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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Translating Polar Functions

How do I rewrite a polar function (expressed in a polar coordinate system $r = F(\theta)$ so the entire curve is shifted right or left $h$ units and up or down $k$ units?
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Converting Polar Equation to Cartesian Equation

Heading ##Convert polar equation to Cartesian equation. $$r= \frac{2}{1-\cos\theta}$$ I tried to answer this and this is how I answered it. Please review if it's correct or not. Thank you! :) ...
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Perimeter of Overlap of $r_1 = 3+2\cos(\theta)$ and $r_2 = 8\cos(\theta)$

I'm trying to find the perimeter of the overlap of the 2 curves. I started off by finding the points of intersection of the two graphs, getting $(4, \pi/3)$ and $(4, 5\pi/3)$. Here's my integral ...
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2answers
44 views

how to compute length in polar coordinates?

The line element $\Delta s^2$ is suppose to be an invariant of Euclidean space. In standard coordinates $\Delta s^2=\Delta x^2+\Delta y^2$ while in polar coordinates $\Delta s^2=\Delta r^2+r^2\Delta ...
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1answer
45 views

integral, show identity

let $t>0$. consider the functions $$F(t)=\int_0^{\infty} e^{-tx^2}cos(x^2)\, dx,\quad G(t)=\int_0^{\infty} e^{-tx^2}sin(x^2)\, dx.$$ i want to show that ...
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$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$

Evaluate $$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$$ My attempt: I've tried to use polar coordinates $x=r\cos \theta, \; y = r \sin \theta$: $$\underset{(x,y) \rightarrow ...
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22 views

Segment direction in polar plane

I have the following situation: Base point (green) and segments, for each segment his vertices represented as polar point with theta angle from base point. The problem: For each segment I have his 2 ...
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1answer
57 views

Integral over a solid angle

I've been reading about energy conservation and radiosity from the perspective of computer graphics. The basic idea is simple enough: For all possible incoming light directions $\vec{l}$ and view ...
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1answer
17 views

Curve shape prediction by changing configuration space (Cartesian to polar)

Let us consider the equation $y=3x+2$ which describes a straight line in the 2D Cartesian space. Is it possible to predict the shape of this curve in the polar ($r,\theta$) space? How? I believe that ...
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35 views

Proving distances of polar coordinates

$r\sin\theta=2, r=\frac{2}{1+\sin\theta}, 0<\theta<\pi$ Line l has the first equation, Curve c has the second. Any point on curve C has polar coordinates (a,$\phi$). The foot of the ...
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1answer
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Line integral of conservative field in polar coordinates

I am solving the vector equation: $$\vec \nabla P(r,\phi) = \vec f(r,\phi)$$ where $\vec f$ is conservative, in polar coordinates. Am I allowed to the following? $$\partial_r P= f_r$$ ...
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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 ...
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2answers
47 views

Finding the coordinates of the vertices of an equilateral triangle.

I have an equilateral triangle. I know the orientation of that triangle(that means I know the angle of one of the sides of the triangle with respect to the origin). I know the coordinates of the ...
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1answer
48 views

Gradient of function in spherical coordinates

How do you find the gradient of the function: $$h(r,\theta,\phi) = \frac{1}{r}e^{r^2}$$ I'm not sure what $h(r,\theta,\phi)$ is supposed to output? Is it coordinates? How do you convert this function ...
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2answers
67 views

Why is the graph of $r = a + b\cos \theta$ the same whether a is positive or negative?

So today's lecture was about polar coordinates, and we were taught about the concept up to limacons. I'd like to know why the graph of $r = a + b\cos \theta$ is exactly the same as the graph of $r = ...
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1answer
42 views

double integral over a circular region in polar coordinates

I have a function $f(x,y) = x$, and I want to find the double integral over the circular region $(x-2)^2 + y^2 =1 $ using polar coordinates. Converting the region to polar, we get $r^2 -4cos\theta ...
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How would you find the eccentricity of this conic section?

$4x^2 - 5y^2 - 16x - 50y + 71 = 0$ Thank you!
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Is this a sound demonstration of Euler's identity?

Richard Feynman referred to Euler's Identity, $e^{i\pi} + 1 = 0$ as a "jewel." I'm trying to demonstrate this jewel without recourse to a Taylor series. Given $z = cos\theta + i sin\theta\; |\;|z| = ...
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38 views

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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26 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
31 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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2answers
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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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1answer
64 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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51 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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40 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
78 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
30 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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34 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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1answer
37 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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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
94 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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1answer
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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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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
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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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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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48 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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26 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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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
92 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
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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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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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80 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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25 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
68 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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94 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: $${r\cos{\theta}+r\sin{\theta}} = 3$$ $$r=0, r=3, \theta=0, \theta=\pi/2$$ ...