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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Change $\int_0^\sqrt{2}\int_x^\sqrt{4-x^2}\sin\left(x^2+y^2\right)\:dy\:dx$ to polar coordinates

This is a homework problem, so please do not give more than hints. I must convert \begin{align} \int_0^\sqrt{2}\int_x^\sqrt{4-x^2}\sin\left(x^2+y^2\right)\:dy\:dx\tag{1} \end{align} to polar ...
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0answers
73 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in Polar coordinate $ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial u} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 u} {\partial \theta^2}$ with ...
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2answers
22 views

How do you compute an expression containing complex numbers with large powers?

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$ $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$ So I have these equations with the answers ...
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1answer
34 views

Rotation group of $20$ degrees

Let $R_{20}$ be a rotation counterclockwise by $20$ degrees in the $xy$ plane. What is this group? Then re-write the group in terms of complex numbers of the form $e^{i\phi}$. Is their a special ...
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0answers
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Derivative matrix in polar coordinates

Considering a vector field in two dimensions, $\vec V(x,y)$, I know that the derivative matrix (Jacobian matrix) is given by: $\nabla \vec V(x,y) = \begin {bmatrix} \partial V_x/\partial x ...
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0answers
22 views

Missing equation in coordinate system transformation?

I want to transform a differential equation from polar coordinates $(r,\theta)$ to the following $(u, v, \phi)$ coordinate system: $$ u = r \cos(\theta - \phi) \\ v = r \sin(\theta - \phi) \\ \phi = ...
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2answers
34 views

When do the curves $r=a(1+\sin\theta)$ $r=a(1-\sin\theta)$ intersect?

By converting the equations to $x$- and $y$-components, and setting them equal, I get they intersect at $\theta=0,\pi$, giving the points $(a,0)$ and $(a,\pi)$. But I don't get the point $(0,0)$--how ...
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2answers
31 views

Where am I going Wrong in this Polar Coordinate Conversion?

Solve the following double integral by converting to polar coordinates first: $\int_{0}^{2}\int_{0}^{\sqrt{4-x^2}}(x^2+y^2)^{3/2}dydx$ My attempt at a solution: $\int\int_{R}dydx$(Cartesian) = ...
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0answers
27 views

Find the Area Inside the Smaller Loop of r = 1- 2sin$\theta$

Using the fact that I need the radius to reach zero $2$ times to enclose the loop, I can write $1-2\sin\theta = 0$, which translates to $\sin\theta = \frac{1}{2}$, or $\theta = \frac{\pi}{6}, \theta = ...
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1answer
25 views

Intersecting polar curves r=1+cosθ and r=1-cosθ

The question was asking for the intersection points of $r=1+\cos \theta$ and $r=1-\cos \theta$ with $0≤ \theta ≤2\pi$, but doing: $1+\cos \theta=1-\cos\theta$ 0=2cosθ 0=cosθ θ=$\frac π2$ or ...
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0answers
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Find the arc length of the curve $r=a \cdot \tanh( \frac{\varphi}{4} )$.

How can I find the length of the loop in polar coordinates: $$r=a \cdot \tanh( \frac{\varphi}{4} ) $$ $$0 \leq \varphi \leq \varphi_{0} $$ Use the formula: $$L =\displaystyle\int \sqrt {r^2 + ...
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3answers
40 views

How might I read “$\cos\left(\theta\right):\sin\left(\theta\right):1::x:y:r$”?

In the book I'm reading, A Course in Pure Mathematics, the author writes the following when introducing polar coordinates in section 22: ...
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0answers
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Find point at distance from other point

what is the basic procedure to find a point on a plane if I know the angle and distance from another point... I think in 2d on you can just use polar coordinates?? but on an arbitrary plane in 3D how ...
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0answers
28 views

Taking the Area of Polar Regions

I understand that the formula for taking the area of any polar graph is $\frac{1}{2}\int_A^B r^2\,d\theta$. A to B is usually where r= 0. But I don't understand exactly how the bounds of A to B work. ...
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0answers
16 views

Double Integration - Special Case Has no Singularity; Is it True in General?

