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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0
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2answers
16 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: $$\cos x=\frac{A}{R}$$ ...
0
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1answer
26 views

How to change complex numbers into polar form? [on hold]

How do I changecomplex numbers, for example $2+3i$ to polar form of $re^{i\theta}$. Thank you for any answers.
0
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2answers
17 views

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. ...
10
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4answers
5k views

Adding two polar vectors

Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?
2
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2answers
25 views

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 ...
-4
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0answers
18 views

Consider the point on the complex plane. [on hold]

$z=r(\cos\theta+ i\sin\theta)$ Where is it? Where is $z^2$, $z^3$, $z^n$?
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0answers
25 views

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
40 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.
-1
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0answers
12 views

Great circle distance on an ellipsoid [on hold]

Let's say I have a set of latitude and longitude (B,L on a reference ellipsoid WGS-84) and I also know the great circle distance (both in radians and meters) from my point to some point X on a sphere ...
0
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2answers
22 views

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 ...
0
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0answers
17 views

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 ...
1
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1answer
23 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 ...
0
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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: $$ ...
1
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2answers
32 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 ...
0
votes
1answer
24 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
6 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!
1
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1answer
40 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. ...
0
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1answer
21 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 ...
0
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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} ...
1
vote
0answers
53 views

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 ...
1
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2answers
39 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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0answers
11 views

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. ...
2
votes
0answers
49 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 ...
0
votes
1answer
19 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, $$\frac{dy}{dx} = \frac{d (r\sin\theta)}{d\theta} = 0$$ ...
0
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0answers
52 views

Tangent undefined for polar curves ($r^2=a^2\sin(s\theta)$)?

I am considering the function $r^2=a^2\sin(2\theta)$ and am trying to find tangents perpendicular to the initial line, so $\frac{dx}{d\theta}=0.$ However, when I take the derivative by implicit ...
1
vote
1answer
23 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 ...
1
vote
1answer
44 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 ...
0
votes
0answers
26 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 ...
0
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1answer
44 views

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} ...
0
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1answer
693 views

Conversion of a complex number into polar form

Below is the complex number that is to be converted into Polar form. I'm facing problem in second part of this number(after + mark not the (b) itself). When I divide them(10/-5+j12) directly, by ...
1
vote
2answers
58 views

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 ...
0
votes
1answer
711 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) $(7, \pi/3 + (2n + 1)\,\pi)$ ...
-1
votes
1answer
44 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 ...
2
votes
3answers
2k views

In polar coordinates, can r be negative?

I'm getting different answers for this. Many websites say that when you get a negative value of r, you flip the coordinate 180 degrees across the pole. However my teacher says that you cannot have a ...
1
vote
1answer
40 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} ...
1
vote
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 ...
0
votes
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. ...
7
votes
5answers
357 views

Finding the orientation of a noisy ellipse

This question comes from a neuroscience study which generates $12$ vectors. The vectors are evenly spaced, $30 n$ degrees for $n=0,\dots, 11$, each with their tail centered on the origin. I am ...
1
vote
0answers
56 views

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 ...
3
votes
1answer
18 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 ...
-1
votes
2answers
139 views

for $z,w\in\mathbb{C}$, $\sqrt{zw} = \sqrt{z}\sqrt{w}$?

for $z,w\in\mathbb{C}$, $\sqrt{zw} = \sqrt{z}\sqrt{w}$ I started by writing $z$ and $w$ in polar coordinates, and writing it out, giving another form for the question: $$ \begin{align} y &= ...
-1
votes
0answers
13 views

Runge-kutta of ode2 in polar coordinates. Newtons law of motion

i am supposed to simplify these equations so that we later can use runge-kutta 4 in matlab. the are supposedly a version of lewtons law of motion in polare coordinates. We have all the initial values ...
0
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0answers
6 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, ...
0
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1answer
33 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 ...
0
votes
1answer
33 views

Area enclosed by a circle and leminscate

Find the area enclosed by a circle $r=4\sin\theta$ and out of $r^2=8\cos 2\theta$ I have tried the following integral ...
1
vote
2answers
66 views

Phase portrait of ODE in polar coordinates

Given the system of ODEs in polar coordinates, $$r' = r(1-r^2)(4-r^2)$$ $$\theta'=2-r^2,$$ one can determine its equilibrium points and limit cycles as follows: $\gamma_1:= \begin{cases} r = 0,\\ ...
0
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1answer
24 views

Can't find the area of a polar region

I've ran into a bit of a stopper on this one problem. I solved this other problem like this yesterday but this one seems to cancel itself out to zero. I'm not sure what I'm doing wrong with this ...
0
votes
2answers
32 views

Finding a circle in polar coordinates

I have converted the system of ODEs, $$x'=x-y-x(x^2+5y^2)$$ $$y'=x+y-y(x^2+y^2),$$ to polar coordinates and got this: $$ r' = r-r^3(1+4\sin^2(\theta)\cos^2(\theta))$$ ...
0
votes
1answer
16 views

Can the limits be applied differently to the following multiple integration?

The question is to change the cartesian form to the corresponding polar form: $$\int_0^a\int_y^a{\frac{x^2\,dx\,dy}{\sqrt{x^2+y^2}}}$$ The limit when applied in the format $\theta =0$ and $\theta = ...
7
votes
2answers
327 views

Spiral of Archimedes area and sketch in polar coordinates

This is an exercise from Apostol's Calculus, Volume 1. It asks us to sketch the graph in polar coordinates and find the area of the radial set for the function: $$f(\theta) = \theta$$ On the interva ...