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 Coordinate System Transform?

What is the fastest way (fewest trigonometric and square root operations) to transform between one radius and angle to that of a polar coordinate system with a different centerpoint? I.e. the polar ...
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Please help me with this polar equation? [on hold]

Give a pair of polar coordinates, with $0°<θ <360°$ and $r \ge 0$, for the point whose rectangular coordinates are $(-1,\sqrt{3})$. (Enter you answer as an ordered pair.)
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How to solve the following Laplace boundary value problem?

I tried to solve the following Laplace Boundary Value Problem by separation of variables method. However, I concluded that these kind of boundary conditions cannot be satisfied! (Maybe I concluded ...
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20 views

Solve DOE system with polar coordinates?

I am studying for a exam and one of model questions is solve a DOE system using polar coordinates. I've research and didn't find any reference about this subject. System in question is $$ ...
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2answers
20 views

Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

While playing with some graphics progamming in OpenGL, I've encounterd this problem: Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle ...
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24 views

Applications of Polar coordinates

What applications exist for Polar coordinates (especially over the more better known Cartesian coordinate system)? Both "applied" applications and applications in pure mathematics may be included for ...
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44 views

Show that a polar equation describes a circle

I want to prove that this polar equation: $$r^2 + 2r(\cos(\theta) - 3\sin(\theta)) = 4$$ describes a circle. I tried converting the equation into a cartesian equation and got $$r^2 + 2x - 6y = 4$$ ...
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34 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
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26 views

Arc length of polar curve

I was trying to determine the arc length of the polar curve $r = f(\theta) = a(1 - \cos \theta)$, and it was going well until I got to the definite integral. I know that $f'(\theta) = a \sin \theta$, ...
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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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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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29 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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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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34 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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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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29 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
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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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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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2answers
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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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25 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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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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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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24 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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35 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
63 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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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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46 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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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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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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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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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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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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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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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
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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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155 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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38 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
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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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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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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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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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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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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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41 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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53 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 $$\prod\limits_{i=1}^{i=n} \Big({(x-a)^2+(y-b)^2-i^2}\Big)=0$$ ...