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 to Cartesian: r = 3 + sin(theta/2)

I am asked to convert the following polar function to cartesian: $$r = 3 + sin(\theta/2)$$ I would be able to do it if it weren't for the fraction. I have already tried substituting the identity $...
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48 views

How do I solve for the volume of a hyperboloid using a double integral in polar coordinates?

Here is the problem text, with my attempts at solving it at the bottom: Suppose you are part of a team designing a water tank in the shape of a hyperboloid. The tank is to have a top radius a of 2 ...
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28 views

Understanding unit normal curvilinear vectors to the surface of an octant of a sphere

I'm supposed to test divergence theorem on an octant of a sphere for a given vector field. The triple integral part was easy. However, I'm stuck with the double integral part. Now, there are four ...
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26 views

Integrating dot product in polar coordinates in the vicinity of pole

I am trying to build finite difference scheme for energy equation for compressible gas, in polar coordinates. Right now i am stuck with integrating the equation over the central cell of the ...
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2answers
39 views

Finding the angle of $-2i$.

Given $z = -2i$, I am to find the exponential form. Now, the radius $= 2$. The angle is derived as $\tan^{-1} \frac{y}{x}= \theta $ . $y$ and $x$follow the form $z = x + yi$. Now, given all this, ...
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Changing polar coordinates: Calculating $\iint_R\dfrac{dxdy}{\sqrt{x^2+y^2}}$ where $R=\{(x,y):1\leq x^2+y^2\leq 2, x\leq0, y\geq0\} $

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $d$ and graded ...
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70 views

Area of a quadrilateral on cartesian plane A(0,0), B(4,0), C(3,${\pi \over 8 }$), D(1, ${3\pi\over 8}$)

I'm having trouble on this question. Could anyone find a solution and answer for this? What is the area of quadrilateral ABCD whose vertices have polar coordinates A(0,0), B(4,0), C(3,${\...
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18 views

Express function in terms of polar coordinates and find residues of poles

The function $f(z)$ is given by $$f(z) = (z + \sqrt{3})^{1/2}ln(z-1).$$ The branch of this function is such that $$-\frac{4\pi}{3}<arg(z-1)\le\frac{2\pi}{3} and -\frac{\pi}{2}<arg(z+\sqrt{3})\...
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1answer
28 views

Width of a spiral

I'm attempting to generate a Archimedes spiral (defined as $r = a\theta$) from a given width $w$ and spacing $a$ between 'arms'. I have plotted Cartesian coordinates generated from my workings, but ...
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2answers
81 views

Polar coordinates of an egg-shaped curve

For a simple 3D-rendering project, I need to get the shape of an egg. (actually, a prism with an egg-shaped based). The idea behind it is to explain how a camshaft in an engine is working. From ...
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50 views

Evaluate the double integral by changing to polar coordinates for $x^2+y^2\leq4$

Change the double integral $\int_{D}\int \sqrt{4-x^2-y^2}dxdy$ where D={$(x,y):x^2+y^2\leq4,y\geq0$} by changing to polar coordinates $r, \phi$ So am I right in thinking the limits would be 0 and 4 ...
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25 views

Verify my calculation of the surface integral without divergence theorem

I have $F=xyi-y^2j+zk$ Over surface $z=0$, $s \le1 $, $x^2+y^2 \le s$ My approach to calculate $ \iint F.ds$ was the outward normal is $k$ the dot product of this with F gives z so integral becomes $...
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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 ...
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1answer
63 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 $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
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24 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 ...
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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 ...
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23 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: ...
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1answer
46 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 ...
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37 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 ...
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34 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 ...
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52 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 ...
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33 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}, \frac{7\pi}{2},\frac{11\...
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38 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 = $\int_{\theta_0}^{\theta_1}\sqrt{(r)^2+(dr/d\theta)^2}...
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3answers
67 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 ...
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96 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, \...
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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 ...
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32 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.
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61 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 $u^1,...
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2answers
44 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 ...
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74 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\\ r_{2}&...
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1answer
36 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 $$r = \cos(k*\theta)$$ ...
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44 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 ...
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22 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 ...
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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 ...
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1answer
40 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 ...
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41 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 ...
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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 Polar ...
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1answer
82 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 $$\begin{...
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1answer
75 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: $$\int_{\Bbb S^{N-1}}f(\sigma)\,d\sigma=N\int_{B_N}f\left(\...
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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 ...
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1answer
46 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),\\ y=r\sin(\...
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73 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: $\exp\left({\...
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1answer
28 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 ...
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136 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 $$-re^{\frac{-r^2}{2}}...
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2answers
121 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 $(r,\...
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1answer
27 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 ...
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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 $$g_a=(x+...
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36 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 ...
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35 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 \leq ...
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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 $$\iint (x^2 + y^2)^{-3/2}dA$$