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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33
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4answers
4k views

Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$

It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why? I got: $2\,\mathrm dy\,\mathrm dx = ...
27
votes
2answers
6k views

Cannabis Equation

How can an equation for the following curve be derived? $$r=(1+0.9 \cos(8 \theta)) (1+0.1 \cos(24 \theta)) (0.9+0.1 \cos(200 \theta)) (1+\sin(\theta))$$ (From WolframAlpha)
27
votes
3answers
2k views

How to force prime numbers into a line?

Inspired by an article on Prime Spiral and Hough transform I tried to analyze patterns created by plotting numbers on spiral (Archimedean?). $$x = \cos( angle ) * radius$$ $$y = \sin( angle ) * ...
20
votes
5answers
3k views

Limit is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of ...
14
votes
4answers
2k views

What are the polar coordinates of the origin?

In polar coordinates, the origin has $r = 0$, but $\theta$ is not unique. what sort of problems does this create, and how can I resolve them? For example, suppose an ant is wandering around a plane. ...
11
votes
4answers
5k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
10
votes
2answers
4k views

Plotting in the Complex Plane

I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$ What is the difference plotting this function in the complex plane or real plane?
9
votes
3answers
219 views

Is Adobe Acrobat's icon a special function?

It looks like a function in polar coordinates. Is it a special function ?
8
votes
3answers
789 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
8
votes
1answer
23k views

Ellipse in polar coordinates

I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - Where $2a$ is the total height of the ellipse and $2b$ ...
7
votes
3answers
333 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
7
votes
3answers
112 views

Area and Polar Coordinates

Would anyone be able to help me with this problem? I think I know the area formula in polar coordinates that should be used: the antiderivative of ((1/2)r^2 dtheta) from alpha to beta but I'm not ...
6
votes
3answers
92 views

What is $\dfrac{dr}{d\theta}$?

Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$ I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations ...
6
votes
4answers
4k views

How to deduce the area of sphere in polar coordinates?

$A = r \int_{0}^{\pi}\int_{0}^{2\pi} e^{i (\alpha+\theta)} d\alpha d\theta = r \int_{0}^{\pi} [\frac{-i}{\alpha+\theta} e^{i(\alpha + \theta)}]_{0}^{2\pi} d\theta = ... ...
6
votes
1answer
216 views

Meaning of Rays in Polar Plot of Prime Numbers

I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, ...
6
votes
2answers
368 views

Laplacian of a Function depending on r in Polar Coordinates

From a bank of exams: Let $u(x,y) = f(r)$ be a smooth function in the plane that depends only on $r = \sqrt{x^2 + y^2}$. Compute $\Delta u = u_{xx} + u_{yy}$ in terms of $f$ and its ...
6
votes
1answer
3k views

polar coordinates and derivatives

Using the standard notation $(x,y)$ for cartesian coordinates, and $(r, \theta)$ for polar coordinates, it is true that $$ x = r \cos \theta$$ and so we can infer that $$ \frac{\partial x}{\partial ...
6
votes
2answers
2k views

Finding a point on Archimedean Spiral by its path length

I've been curious about Archimedean Spirals and their relations to Sacks Spirals and prime numbers. I would like to draw some visualizations of the points with a given distance from the center, ...
6
votes
2answers
972 views

Integration of radial functions?

Let $f(|x|)$ be a integrable radial function in $\mathbb{R}^n$ ($|\cdot|$ denotes the euclidean norm as in convention). The following identity is used to simplify computations ...
6
votes
2answers
11k views

Find the area of the region inside: $r= 6\sin(\theta)$ but outside of $r = 1$

How do we find the area of the region inside $r = 6 \sin(\theta)$, but outside $r = 1$? So, here's my work thus far: First off, we know: $r^2 = x^2 + y^2$ and $\mathrm{sin}(\theta) = y/r$ ...
6
votes
2answers
47 views

Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
6
votes
1answer
290 views

Evaluate the integral by converting to polar coordinate

$$ \int^{\pi/2}_{\pi/4} \int^{\sqrt{2-y^2}}_y 3(x-y) dx dy$$ I attempted the following: $$ \int_{\pi/4}^{\pi/2} \int_{0}^{1} 3r^2 (\cos\theta - \sin\theta) dr d\theta $$ which is wrong apparently. ...
6
votes
3answers
310 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...
5
votes
6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
5
votes
5answers
156 views

Points on $(x^2 + y^2)^2 = 2x^2 - 2y^2$ with slope of $1$

Let the curve in the plane defined by the equation: $(x^2 + y^2)^2 = 2x^2 - 2y^2$ How can i graph the curve in the plane and determine the points of the curve where $\frac{dy}{dx} = 1$. My work: ...
5
votes
4answers
181 views

Find the maximum value of $r$ when $r=\cos\alpha \sin2\alpha$

Find the maximum value of $r$ when $$r=\cos\alpha \sin2\alpha$$ $$\frac{\rm dr}{\rm d\alpha}=(2\cos2\alpha )(\cos\alpha)-(\sin2\alpha)(\sin\alpha)=0 \tag {at maximum}$$ How do I now find alpha? ...
5
votes
4answers
145 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?
5
votes
2answers
129 views

Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?

