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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polar moment of area for nonplaner circle (cup)

Can somebody tell me the polar moment of area of chord for a sphere. for example when you cut a sphere at a point other than from center? Also polar moment of area for curved axis symmetry ?
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Integrate in cylindrical coordinates

$\vec{\nabla} p = \rho \vec{f}$ How do you solve this in polar coordinates? I can't find a way to insert g in my equation. I would have to split it in $r$ and $\phi$. So $y=sin(\phi) * r$ but I ...
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1answer
50 views

Find centre of mass of a circle when one half is heavier than the other half?

I have a problem which simply states: Consider a circle (lamina) of radius 1 with centre (0,0) where the left half is twice as heavy as the right. Find its centre of mass. Extend your solution to ...
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1answer
25 views

Integral: Area of figure described in polar coordinates

Lets say we have a figure described as follows: $r=2\cdot\sqrt{\cos(2\theta)}$ Click here to see a plot. Now lets say that we want to calculate the area of this rotated eight. I'd like to ...
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28 views

Triple Integration: from Cartesian to Polar Coordinates

I have to evaluate $\iiint_Q (x+y)^2 dV $, where $Q$ is a solid hemisphere within the bounds $z \ge 0$, $\space x^2+y^2+z^2 \le 4$. I am assuming that in order to solve the above integral I have to ...
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21 views

Rewriting Integral Forms

I am asked to (1) Rewrite the integral $ \iint \limits _R f(x,y) \space \Bbb dx \Bbb dy$ in different coordinates $u,v$. (2) Hence, derive the form the integral $\iint \limits _R f(x,y) \space \Bbb ...
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1answer
33 views

Area calculation

How could we best approach calculating the area inside $r=\cos^{2n-1}(x)+\sin(x)$, $0\leq x\leq \pi$, for $n=1,2,...$? For $n=3$ we get the following "potato/bean" graph: and for $n=51$ we get ...
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3answers
48 views

What is the first step I should take in solving this equation?

I have to change this polar equation and put it in terms of $x$ and $y$. $$r = \frac{5}{5\cos(\theta) + 6\sin(\theta)}$$ I was guessing that I should multiply all the terms by r and then convert ...
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2answers
47 views

Graphing polar equation $r\sin \theta = 1$?

How would you graph $r \sin \theta = 1$? I know that $r\sin \theta$ is equal to $y$, but the place where I'm told to graph this function on is a polar graph. How should I go about this?
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0answers
18 views

2D finite difference boundary conditions for radial direction

I am trying to solve Poisson's equation in an axisymmetric cylindrical domain using finite difference. So I start with my differential equation and boundary conditions and discretize them. However, ...
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0answers
25 views

The gradient in $n$-dimensional spherical coordinates

I am in the middle of a computation where I need to work with the formula of the gradient in spherical coordinates in $\Bbb R ^n$ (no preferred convention for the angles). I could patiently and ...
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1answer
13 views

Identical transformation about integrals

\begin{align} I &=\int_0^1dr\int_0^{2\pi}\left(cos\theta\cdot\frac{\partial f}{\partial x}+sin\theta\cdot\frac{\partial f}{\partial ...
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4answers
37 views

convert rectangular coordinate (-3,0) to polar coordinate

I'm trying to convert (-3,0) to polar coordinate. I can get r=$\sqrt {(-3)^2 +(0)^2}$ =3, but when computing for the angle $\theta$=$\tan^{-1} (\frac {0}{-3})$=0 but the answer for the ...
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1answer
26 views

Equations of Motion in Polar Basis

A particle of mass m moves under a central force field $ \mathbf{F}=-k\mathbf{r}$ where k is a constant with dimensions $ N m^{-1} $. Assuming that the particle moves in the equatorial plane ( ...
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1answer
24 views

Region bounded by a Polar Curve

For a National Board Exam: Find the area of the region bounded by a polar curve $r^2 = a^2 \cos(2\theta)$ Answer = $a^2$. So I cheated a bit and plotted the curve on wolfram so i could ...
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2answers
18 views

Area inside a curve and outside a Cardoid

For a National Board Exam: Find the area which is inside the curve r=3cos(theta) and outside the cardoid r=1+cos(theta) Answer is pi Ok I am trying to setup the right definite integral for ...
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1answer
37 views

How can $r$ be negative when dealing with polar coordinates?

