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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43 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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18 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. ...
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1answer
22 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 ...
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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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57 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 ...
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30 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 ...
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1answer
46 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
45 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} ...
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2answers
60 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 ...
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1answer
51 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
41 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} ...
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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
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 ...
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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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8 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
36 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 ...
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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 ...
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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 ...
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2answers
33 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: $$ r' = r-r^3(1+4\sin^2(\theta)\cos^2(\theta))$$ ...
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71 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,\\ ...
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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 = ...
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1answer
18 views

Jacobian matrix for ellipsoid

ive been asked to fine the jacobian matrix for an ellipsoid $$x^2/a^2 + y^2/b^2 + z^2 / c^2 = 1$$ ive been looking online for the parametric equations and i get two different answers ...
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2answers
34 views

Differentiation with polar coordinates

I'm sorry if this is supposed to be something basic but I'm not being able to understand if r is as given above, how have they worked out r dot? What have they differentiated the x,y and z ...
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2answers
23 views

Parabola equation from cartesian to polar representation

I've got the following equation: 0) $ \frac{(y-y_p)^{2}}{4\cdot(x-x_p)} = p $ I'd like now to convert this expression to a polar representation. For this I got back to the basic rules: 1) $x = ...
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2answers
25 views

Polar Equation to Rectangular

$$r=\frac{9}{4 \cos θ − 3 \sin θ}$$ How do I do this? (Equation is in polar form.) I have already tried to do this, but I don't know how to finish it.
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31 views

Help needed on evaluating the following double integral

Use polar coordinates to evaluate $\int\int_{D}\ x\ dA$ where $D$ is the region inside the circle $x^2 + (y-1)^2 = 1$ but outside the circle $x^2 + y^2 =1$. (It's like a crescent moon facing the ...
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1answer
28 views

how to write floor function vectors in polar coordinates

let $$\lfloor{x}\rfloor=y$$ And $$z=x-\lfloor{x}\rfloor$$ Plot the following vector in polar coordinates: $$x\hat{\imath}+(y/z)\hat{\jmath}$$ I know that while transforming from cartesian to polar we ...
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26 views

Find the center of mass of homogeneous object

I am asked to find the center of mass of this homogeneous object: Let's say that it's density is $k$ so the mass is $$ m = \int_{0}^{\pi} \int_{a}^{2a} k r drd\theta = \frac{3\pi a^2 k}{2} $$ So ...
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1answer
22 views

Find the area of this figure using polar coordinates (possible textbook mistake)

I am required to find the area of this region using polar coordinates: My setup is $$ A = \frac{1}{2} \int_{0}^{\phi} \left[ R \sin(\theta) \right]^2 d\theta = \frac{R^2}{4} \int_{0}^{\phi} \left[ ...
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1answer
20 views

How to show that a function is in a Sobolev space

This question is about the solution of exercise 1.20 in Elman, Silvester, Wathen. Finite Elements and Fast Iterative Solvers. (The first Chapter of the book is open access and available, for example, ...
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1answer
31 views

Find a polar equation for the curve of the given Cartesian equations: $y=x$, $4y^2 = x$ and $xy=4$

I am asked to find a polar equation for the curve of the given Cartesian equations: $y=x$, $4y^2 = x$ and $xy=4$. What I got here so far is $$ y = x\\ r \sin(\theta) = r \cos(\theta)\\ ...
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3answers
45 views

How to convert $\theta = \pi/3$ into cartesian form?

How can I convert $$\theta = \frac{\pi}{3}$$ into cartesian form? What I get is $$ \theta = \frac{\pi}{3}\\ cos(\theta) = \frac{x}{r} = \frac{1}{2}\\ x = \frac{r}{2} $$ and I'm not sure what the ...
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1answer
37 views

Integrate $f(r,\theta) = r$ over the region between $r = a(1+cos\theta)$ and $r = a$

I am asked to integrate the function given in polar coordinates $f(r,\theta) = r$ over the region between $r = a(1+\cos\theta)$ and $r = a$. My answer is $$\int_{-\pi/2}^{\pi/2} \int_{0}^{a} r\cdot ...
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2answers
85 views

