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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2answers
113 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?
1
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1answer
134 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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1answer
19 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
282 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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2answers
45 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
27 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
56 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 ...
2
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1answer
52 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
28 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
54 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 ...
4
votes
2answers
99 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 ...
2
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0answers
39 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
86 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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1answer
134 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
21 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
29 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
85 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
103 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
29 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
33 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 ...
1
vote
1answer
62 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
76 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
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?
3
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2answers
148 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
16 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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0answers
47 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
95 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
49 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
94 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
76 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
50 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 ...
4
votes
1answer
91 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 $$\prod\limits_{i=1}^{i=n} \Big({(x-a)^2+(y-b)^2-i^2}\Big)=0$$ ...
2
votes
2answers
161 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
51 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
36 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 ...
0
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2answers
143 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 ...
2
votes
1answer
159 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
46 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}} ...
0
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0answers
56 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 ...
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1answer
15 views

A Conceptual Polar Curve Question

A polar curve has $r=f(\theta), 0\le \theta \le 2\pi$ has a length of $L$ and is closed by a region that has an area $A$. How can I find the area of a region closed by polar curve say $r=4f(\theta)$ ...
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0answers
38 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
0
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1answer
37 views

How to find the position on a circle that satisfies two constraints?

Say I'm given an point P1 at coordinates $(x_1,y_1)$, and another point $P_2$ at coordinates $(x_2,y_2)$. Then I have a point $P_0$ that needs to be at coordinates $(x,y)$ such that it is a fixed ...
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0answers
25 views

Speed and velocity in x-direction of a point in polar coordinates

I have a list of values that describe the angle (a) of the polar coordinates to a time (t). The radius is 1. I was asked to estimate the speed of the point in the x-direction at time t(3)=2.4° My ...
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2answers
69 views

Calculus - finite integration of $e^{y^3}$ in double integration

i have this problem that bugs me for 3 hours now. I searched the internet and did not find a solution to this specific problem which was asked in our final: $$\int_0^3 \;\int_{\sqrt{x/3}}^r ...
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3answers
48 views

Polar coordinates in double integral of two circles

Use polar coordinates to calculate the integral $\int\int_R(x²+y²)\,dx\,dy$ where $R$ is the region inside $x²-4x+y²=0$ and outside $x²-2x+y²=0$. This is the graphic of the region: ...
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1answer
54 views

Is it a good way to find polar equations of curves?

When I was in my first year of Prepa classes it was not at the program but we have to see it on an example and our maths teacher did it with hypocycloïd and epicycloïd too for fun, well it was very ...
2
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1answer
34 views

Transform the following cartesian equations in polar equations

$$4y^2-20x-25=0$$ The answer given by the textbook is $r=\frac{5}{2(1-\cos \theta)}$ and I couldn't get to this result. I have done $x=r\cos\theta$ and $y=r\sin\theta$ and it leads to ...
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0answers
28 views

Derivative of angular function by cartesian coordinates using Legendre polynomials?

I'm programing some numerical evaluation of force dependent on angle $\phi$ between vector ${\vec a}=(x,y)$ and normalized direction vector ${\hat d}$. To achive maximal performance I wan't to avoid ...
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1answer
34 views

Formula for area in a special occasion in polar coordinates.

I know that the area of a curve given in polar coordinates is $$\int_{\theta_1}^{\theta_2}\frac{r^2}{2}d\theta$$. But what is the area outside one curve and inside another, when one of them is not ...
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0answers
62 views

Unit vector of an angle in plane polar coordinates

I'm struggling to find any information, about how the tip of a unit vector of an angle in plane polar coordinates, $\hat u_{\theta}$, describes a circle - if $P$ is a moving particle - with an angle ...