I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are ...
5
votes
1answer
2k views

Area Bounded by Polar Curves

I am answering sample exams for my Calculus class and my attention was caught by the following item. Set-up the definite integral or sum of definite integrals equal to the area of the region above ...
5
votes
2answers
2k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
5
votes
1answer
76 views

Changing operator to polar coordinates

Let $$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ be the Laplace operator on the $(x,y)$-plane. Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$. ...
5
votes
2answers
800 views

How do I write the 2D Dirac delta in a manifestly rotationally invariant form?

Consider the following integral over a 2D plane, $$\iint \mathrm{d}^2\mathbf{k}\ e^{i\mathbf{k}\cdot\mathbf{r}} = 4\pi^2\delta^2(\mathbf{r})$$ This is a Fourier transform of a distribution which is ...
5
votes
1answer
75 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
5
votes
1answer
64 views

Introducing $\mathrm π$ and polar coordinates in real analysis

From time to time, I think about how material from introductory courses like real analysis or linear algebra can be structured in a way I would have liked to see in my freshman days. So recently, I ...
5
votes
0answers
56 views

Area of two polar regions

I'm trying to find the region inside r=sinθ and outside r=1+cosθ. My issue is my limits of integration. I get an intersection at $\frac π2$ and one at the pole. What are my limits for the integral? ...
5
votes
1answer
65 views

Polar Coordinates — Equation of a line

Hey, does anyone know how to tackle this question? I've tried using the formula r=d*sec(theta-alpha)but I'm not sure what each of the variables are equal to. If anyone can offer any help, it would ...
4
votes
3answers
3k views

Writing Polar Equations In Parametric Form

For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
4
votes
2answers
1k views

Finding the volume of water in a swimming pool

Reposting from stackoverflow :) - Told to go here :) So I've got a issue with a math assignment, and I were hoping someone could at least point me to what I should be doing, because currently I'm ...
4
votes
1answer
1k views

Heat equation in polar co-ordinates

I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge $r=a$ of a circular plate is kept at temperature $f(\theta)$. The plate is insulted so that there is ...
4
votes
3answers
260 views

Transform to polar the following integral $\int_0^6\int_0^y x \, dx \, dy$

I need to transform this integral $\int_0^6\int_0^y x \, dx \, dy$ to polar and then find its value. I'm stuck finding the r-limits of integration.
4
votes
4answers
476 views

How can the trefoil knot be expressed in polar coordinates?

From Wikipedia, the parametric equations for a trefoil knot are \begin{align*} x(t) &= \sin t + 2\sin 2t \\ y(t) &= \cos t - 2\cos 2t \\ z(t) &= -\sin 3t. \end{align*} I am only ...
4
votes
1answer
1k views

Jacobian for a Cartesian to Polar-Coordinate Transformation

I have a simple doubt about the Jacobian and substitutions of the variables in the integral. suppose I have substituted $x=r \cos\theta$ and $y=r \sin\theta$ in an integral to go from cartesian to ...
4
votes
3answers
576 views

Trying to understand the meaning of symmetry

The picture below is the solution to the following problem as presented in my book: Find the area of the region that lies inside both curves $$r = 8 + \cos \theta \\r = 8 − \cos θ$$ According to ...
4
votes
2answers
474 views

Kepler's First Law in 3D

Kepler's First Law in 2D polar is $$ r = \frac{p}{1 + \varepsilon\cos(\nu)}. $$ How can this be written to consider ellipses in ...
4
votes
3answers
1k views

Need hint: Prove that $r = a (\sin t) + b (\cos t)$ is a circle, where $ab \neq 0$

"Show that the polar equation $r = a \sin(t) + b \cos(t)$, where $ab \neq 0$, represents a circle, and find its center and radius." I don't need the answer - I just need a hint to nudge me on. I've ...
4
votes
2answers
402 views

question about continuity: using polar coordinates

Given a function $f\colon\mathbb R^2\rightarrow \mathbb R$ I want to study continuity. So I know the $\varepsilon-\delta$ and sequence criterion. Now we had polar coordinates in lectures: set ...
4
votes
2answers
12k views

Why is $dy dx = r dr d \theta$ [duplicate]

Possible Duplicate: Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$ I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we ...
4
votes
1answer
51 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 $$\prod\limits_{i=1}^{i=n} \Big({(x-a)^2+(y-b)^2-i^2}\Big)=0$$ ...
4
votes
2answers
55 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : ...
4
votes
2answers
66 views

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and ...