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 ($\theta$).

learn more… | top users | synonyms

0
votes
0answers
31 views

Conversion of polar equations when you change the position of the origin

I'm working on a physics problem that is described as follows: "I am standing on the ground beside a perfectly flat horizontal turntable, rotating with constant angular velocity w. I lean over and ...
0
votes
1answer
48 views

How do you find the angle of intersection between two given polar curves?

How does one find the angle of intersection between two given polar curves? For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
1
vote
1answer
23 views

Is this a viable way of trisecting an angle in polar coordinates, using an Archimidean spiral?

Say you plot $ r = \theta $ from [0, 2pi] Consider an arbitrary angle $\alpha$ The length $r(\alpha)$ can be trisected using a ruler and compass. Arcs can be "swept out" from the points of ...
0
votes
0answers
10 views

Time for finite beam to cross a point in circular region

I'm trying to find the time a finite width beam takes to cross a point in circular region. Assuming the beam width at distance $r$ from the center is some constant times $r$, $kr$. I have calculated ...
0
votes
0answers
21 views

Calculate flux of vector field

I want to calculate the flux of the vector field $$X(x,y)=y\partial_x-x\partial_y$$ in $\mathbb R^2$ written in polar coordinates ($\partial_x:=\frac{\partial}{\partial x}$ and so on). Step 1: ...
1
vote
1answer
43 views

Transformation matrix in polar coordinates

I'm trying to write a software widget that allows the user to resize the component, so I can write a transformation matrix $\mathbf T_\text{xy}$ that will map $(x,y)$ to a transformed $(x',y')$, that ...
0
votes
1answer
32 views

Transforming ODE into polar form

Let $z=\rho e^{i\phi}$ be a complex number and $\alpha$ some parameter. I determined the following ODE $$ \dot{\rho}e^{i\phi}+i\rho\dot{\phi}e^{i\phi}=\rho e^{i\phi}(\alpha+i-\rho^2). $$ How to get ...
1
vote
2answers
29 views

find coordinates from known angles and length in 3d

Suppose I have 3 vectors with length a,b,and c. They are oriented in 3D space such that the angles between the three vectors are $\alpha$, $\beta$, and $\gamma$ (suppose all less than 90 degrees). If ...
1
vote
4answers
40 views

Polar equation of an ellipse given the origin coordinates and major and minor axis lengths?

I've been trying to create a polar equation that will give me all points on an ellipse with the independent variable being theta and the dependent variable being the radius, but I'm having a great ...
0
votes
2answers
29 views

find the equations of the tangents at the pole.

For the graph with polar equation $r = 1 + sin 3\theta$, find the equations of the tangents at the pole. My attempt, When $r=0$, $\sin3 \theta=-1$ $\theta=\frac{\pi}{2}, ...
0
votes
1answer
34 views

What is the polar coordinate equation for an Archimedean spiral with arc length known relative to theta?

What is the equation for the radius of a polar coordinate for an Archimedean spiral with the arc length known relative to theta? arc length: L = ...
0
votes
3answers
60 views

Prove a function is harmonic

This problem is from Conformal Mapping by Zeev Nehari: If $u(x,y)$ is harmonic and $r=(x^2+y^2)^{1/2}$, prove $u(xr^{-2}, yr^{-2})$ is harmonic. The hint is obvious: "Use polar coordinates." I ...
6
votes
2answers
83 views

Interval for area bounded by $r = 1 + 3 \sin \theta$

I'm trying to calculate the area of the region bounded by one loop of the graph for the equation $$ r = 1 + 3 \sin \theta $$ I first plot the graph as a limaçon with a maximum outer loop at $(4, ...
0
votes
2answers
32 views

Question about integrals in polar coordinates

I've just made on question where is asked the area of a region enclosed by one loop of rose $r=\cos3\theta$ and had one uncertained. In this case, the figure is the following: Suppose if it is ...
0
votes
0answers
23 views

Polar co-ordinates dr/dtheta

How can you visualise what is the curve doing by calculating Dr/dtheta in polar co-ordinates form. Also, what will it mean for Dr/dtheta to be zero? Thank you.
0
votes
1answer
57 views

Contravariant vector example with polar coordinates

My book gives me this definition for contravariant vector: Let an n-tuple of real numbers $a^1,a^2, \dots, a^n$ be associated with a point P of an n-dimensional Riemannian space with coordinates ...
1
vote
2answers
37 views

Converting Polar Equation to Cartesian Equation: general form solution

I'm trying to find the Cartesian equivalent of the general equation $$r=a\cos(q\theta) + c; q\in\mathbb Q, a\gt c \in\mathbb R$$ if it exists. My memory of calc is a bit hazy, and I haven't been able ...
1
vote
0answers
62 views

How to Trace a Real-Life Flower Using Polar Equations?

