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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Rectangular coordinates transform in polar coordinates $\displaystyle\int_{0}^2\int_0^{x} f(x,y) \,d y\,d x$

If i have this integral in rectangular coordinates and i want to transform in polar coordinates $$\int_{0}^2\int_0^{x} f(x,y) \,dy\, d x$$ The limits in $\displaystyle 0\le\theta\le\frac\pi4 $ but ...
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1answer
76 views

Find the area inside the lemniscate $r^2 = 8 cos 2\theta$ and outside the circle $r = 2$.

Fooplot graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. This ...
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0answers
28 views

Find the area inside the circle $r = 10 \sin \theta$ and above the line $r = 2 \csc \theta$.

Fooplot graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. This ...
0
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0answers
28 views

Find the area common to the inside of the cardioid $r = 1+\sin \theta$ and the outside of the cardioid $r = 1 + \cos \theta$.

Fooplot graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. This ...
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1answer
41 views

Find the area of the small loop of the limacon $r = 1+2\cos(\theta)$

Find the area of the small loop of the limacon (graph): $$r = 1+2\cos(\theta)$$ What I tried: Set $r=0$ to get $\theta = 2\pi/3, 4\pi/3$. Then $$A = \frac 1 2 \int_{2\pi/3}^{4\pi/3} r^2 d\theta$...
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1answer
41 views

$r = \sin(3\theta)$ Manual graph

What is the thought process behind the second line of the following, namely that the sin of 3 theta equals +/- 1? Thank you. $r = \sin(3\theta)$ $\sin3\theta = +- 1$ $3\theta = (2k+1)\frac{\pi}{2}$...
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2answers
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Find the polar equation of the circle with center on the line theta = pi, of radius 1, and passing through the origin. [closed]

Question Find the polar equation of the circle with center on the line theta = pi, of radius 1, and passing through the origin. i set a point (a,pi) on the line and going to this equation a = rsin(...
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0answers
22 views

Infinitesimal area element in polar coordinate

We know, that the infinitesimal area element in Cartesian coordinate system is $dy~dx$ and in Polar coordinate system, it is $r~dr~d\theta$. This inifinitesimal area element is calculated by measuring ...
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1answer
27 views

Convert ODE to polar coordinates.

$$k \frac{d}{dx}[A(x)\frac{dT(x)}{dx}] - hP(x)[T(x) - T] = 0 $$ What I had in mind was: $$x = rcosϴ, r = \frac{x}{cosϴ} , \frac{dr}{dx} = \frac{1}{cosϴ} $$ $$\frac{dA(x)}{dx} = \frac{dA(r)}{dx}\frac{...
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2answers
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Polar Coordinate Conversion (Integration)

I want to convert some integrals to use polar coordinates as my differentials, my problem is getting the limits. So this is the first concept I am not understanding: If I have a circle in the xy-...
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0answers
4 views

default Metric Unit Circle in plane with polar coordinates - NAME

Quick question: Is there a certain name for this metric? $d(\theta_1,\theta_2) = \left\{ \begin{array}{ll} |\theta_1-\theta_2| & \mbox{if } |\theta_1-\theta_2|\le \pi \\ 2\pi-|...
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0answers
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Direction of a ray in the hemisphere

If my surface has normal (0,0,1), and I center a hemisphere about that normal, how do I compute the ray that is cast in direction $[\theta, \phi]$ within that hemisphere?
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1answer
20 views

Rectangular to polar conversion

I am trying to write this fraction in polar form (4+10i)/(24i-5) . I am having trouble to get the angle of the polar conversion. I know that in order to get the angle I need to write arctan(10/4)-...
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3answers
43 views

Polar coordinate form of circle equation

I am trying to convert the equation $$(x-0.9)^2+y^2 = 0.1^2$$ into polar coordinates. Using $x = r\cos \theta$ and $y = r\sin \theta$ I get that $$ r = \frac{1.8 \cos\theta \pm \sqrt{(1.8\cos\theta)^...
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1answer
41 views

Convert Integral Rectangular to Polar

How can convert this problem $$ \int_0^2 \int_x^\sqrt{8-x^2} \left(x^2+y^2\right)^{3/2} dydx $$ I convert limits and funtion to polar cordinates as follows: $$ \begin{split} r^2 &= x^2+y^2\...
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2answers
24 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ $$...
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1answer
28 views

