found. Hi im having a hard time understanding ...
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0answers
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Finding flux using polar coordinates.

I solved exercise using polar coordinates. I thought that $x=rcos(\alpha)$ and $y=rsin(\alpha)$. In my exercise given. Given vector field and I solved it using divergence theorem. Result is: $\iiint$...
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1answer
65 views

Find the area of the region that is bounded by $r = \cos(\theta)$ from $0$ to $\pi/6$

Find the area of the region that is bounded by $r = \cos(\theta)$ from $0$ to $\pi/6$. I'm pretty confident that I know what I am doing for this problem but I just want to make sure I have it down. ...
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2answers
39 views

Find the area of the region that is enclosed by:

$r = 2\cos(\theta)$ and $ r = 1$ I went ahead and tried it and my answer was just $2\pi$. I was wondering if someone could check if I got it right, and if I didn't, tell me what I did wrong? ...
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1answer
948 views

Find all polar coordinates of point $P$ where $P = (7, \pi/3)$.

I don't know where to go from here. Answer choices are: a) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + 2n\pi)$ b) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + (2n + 1)\,\pi)$ c) $(7, \pi/3 + (2n + 1)\,\pi)$ ...
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2answers
25 views

Eliminate the parameter

Given the parametric equations: $x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$ Eliminate the parameter. I am completely lost. Please help.
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1answer
26 views

Converting coordinates

I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ...
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3answers
45 views

Find polar coordinates $(r, \theta)$ of the point, where $r > 0$ and $0 \leq \theta < 2\pi$

Given these Cartesian coordinates: $(2,-3)$ This is my fourth problem of this type, I solved the other 3, but this one has weird numbers and I don't know what to do. $\tan\theta = -\frac{3}{2}$ ...
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line integral problem, stuck in the curve

Problem: A force field is given in polar coordinates by the equation $$F(r,\theta)= (-4\sin (\theta), 4\sin (\theta)).$$ Compute the work done in moving a particle from the point $(1,0)$ to the ...
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2answers
40 views

Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?

Is it possible to convert the polaer equation $$\ r = k \cos (\theta n) + 2$$ into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$. The ...
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2answers
38 views

Changing the order of integration on a rectangular and polar region

change the order of integration for the following integral from dydx to dxdy, and from dydx to polar coordinates. $$ \int \int f(x,y) dydx$$ where $$ 0≤y≤(-x^2)+2 $$ $$ 0≤x≤1$$ From dydx to dxdy $...
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Eliminate the parameter to find a cartesian equation for the curves

For the first part I am just unsure as to how the book has a different answer than mine. The book has the answer $y = \frac{3}{4} x - \frac{1}{4}$ but given the functions $x(t) = 3 - 4t$ and $y(t) = ...
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1answer
44 views

Cauchy rieman equation. What is u?

Using cauchy rieman equation, i want to show the function is analytic. So i want to decompose from f(z) to two term (real part and imaginary part) With rectangular form or polar form. But it is so ...
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2answers
61 views

Why does $\sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}?$

The relation between $p$ and $r$ where $p$ is the length of the perpendicular from the fixed point $O$ on the tangent to the curve at any point $P$ is called pedal equation of the curve. I want to ...
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4answers
70 views

How do I find the Integral of $\sqrt{r^2-x^2}$?

How can I find the integral of the following function using polar coordinates ? $$f(x)=\sqrt{r^2-x^2}$$ Thanks!
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1answer
82 views

Solve differential equation by using polar coordinates

For $\alpha, \beta>0$ the differential equation, I am trying to solve, is given by $$\begin{pmatrix}\dot x_1\\\dot x_2\end{pmatrix}=\alpha\sin(x_1^2+x_2^2)\begin{pmatrix}x_2\\-x_1\end{pmatrix}+\...
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0answers
15 views

Polar Coordinates Volume of Solid, Angle for integration?

I'm trying to understand how to find the angle for the integration in polar coordinate form for a solid. Here's an example of what I'm trying to solve: Find the volume of the solid bounded by the ...
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24 views

Cartesian and Polar Coordinates, proving the same number-pair

The question states: Prove that a necessary and sufficient condition for a point to be represented by the same number-pair (a,b) both cartesian and polar coordinates is that it lies on the initial ...
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3answers
448 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
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0answers
21 views

Calculating a likely position in 0.5 seconds with only the knowledge of the last second (x,y coordinates)

I have x,y coordinates for football players (22) at a rate of 10 records per second - for an entire football match. I have created an animation of the match with the player locations updates at a ...
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Solve a Laplace in Poolar Coordinate under two minutes?

I trouble with calculating the following example from previous exam with short solution on this Link. OP says there is a Laplace ٍPoolar Coordinate: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\...
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2answers
81 views

Solve $z^5=-32$ and draw its solutions in complex space, then describe their characteristic geometrical property.

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
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2answers
42 views

Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?

Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?
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1answer
51 views

How to think in terms of Polar Coordinates?

I am currently studying Polar Coordinates and many times I've noticed that one converts polar equations in cartesian form to do further analysis. Is there a way to think in terms of Polar Coordinates? ...
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1answer
78 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
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1answer
16 views

Polar angle of a given point

How to find out polar angle of a given point $A(x_1,y_1)$ relative to another point $B(x_2,y_2)$ in a 2D space?
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48 views

Convert Polar Equation to Rectangular Equation

I have been trying to convert the polar equation to rectangular. $U(1,\theta) = 10+3*sin(\theta)-10*cos(2*\theta)$ I started off my multiplying everything by r and using the trig identity for cos($...
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2answers
32 views

Intersection between roses with given polar equations

$ r_1 \ = \ 4 \sin(3 \theta) \ $ and $ \ r_2 \ = \ 3 \cos(3\theta) \ $ a) find the solutions to the system using polar coordinates I was able to solve this by setting $ \ r_1 \ $ and $ \ r_2 \ $ ...
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1answer
1k views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * \...
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2answers
30 views

Converting polar equation to cartesian coordinates

I have been trying everything to convert the polar equation $r=\frac{2}{1-\cos(\theta)}$ to cartesian coordinates but I simply didn't manage to know the right answer. Please help me..