Here is the flower I'm trying to trace: $\hskip2cm$ How can I trace this flower using polar equations? I currently have the formulas \begin{align} r_{1}&=1.75\sin(10\,\theta + 18) +3\\ ...
0
votes
1answer
34 views

Polar equation for a k-leaf rose: is it possible to define an inner radius?

Is it possible to define a polar equation for a k-leaf rose with an inner radius for a k-leaf rose (as in this image)? I'm familiar with the general equation for a k-leaf rose $$r = \cos(k*\theta)$$ ...
2
votes
1answer
41 views

Convert $y^2 = 4(x + 1)$ to a polar equation

I'm trying to convert the rectangular cartesian equation $$ y^2 = 4(x + 1) $$ to a polar equation. After replacing $y = r \sin \theta$ and $x = r \cos \theta$, I get $$ r^2 \sin^2 \theta = 4(r \cos ...
0
votes
0answers
21 views

Cylindrical coordinate derivative of a vector field.

Considering the following identity transformation in cylindrical coordinate: $$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z} $$ Taking its derivative ...
0
votes
0answers
20 views

How to compute this integrale $\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2} } dx$

I would like to calculate the following integral $$I(a,y)=\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2}} dx, \quad a>0, y\in \mathbb R^3 .$$ Here's what I did: In ...
3
votes
1answer
38 views

Polar coordinates integration

Compute the following integrals over $R$ $f(x,y)\,dx\,dy$ over the area $R$ where: $f(x, y) = x$ and $R$ is given by $0 ≤ r ≤ \cos θ$ and $f(x, y) = x$. I understand polar coordinates is probably ...
0
votes
0answers
36 views

World To Screen Game math

I have a Coordinate system, I have my XYZ, pX,pY,pZ and the other player, eX, eY, eZ and I want the Pitch and YAW First the YAW: I first take VectorX = eX - pX VectorZ = eZ - pZ then I ...
1
vote
0answers
14 views

Hint for setting up this surface integral

\begin{equation} \iint_S z+x^2y \,\, dS \end{equation} Where S is the part of the cylinder $y^2+z^2=1$ that lies between the planes $x=0$ and $x=3$ in the first octant. I tried to convert to ...
2
votes
1answer
76 views

The Plot of a Leaf

Motivation Recently, when I was doing some searches for some syntax in the help pages of my Computer Algebra System (CAS), accidentally, I found this parametric curve in polar coordinates ...
2
votes
1answer
41 views

About polar coordinates in high dimensions

I'm trying to understand a proof in Michel Willem, Functional Analysis -- Fundamentals and Applications, Birkhäuser. The book defines: And then goes on to proving: The first inequality chain ...
1
vote
0answers
27 views

Polar form of generalized superellipse

I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have ...
2
votes
1answer
45 views

Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$ [closed]

$$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$ I think you need to be solved by the transition to polar coordinates: \begin{cases} x=r\cos(\phi),\\ ...
0
votes
1answer
69 views

Simultaneous equations in polar coordinates

I want to find the intersections of pairs of curves in polar coodinates. As an example, I have three circles with different offsets in a plane which you can see here. The offsets are: ...
1
vote
1answer
27 views

Find limits of integral for plane polar co-ordinates question

Use plane polar co-ordinates or otherwise to evaluate the integral $$\int\int_D^\ \frac{x^2-y^2}{x^2+y^2} dA$$ where D is the part of the x,y plane bounded by the parabola $y^2=4(1-x)$ and the ...
8
votes
3answers
132 views

how to integrate this $\int_0^{\infty} r^2 e^{\frac{-r^2}{2}} \, dr$?