How to change complex numbers into polar form? [closed]

How do I changecomplex numbers, for example $2+3i$ to polar form of $re^{i\theta}$. Thank you for any answers.
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2answers
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Converting specific equations from Polar to Cartesian

These different equations are given in Polar and my goal is to plot them in Cartesian coordinate system: $r = \cos(4φ)$ $ φ = \dfrac r {r-1}$, $r > 1$ I am aware of: $x = r \cos( φ )$ $y = r \...
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0answers
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Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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2answers
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How can I find the limits of this iterated polar integration?

How can compute the area of the triangle whose corners are at the origin, (1,0) and (1,1). I solved this with r integral first but I could not find the correct limits for theta integral first order. ...
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2answers
45 views

Polar equation of the curve y = sinx

I am looking for the polar equation of the following curve given in Cartesian Coordinates. y = sinx Any kind of hint or help is appreciated.
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Quick Question regarding De Moivre's Theorem

If $z^2 = 2 - 2i$ find z using the theorem of De Moivre For this question, i first expressed it in polar form which is $$2\sqrt{2}\left(\cos{\frac{7\pi}4} + i\sin{\frac{7\pi}4}\right)$$ Now because ...
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0answers
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Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
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0answers
15 views

Proof of alternate cartesian to polar transformation of theta

My vector calculus lecturer has claimed that rather than the angle $\theta$ in the transformation from cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$ can not only be given by: $$ \...
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1answer
36 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r <a,\...
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2answers
35 views

Transforming integral in cylindrical coordinates into cartesian.

I am trying to transform the following integral to an integral in cartesian coordinates. $$\int^{2\pi}_0\int^1_0\int^{\sqrt{1-r^2}}_0r \ dzdrd\theta$$ I cannot really visualise how the region enclosed ...
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1answer
28 views

$\sin(z)$ in polar coordinates

the following formula can be found in the literature: $\vert \sin(z) \vert^2 = \sin(x)^2 + \sinh(y)^2$, $z=x+iy;$ $x,y\in\mathbb{R}$. I am wondering if there is a similar formular in polar ...
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0answers
8 views

Third order partial derivatives in cylindrical coordinates

Do you know, where I can find formulas for third order partial derivatives in cylindrical coordinates? All I can find are second order partial derivatives. Thanks!
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1answer
55 views

Finding area inside circle and outside another circle

I was trying to find the area inside the circle $r=-2\cos(\theta)$ and outside $r=1$ and the upper bound was $2\pi/3$ while the lower bound is $0$. Is this correct? If not please help me set this up. ...
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1answer
36 views

Help calculating area inside circle and outside cardioid

I've calculated the area inside the circle $r=3a\cos(\theta)$ and outside the cardioid $r=a(1+\cos(\theta))$ and I got two answers: $a^2\pi + a^2\frac{\sqrt{3}}{2}$ and the answer: $a^2 \pi$. Can ...
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0answers
18 views

Double integral using polar coordinates, under a region R

The question I am asking along with the answer I am stuck on I set up my limits of integration to be 0 to 2pi and 2 to 9. $$ \int_{0}^{2pi} \int_2^9 \sin(r^2)rdrd\theta\\ \frac{1}{2}\int_{0}^{2pi} ...
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2answers
44 views

Explain finding the area of a region?

How is the area of the region inside the lemniscate $r^2 = 6\cos(2\theta)$ and outside the circle $r = \sqrt3$ equal to $(3(\sqrt3) - \pi)$? Thank you for anyone that helps.
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How to determine the smallest interval to vary $\theta$ to produce an entire polar graph?

My textbook's method: [For $r=2+cos(5\theta2)$] To find such an interval, we will look for the smallest number of complete revolutions that must occur until the value of r begins to repeat. ...
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1answer
28 views

Tangent parallel to the initial line for polar equation =, can r^2 be used instead?