If by definition $r=\sqrt{x^2 + y^2}$, then why do we allow $r$ to be negative? Relatedly, I do not understand the last section of this conversation discussing points being represented by multiple ...
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1answer
23 views

Area enclosed by polar curves

Given $$r_1(\theta)=2(1+\cos\theta) \\ r_2(\theta)=2(1-\cos\theta)$$ I want to find the area of the region resulting from the intersection of those curves. Is the following integral correct? $$ 2A= ...
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1answer
34 views

Square inside a Polar coordinate system

I have a square lying on a polar coordinate. Is there any general relationship between radius and angle, which may be derived along the side of square. More generally put, given the coordinates of the ...
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2answers
62 views

From Gravity Equation-of-Motion to General Solution in Polar Coordinates

I'm having trouble getting the general solution of this differential equation. The gravitational equation of motion is, for constants $M$ and $G$ and position vector $\vec{r}$, $$\frac{d^2}{d ...
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0answers
19 views

Diffusion equation in polar coordinates with non-zero boundary conditions (BC)

I'm trying to solve the diffusion equation in polar coordinates: $$c_t = \frac{D}{r^2}[2r\,c_r + r^2\,c_{rr}] = \frac{D}{r}[2\,c_r + r\,c_{rr}] \tag{1}$$ with the following BC: $$c(0,t)=0, \quad ...
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2answers
45 views

Find the maximum radius for given theta and phi (spherical coordinates) that will fall within a cuboidal boundary

I have a cuboid with measurements (width, depth, height) which is my boundary. The origin is the center of the cuboid. Given a theta(Azimuth) and phi(elevation), how do I find the highest radius that ...
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0answers
38 views

polar coordinate transformation

If we have an equation $\mathcal{L_I}=\prod \mathrm{exp}\bigg(-\lambda_j \displaystyle\sum\limits_{m=1}^{\Psi_{j}}\binom{\Psi_j}{m} ...
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1answer
36 views

Line segment equation in polar coordinates

I have a line segment given by two points $A$ and $B$. $$A+u(B-A), u\in[0,1]$$ when doing calculations with this segment, it would be advantageous to have it written in polar coordinates around some ...
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1answer
15 views

Finding the horizontal and vertical tangents of a parametric equation.

Find the points at which the polar curve $r=2+2\sin{(\theta)}$ has a horizontal or vertical tangent line. Translate the parametric equation to Cartesian coordinates: $$ r^2=2r+2r\sin{(\theta)} ...
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1answer
20 views

Polar conversions of coordinates and parametric equations

Express the polar coordinates $P\left(6, -\dfrac{\pi}{4} \right)$ in Cartesian coordinates. $\displaystyle x=r\cos{(\theta)} ,\ y=r\sin{(\theta)} \implies x^2+y^2=r^2 \wedge \theta = ...
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3answers
21 views

Eliminate the parameter of a

Eliminate the parameter to find a description of the following circles or circular arcs in terms of $x$ and $y$. Give the center and radius, and indicate the positive orientation. ...
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1answer
63 views

Find the area using double integral and polar coordinates.

I need to find the area using double integral and polar coordinates. $$y=3-x$$ $$y^2=4x$$ This is what i figured already: $${r\cos{\theta}+r\sin{\theta}} = 3$$ $$r=0, r=3, \theta=0, \theta=\pi/2$$ ...
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0answers
23 views

Converting cartesian to polar integral

I feel like I almost have a grasp on regions of integration, I am a bit frustrated that I haven't fully gotten it but because I feel like I'm almost there. In this particular homework problem I have a ...
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2answers
16 views

Setup region of integration for polar coordinates

I've been working on a homework set for Calc III, right now we're emphasizing double integration and polar integrals. I keep having problems conceptualizing where to actually create my region of ...
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1answer
32 views

Equation to place points equidistantly on an Archimedian Spiral using arc-length

I am looking for a way to place points equidistantly along an Archimedes spiral according to arch-length (or an approximation) given the following parameters: Max Radius, Fixed distance between the ...
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1answer
53 views

Is this a valid example of a non-euclidean Sierpinski attractor?

I am learning the basic concepts about the Chaos Game (I did a previous question about the same topic here), the method to create fractals elaborated by professor Michael Barnsley. The basic example ...
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4answers
139 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?
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2answers
37 views

Volume of a cube in spherical polars

Let us calculate the volume of the cube using spherical coordinates. The cube has side-length $a$, and we will centre it on the origin of the coordinates. Denote elevation angle by $\theta$, and the ...
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1answer
12 views

Perpendicular distance from a 3D point to a vector in spherical polar coordinates.