What am I doing wrong when integrating this? $$\int_0^{\infty} r^2 e^{\frac{-r^2}{2}} \, dr$$ I used integration by parts and set $u=r^2$ and $dv=e^{\frac{-r^2}{2}}dr$ and I get ...
2
votes
2answers
91 views

The function $f(r,\theta)=(r\cos\theta,r\sin\theta).$

Consider the function $f:\mathbb{R}^{2}\rightarrow\mathbb{R^2}$ given by $$f(r,\theta)=(r\cos\theta,r\sin\theta)$$ I like to show that $f$ is one-to-one in some neighborhood of any non zero point ...
1
vote
1answer
24 views

${\int\int\int}_B dxdydz$ where $B$ is the region delimited by $x²+y²+z² = 4$ and $x²+y²=3z$

Take the following integral over the specified region: ${\int\int\int}_B dxdydz$ where $B$ is the region delimited by $x²+y²+z² = 4$ and $x²+y²=3z$ (i'm answering my own question because I was ...
1
vote
0answers
23 views

Integration problem in polar

How to integrate double integral $$\int_{0}^{\infty}\!\int_{0}^{2\pi}\ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right)g_m \bar{g_n} , d\theta dr$$ where ...
0
votes
0answers
33 views

Ways of representing “half-way” between applying a homeomorphism?

Something I am particularly interested in is finding a potential way to create an animation illustrating smoothly how points in one 2D space map to another. In particular, I would like to show ...
3
votes
0answers
32 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 ...
0
votes
2answers
41 views

Evaluating area D using polar coordinates

Let $D$ be the region in the xy-plane bounded on the left by the line $x=2$ and on the right by the circle $x^2 + y^2 = 16$. Evaluate $$\iint (x^2 + y^2)^{-3/2}dA$$
0
votes
1answer
48 views

Line Integrals - Calculus

I have a problem asking me to find $\int_C \textbf{f} \cdot d\textbf{r}$ where $\textbf{f}$ = $(\sin y,x\cos y)$, and the curve $C$ is any closed circle. I'm struggling with this, so far I have found ...
2
votes
1answer
50 views

How to find the domain of each petal in a Polar graph?

Given the equation $r=4\cos(3\theta)$, how can I find the domain of each petal? Help!
0
votes
1answer
41 views

Non-simultaneous intersections of $r = 4\cos\theta+1$ and $r = 2\cos\theta+1$

$$r = 4\cos\theta+1$$ $$r = 2\cos\theta+1$$ This system has simultaneous solutions at $(1, \frac\pi2)$ and $(1, \frac{3\pi}2)$. But looking at the graph, there are non-simultaneous intersections at ...
2
votes
1answer
67 views

Applied Mathematics: Spherical Polar Coordinates and Newton's Second Law

I've been attempting this question but can't seem to find a solution. Question: A particle of mass $m$ moves under the influence of a force which, in spherical polar coordinates, only acts in the ...
0
votes
1answer
43 views

Find the points on the given curve where the tangent line is horizontal or vertical.

Please help! I don't know how else to do this question. Thank you!!
0
votes
1answer
18 views

Max and minimum value that function $x*e^{x^2+y^2}$ can take on D

So I have to find the maximum and minimum value that the function $~xe^{x^2+y^2}~$ can take on: $$ D = \bigl\{(x,y) :\, 9 \leq x^2 + y^2 \le 16,~ y \geq 0\bigr\} $$ I've converted the Cartesian ...
0
votes
2answers
24 views

Exponential to polar form

I have exponential form $$ je^{-j\pi/2} $$, where $j = \sqrt{-1}$ I want to convert this to polar form $$j(\cos\pi/2 + j \sin \pi/2)$$ is it correct?
0
votes
1answer
32 views

Polar coordinates: what is the area of the region inside the inner loop of $r = \cos (\theta) - \frac12$?

I'm struggling plotting $r = \cos (\theta) - \frac12$. I've done it in Cartesian but I can't quite get in polar coordinates. I know it is supposed to be a loop but how do I get it? Being that I have ...
1
vote
1answer
26 views

Change of variables - Double integrals

I have trouble understanding how the limits work regarding polar coordinates in a double integral. For example, say if I had the equation $$(x-2)^2 + y^2 = 1.$$ This is a circle centred at (2,0) with ...
0
votes
1answer
75 views

Evaluate the double integral by changing to polar coordinates

I experience some difficulty with converting to polar coordinates in integrals. So the question I'm struggling with is Evaluate the double integral $$\int\int x^{6}y\, dA$$ where $D$ is the top ...
1
vote
1answer
12 views

What PC programs or iPad applications are there which allow you to plot cylindrical/spherical polar graphs?

I've been trying to get my head around the use of cylindrical and spherical polars to plot graphs. I feel that the easiest way to do this would be to try plotting some, but I'm struggling to find a ...
0
votes
1answer
14 views

How would you use cylindrical polar coordinates to find the area of a cone (and why does my method not work?

The following question was recently asked in a lecture: Using cylindrical polar coordinates find the area of the curved surface of a cone of height $h$ and radius $a$. My attempt to do this was ...