Imagine we want to do a double integration in two-dimensions with all variables $x,x',y,y'$ confined to the surface of a circle $x^2+y^2-1=0$: $f(x,y)=\int dx' dy' ...
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0answers
27 views

Showing a hyperbola in polar form approaches two asymptotes

Consider a curve given in polar coordinates by $r(θ)=\frac{1}{1+e\cos\theta}$, where $e≥0$. When $e>1$, show that the curve approaches two asymptotes, find them and sketch the curve. Hint: If the ...
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2answers
42 views

Line equation in polar coordinates does not hold

I am having trouble understanding how the Line equation in polar coordinates holds. If I have 2 points on same line, (1,1) and (3,3) then for the equation $$b=y-mx$$ b=0 and m=1 holds for the two ...
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0answers
64 views

Polar representation of conic sections $r(\theta)=\frac1{1 + e \cos\theta}$

Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$. a) Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant ...
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1answer
47 views

MVT : Integration of harmonic function over boundary of a disc

$u$ is a harmonic function in a domain $\Omega \subset \mathbb{C}$ and $ u : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ suppose $\bar{D{(a,R)}}$$ \subset \Omega $ then To show that $u(a) = \frac{1}{\pi ...
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1answer
38 views

Is polar coordinate right?

Use polar coordinates to evaluate $$\iint_{D}^{} x \ dA$$where D is the region inside the circle, $x^2+(y-1)^2=1$ but outside the circle $x^2+y^2=1$ this what i have got so far: $A = ...
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2answers
40 views

Trouble finding the limits of integration for polar coordinates

Use polar coordinates to evaluate $\iint_D x \, dA $, where D is the region inside the circle $x^2 +(y-1)^2 = 1$ but outside the circle $x^2 +y^2 = 1$ as shown below. Hi all, i'm stuck on finding ...
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3answers
43 views

Trignometric Equation Solution

Question : On the interval $[0,2\Pi]$ there is one point on the curve $r = \Theta - 2cos\Theta$ whose x-coordinate is 2. Find the y-coordinate there. The solution simply states: Solving $(\Theta - ...
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2answers
42 views

Transformation of xy plane to polar coordinates. (What would be the bound of polar coordinate?)

I have a double integral $$\int_0^a \int_0^x (x^2+y^2)^{1/2} \operatorname d y \operatorname d x$$ So, I am double-integrating $r^2$ What would be the region of the polar coordinate..?
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2answers
40 views

What is the area of $[r = \frac{4}{2 - \cos \theta}]$?

It makes an ellipse, but I'm unsure where to go from here.
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1answer
39 views

Second order linear PDE

I have the system with the following partial differential equation. $\\ \frac{\partial u}{\partial t}=\frac{3 a}{4r^ 2}\frac{\partial^ 2 u}{\partial r^ 2}\\$ How can I solve this?
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1answer
18 views

Find the intersection between two lines in a polar notation

I've a polar chart in an application, which displays a curve: I would like to add a functionality when I click on the plot. When I click(at the point M here), I know the orientation and amplitude ...
0
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1answer
19 views

converting a circle's equation not touching axis to polar from Cartesian system for integration

I am having a really hard time figuring out how to convert this circle to polar coordinates, I am to use double integration after converting it. I know that $\theta$ has to be between $0$ and $\frac ...
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0answers
15 views

convert vorticity from cylindrical to cartesian coords

I have a quick question. I am currently working in cylindrical coordinates studying fluid flow and need to calculate the vorticity. I am able to compute the vorticity in cylindrical coordinates ...
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0answers
32 views

Real world Geometry - Finding the Location of a point of intersection between two known locations and one angle

This is a real world problem I'm trying to solve. If I know the locations of two points A and B have a certain latitude, longitude and elevation (GPS coordinates). I also know that a vector from ...
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1answer
22 views

I'm struggling on getting the limits right for this cartesian to polar double integral

I've been working on this homework problem for a while now and I'm just not getting it right. I'd like for some extra eyes to look at this and hint to me where I'm going wrong. The cartesian for is: ...
2
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2answers
20 views

Oblique asymptote polar equation

I have the polar equation $r(\theta)=\frac{1}{\theta-\frac{\pi}{4}}$. I can see that it has an oblique asymptote for $\theta \rightarrow\pi/4+$, but what is it in Cartesian form ?
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1answer
39 views

Name of this type of plot? Does anyone know how to produce it

Does this type of polar plot have a name? Does anyone know how to produce it in octave 3.8.1 which is compatible with matlab? Link to site
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1answer
17 views

When changing variables to polar coordinate and deriving a differential equation for r

The system is $$\dot{x}=-y+ax(x^2+y^2)$$ $$\dot{y}=x+ay(x^2+y^2)$$ and the variables are changed to $x=r\cos\theta$, $y=r\sin\theta$, and when you note $x^2+y^2=r^2$, why can you say ...
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0answers
31 views

Complex integral over sphere in polar coordinates

I have trouble evaluating the integral: $$\int_{B(0,\frac{3R}{|h|})} \frac{1}{(r e^{2i a}-e^{i a})}dr da$$ In fact I just need to estimate it from above in terms of $|h|log (\frac{1}{|h|})$, where ...
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0answers
52 views