Given a formula for a polar equation: $$\ r^2 = a^2 \cos^22 \theta $$ It could be said that to find the points parallel to the initial line, $$\frac{dy}{dx} = \frac{d (r\sin\theta)}{d\theta} = 0$$ ...
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1answer
24 views

complex number multiplication by a real number [closed]

I'd like to multiply a complex value by a real integer. I know that multiplication of complex numbers is similar in the polar form, but the way I know and have been taught is to multiply the two real ...
2
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0answers
50 views

Solving exp integral in closed form?

I am trying to solve the following integrals: 1) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 m^2})} dxdy $ 2) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 n^2})} dxdy $ 3) $\int \int ...
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Can I switch to polar coordinates if my function has complex poles?

You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta ...
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0answers
31 views

General polar equation for an off center ellipse?

For a centered ellipse I can plug in r(θ)cosθ and r(θ)sinθ in to the base ellipse equation, getting $r=\frac{ab}{\sqrt{(b\cosθ)^2+(a\sinθ)^2}}$ However, for a noncentered ellipse I get stuck on $$a^...
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1answer
47 views

Constructing a Poincare Map

I need to construct a Poincare Map of the following dynamical system: $\dot x = x-(x+y)(x^2+y^2)$ and $\dot y = y + (x-y)(x^2+y^2)$ I changed the system to polar coordinates which gives me: $\dot ...
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1answer
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Find the volume of bullet shape solid.

Bullet function is given by $y = 16 - x^2 - z^2$ to the right of the $xz-$plane. I have set up the following integral but not sure whether it is true or not. $\int_{-4}^{4} \int_{0}^{2π} \int_{0}^{4}...
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2answers
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inverse of the function $f(r,\theta) = (r\cos \theta, r \sin \theta)$

inverse of the function $f(r,\theta) = (r\cos \theta, r \sin \theta)$ set $x = r \cos \theta$, $y = r \sin \theta$ then we have $ x^2 + y^2 = r^2$ so $r = \sqrt{x^2+y^2}$. Now $y/x = \tan \theta$ so ...
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1answer
56 views

How to convert dynamical system to polar coordinates? [closed]

I have a dynamical system on the plane given by $$\dot{x}=-y+x\left(1-\sqrt{x^2+y^2}\right)\\ \\ \dot{y}=y+x\left(1-\sqrt{x^2+y^2}\right)$$ I want to convert this into polar coordinates as it will be ...
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1answer
42 views

Find the length of the polar curve

How do I find the exact length of the polar curve $$r = 1+sin(\theta)$$ from $$\frac{\pi}{3} \leq \theta \leq \pi $$? I had originally setup my equation as $$\int_{\frac{\pi}{3}}^{\pi} \sqrt{(1+\...
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1answer
22 views

differential equation system to polar coordinates

pic of the question I am having trouble showing that $y(t)=(2\cos(2t), \sin(2t))$ is a periodic solution of the system: $$\frac{dx}{dt}=-4y+x\left(1-\left(\frac{x^2}{4}\right)-y^2\right)$$ and $$\...
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1answer
20 views

Exact length of a polar curve

I have the following problem: Find the exact length of the curve: $$r = 2(1 + cos(\theta))$$ How should determine the intervals. I used the graph but it is a cardioid and i do not know how to proceed.
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0answers
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Find imaginary part of complex expression

Given the system of ODEs, $$x'=x^3-3xy^2$$ $$y'=3x^2 y-y^3,$$ it can be shown that the system may be written as $z'=z^3$, where $z=x+iy$. However, I don't seem to get how to show that $\Im m\{\frac{1}{...
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1answer
20 views

Rewriting basic functions in polar form

I've been exploring how to rewrite common parent functions ($x^2, \sqrt x$,...) in polar form. Is it possible to rewrite natural log or trig functions in polar form as a function of $\theta$? For ...
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10 views

derivation of polar planimeter - polar coordinates, finding partial derivatives

I'm working through a derivation of the equations for a polar planimeter, source https://www3.amherst.edu/~tleise/HomePage/LeisePlanimeter.pdf, and I'm stuck at this point of the derivation. (Page 5, "...
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1answer
37 views

Polar Coordinates Double Integral Question

Evaluate $\int(x^2+y^2)^{1/2}dA$ where $D$ is region enclosed by the two circles: $x^2+y^2=64$ and $x^2+(y-4)^2=16$. I'm confused on what the limits of integration for the corresponding double ...