I have a point $(r, \theta, \phi)$ and a direction vector with angles $(\theta', \phi')$. What would be the method to calculate the shortest distance from the point to the vector?
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45 views

Vector Calculus - Polar Co-ords

I am having a lot of difficulty finding an approach to solving the following question: A dyon is a particle with both electric and magnetic charge; in suitable units $$\mathbf{E} = ...
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1answer
60 views

Express -i in polar exponential form

Express $-i$ in form $r\cdot e^{i\cdot \theta}$ $r=1$ is simple enough. As on an Argand diagram, $-i$ will be at $(0,-1)$. Does $\theta = 3\pi/2$ here? Or -$\pi/2$ to get it $-\pi < \theta < ...
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1answer
41 views

Surface area of the circle

I was told to calculate the surface area of the following circle by the integration method (monte carlo) $x^2 + y^2 = 1$ The area of this circle is determined by the following inequalities: $-1 ≤ x ...
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1answer
28 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system (r, θ, φ), θ can range from 0 to 2pi, but φ only varies from 0 to pi. Why is that?
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3answers
53 views

Calculate the divergence of the polar coordinate vector field $\partial_\phi$ [closed]

I have to solve this problem: $v=\partial_\phi$ on $M=\mathbb{R}^2\backslash{0}$ where the components of $v$ are in polar coordinates. Calculate the divergence of $v$. Even with the help of ...
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1answer
38 views

Integration of a generic radial function in polar coordinates

I need to perform the following integral $\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$ using polar coordinates. I think the result should depend on some Bessel function, but ...
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1answer
50 views

Plot of $n$ concentric circles at once?

While we plot the Equation of $$(x^2+y^2-1)=0$$ we get: While we plot $$(x^2+y^2-4)=0$$ we get: So What will happen if we plot $$\prod\limits_{i=1}^{i=n} \Big({(x-a)^2+(y-b)^2-i^2}\Big)=0$$ ...
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0answers
42 views

How do I plot a function like this in cylindrical polar coordinates?

Considering that $R = (x^2 + y^2)^{1/2}$, $x = R\cos(\phi)$, $y = R\sin(\phi)$, plot: $$R(\phi) = \left(\dfrac{(\cos\phi)^{1/2} + (\sin\phi)^{1/2}}{ \cos\phi + \sin\phi}\right)^2$$ Have tried to ...
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2answers
82 views

Is Wolfram-Alpha giving me a wrong result?

I have to calculate: $$\nabla^2 \frac{e^{ikr}}{r}$$ which I know to be $\displaystyle -k^2 \frac{e^{ikr}}{r} $ (from a lecture). Doing it by hand: $$ \nabla^2 f(r) = \frac{1}{r^2} ...
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3answers
38 views

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$ The answer should be: $y = \frac{1}{12} x^2 -3$ But how to arrive at the answer? I tried replacing r with $\sqrt{x^2 + y^2}$, then ...
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1answer
30 views

Graphing function with polar coordinate

I am studying polar coordinates and I am not understanding what's the practical method for graphing this relation: $$r = \frac{1}{2} + \sin \theta, \text{for } 0 < \theta < 2\pi$$. I plotted ...
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2answers
71 views

Find the area of the entire region that lies between $r=1+\sin\theta; r=1+\cos\theta$

I have to find the area of the region that lies between the curves $r=1+\sin\theta; r=1+\cos\theta$ . The answer the book gave was $\frac {3\pi}{2}-2\sqrt{2}$ . I tried generating the curve for ...
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1answer
35 views

Circles limits of integration with polar coordinates

Footnote: Got caught up thinking it asked for a 'mutual region' in both functions, while the question actually asked for area of the second function not covered by the first function. I have two ...
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1answer
24 views

Integration with Polar Coordinates

I want to integrate this integral with polar coordinates: $\int \sin x \ dA$ on the region bounded by $ y=x, y=10-x^2, x=0$. So far I've got that $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} ...
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25 views

Why doesn't line fitting seem to work in polar coordinates

I have 2 points, $(r_1, \theta_1)$ and $(r_2, \theta_2)$. They are plotted and I'm trying to find a curve in the form of $r=\theta\beta_1+\beta_2$ to connect both of them. This is basically performing ...