Calculate Coordinates on Arc, Based on Time of Day

Hopefully someone can help me out with this. I'm trying to calculate the position of a point on an arc, based on a percentage of distance along the circumference (% time of day). Sidenote - I'm ...
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1answer
30 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
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1answer
42 views

Riemannian metric given in polar coordinates

the Riemannian metric of the euclidean plane is given in polar coordinates as \begin{align*} ds^2=dr^2+r^2d\theta^2. \end{align*} Consider more generally, \begin{align*} ds^2=dr^2+\psi(r)^2d\theta^2, ...
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1answer
14 views

Volume of a Cone using Cylindrical Coordinates

I'm aware of the usual method for calculating the volume by expressing the integrals for $dr$ and $dz$ in terms of $z$ to get the correct answer but when I attempted to solve it expressing everything ...
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19 views

Transforming points between two polar coordinate systems

I have 2 dimensional points (r, theta) defined in a polar coordinate system A, and a second polar coordinate system B with a known homogeneous transform T transforming between A and B in a Cartesian ...
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2answers
28 views

Integrate function $f(x,y) = x^2+y^2$ in circle $(x-a)^2+y^2 < a^2$

I am asked to integrate function $f(x,y) = x^2+y^2$ in circle $(x-a)^2+y^2 < a^2$ My answer is $\frac{3 \pi a^4}{2}$ but for some reason the answer of the textbook is $\frac{32 \pi a^4}{2}$. Does ...
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1answer
27 views

Integrate function $f(x,y) = y^2$ in $x^2+4y^2 \leq a^2$

I am asked to integrate function $f(x,y) = y^2$ in $x^2+4y^2 \leq a^2$ To do that using polar coordinates, how may I find the boundaries for $r$? Is there a procedure that always works (for ellipses ...
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3answers
52 views

Integrate $f(x,y) = e^{-x^2-y^2}$ in a circle with radius $1$ and center at $(0,0)$

I am asked to integrate $f(x,y) = e^{-x^2-y^2}$ in a circle with radius $1$ and center at $(0,0)$. The setup of the integral (in my solution) is $$ \int_{0}^{2\pi} \int_{0}^{1} e^{-r^2} r dr ...
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36 views

Integrate $f(x,y) = \sqrt{4-x^2-y^2}$ inside circle with radius 2 and center $(2,0)$

An exercise asks to integrate $$f(x,y) = \sqrt{4-x^2-y^2}$$ inside circle with radius 2 and center $(2,0)$. When I set up the double integral I get $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} ...
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27 views

Converting region $\int_{0}^{2} \int_{0}^{x} f(x,y) dydx$ from rectangular form to polar form

How can I convert the region in $$ \int_{0}^{2} \int_{0}^{x} f(x,y) dydx $$ (which is basically a right triangle contained in the first quadrant) to a polar form? Im sure that $$0 \leq \theta \leq ...
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1answer
22 views

Why the degree change of the following polar form

From the following image: Why was $\theta$ changed to 3.08 instead of -3.12? Confused about that point. Details about the image: converting rectangular form to polar = $5^5$ ∠ $5 * -0.64$
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2answers
21 views

Converting this complex number to polar form?

Given $ z = -1 - i$ ,I converted it to polar form, resulting r =$\sqrt 2$. And $\theta = \tan^{-1} (\frac{-1}{-1}$) = 0.785 rads, which seems incorrect with the solutions of my instructor I don't ...
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2answers
68 views

Converting $r=\sec^2(\theta)$ to Cartesian

I encountered this problem on my Calculus test today and am struggling to figure it out: Write $r = \sec^2(\theta)$ as a Cartesian equation. I have tried using all sorts of tricks on it ($x^2 + y^2 ...
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2answers
55 views

Find a Cartesian equation for the curve and identify it. $ r^2 \cos 2\theta = 1$

Find a Cartesian equation for the curve and identify it. $$ r^2 \cos 2\theta = 1$$ I'm confused by the $2\theta.$ I isolated $r^2$ to get $r^2 = \frac{1}{\cos2\theta}$ Now, normally if it was ...
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1answer
44 views

Dynamical System in Polar Coordinates

I have a dynamical system defined by : $ \dot x = {(x+iy)^n + (x-iy)^n \over2}$ and $\dot y = {(x+iy)^n - (x-iy)^n \over2i}$ Converting the system to polar coordinates gives the system: $\dot r = ...