Definite Integral of $\sqrt{(x^2+y^2)^k+B}$

I'm trying to evaluate the integral $$ \int_{-1}^1 \sqrt{(x^2+y^2)^k+B} \, \mathrm{d}y $$ WolframAlpha doesn't return a response even for simplified versions of this, but I believe it can be ...
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1answer
83 views

How to calculate shortest distance in polar coordinates when approaching a pole

Given a distance (generally, a large one, say of 850km), a polar coordinate on the earth, and a bearing (with respect to the north pole), I'm using the Haversine formula to calculate a second ...
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2answers
68 views

Converting an integral from polar to cartesian

Question concerning definite integrals. Lets say we have some integral in cartesian coordinates (like the integral of $$ \int^{3/4}_0 \sqrt{1+y^2} \space dy$$ I completely understand how to ...
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2answers
38 views

Need help with polar double integral problem

this is an even problem in my textbook. So this is one of the few places I can check my answer. $$\int _{-1}^1\int ...
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2answers
37 views

How to calculate the argument and its limit for the sequence $z_n=-2+i\frac{(-1)^n}{n^2}$

I am trying to show that the limit of the sequence $$z_n=-2+i\frac{(-1)^n}{n^2}$$ exists, using the polar representation. Note that $\lim_{n\rightarrow \infty }z_n=-2$. $$$$I am finding difficulty in ...
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0answers
50 views

Area under a curve with polar coordinates. Seems to be too simple?

Curve is given by equation: $$r^2 = 2a^2|\cos \phi|$$ I would like to use the formula: $$A = \frac{1}{2}\int_a^b (f(\phi))^2 \, d\phi$$ So, since equation is already squared, i can put the right ...
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1answer
21 views

Polar Represantation of Shifted Disk

How to represent a shifted circle or disk (I mean the center of the circle is not at origin) in polar coordinate? For example I have a circle/disk in z-Domain like this: I thought this: $z = ...
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0answers
16 views

Tangential and normal components of acceleration of a point moving along a curve

If a point is moving along a curve in polar coordinates, is the tangential component of its acceleration given by $r\left(d^2\theta \over dt^2\right)$ and the normal component by $r\left(d\theta \over ...
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0answers
25 views

Single integral of 2 variable function in polar coordinates

I have a function $f(x,y)$ and I want to integrate it in polar coordinates, but only along one variable. What is this integral equal to? $$\int f(x,y) dx \overset{?}{=}\int f(r\cos\theta,r\sin\theta) ...
2
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1answer
37 views

Vanishing of the Riemann tensor

The Riemann tensor in a coordinate basis is $$R^{i}_{\,jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - \Gamma^m_{jk}\Gamma^i_{ml}$$ Consider $\mathbb{R}^2$ ...
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1answer
39 views

Is this function continuous? Polar coordinates “identity”

Is the function $f:\mathbb D\to S^1\times I$ given in polar coordinates by $f(r,θ)=(θ,r)$ (or to be precise: $f(r\cos\theta,r\sin\theta)=((\cos\theta,\sin\theta),r)$) continuous? How would one prove ...
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1answer
20 views

Polar form of equation of line in $xy$-plane

Urgent help requested!! Anything I can do to get an answer faster, in terms of my question?? The question, diagram, and my work are attached. Any help or suggestions or hints are extremely welcome ...
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0answers
45 views

complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
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1answer
32 views

Computing $\iint \limits_R \frac{xy}{x^2 + y^2} \mathrm{d}x \, \mathrm{d}y$ where $R=\{ (x,y) \in \mathbb{R} : y \geq x, 1 \leq x^2 + y^2 \leq 2 \}$

Homework question, so just hints please Sketch the region $$ R=\{ (x,y) \in \mathbb{R} : y \geq x, 1 \leq x^2 + y^2 \leq 2 \} $$ and, by changing to polar coordinates, compute $$ \iint ...
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0answers
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Question about rewriting polar to rectangluar coordinates

I'm asked to rewrite the function $$ f(\alpha,r) = \left\{\begin{aligned} &\frac{1}{2}\sin(2\alpha) &&: r\not=0 \\ &0 &&: r=0 \end{aligned} \right.$$ to rectangluar ...
2
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1answer
28 views

Need help converting this to Polar integral and evaluating it

I have to convert this to polar integral and evaluate it. $$\int _{-1}^0\int _{-\sqrt{1-x^2}}^0\:\frac{2}{1\:+\:\sqrt{x^2\:+\:y^2}}\:dy\:dx$$ I attempted the conversion and ended up with this ...