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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3
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2answers
83 views

show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. ...
1
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0answers
53 views

Shifting to polar coordinates

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2,\ a,b\in\mathbb{R}, a<b$, be a $C^{\infty}([a,b])$ application (smooth). Is it true that we can always find two functions $r,\varphi :[a,b]\to\mathbb{R}$, $r\in ...
2
votes
1answer
34 views

Area under the curve described by θ=ar

I'm interested in finding the area under the curve described by θ=ar, which is a linear curve with slope 'a' in polar coordinates. Here is what the curve looks like: ...
3
votes
1answer
19 views

Differential length of a logarithmic spiral

I am working on a problem that asks to find the magnetic field at the origin of a logarithmic spiral $r = e^\theta$ from $\theta = 0$ to $\theta = 2\pi$, where the angle $\theta$ is measured ...
0
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0answers
13 views

Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
1
vote
3answers
16 views

Converting unit square domain in (x,y) to polar coordinates

I have the following double integral $\int_{0}^{1}\int_{0}^{1}\frac{x}{\sqrt{x^2+y^2}}dxdy$ The integrand is fairly simple: $\frac{x}{\sqrt{x^2+y^2}}dxdy=\frac{rcos(\theta ...
1
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2answers
61 views

Area inside loop of polar equation, unsolvable problem?

Is this problem solvable? "Please find the area inside the first loop of the following equation (using polar coordinates): r = cos$(\theta)$ - sec$(\theta)$." From what I can tell, this function ...
2
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0answers
32 views

Why is e used for polar form of complex numbers? [duplicate]

This is a real basic question. Why is $e$ the base for polar form of complex numbers? In high school maths I learned that e is useful in derivatives etc. And it's conventional to use it for ...
2
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1answer
54 views

Describing polar coordinates in a window

I'm having trouble with the following problem and have no idea what to do. I tried drawing a horizontal and vertical line down the middle of the window but got nowhere. A window is in the shape of a ...
-3
votes
2answers
41 views

$Arg(z+1) = \frac{π}{6}$ and $Arg(z-1) = \frac{2π}{3}$ [closed]

I'm really stuck I need to find z when $$Arg(z+1) = \frac{π}{6}$$ and $$Arg(z-1) = \frac{2π}{3}$$ Please help!!!!
2
votes
1answer
33 views

Cartesian into polar integral.

I have set up an double integral to prove gauss theorem in physics for a gaussian surface of cube of edge $a$ which is as follow. I supposed that mid point of cube is at origin and a charge is placed ...
0
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2answers
40 views

Argument for $(a+bi)^2$

I found out the modulus for $(a+bi)^2$, which is $$a^2+b^2$$ but I am unable to find the argument. I found out that $$\theta = \frac{2ab}{(a-b)(a+b)}$$ I don't know how to simplify further! Please ...
0
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1answer
21 views

Point in a spherical triangle test

Given three latitude/longitude coordinates on a sphere forming a triangle, how do I test if a point p is inside that triangle? I know latitude and longitude implies Earth and Earth is not perfectly ...
2
votes
1answer
40 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
1
vote
0answers
29 views

Why is the problem in polar coordinates in that form ?

We have the initial and boundary value problem $$u_{xx}(x,y)+u_{yy}(x,y)=0 , x^2+y^2<1 \\ u(x,y)=0 \\ u(1, \theta)=\sin{\theta}, 0< \theta< \pi$$ $$U_{\rho \rho}(\rho, \theta)+ ...
0
votes
2answers
24 views

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 ...
8
votes
3answers
279 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
0
votes
2answers
23 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 ...
0
votes
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 ...
1
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0answers
18 views

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
23 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 = ...
1
vote
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 ...
0
votes
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) = ...
0
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0answers
34 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 = ...
3
votes
1answer
35 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
44 views

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 + ...
2
votes
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: ...
0
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0answers
15 views

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 ...
0
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0answers
30 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
17 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' ...
0
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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 ...
0
votes
2answers
44 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 ...
4
votes
0answers
65 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 ...
0
votes
1answer
49 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 ...
1
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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 = ...
2
votes
2answers
42 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 ...
1
vote
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 - ...
1
vote
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..?
0
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2answers
42 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.
3
votes
1answer
40 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?
1
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1answer
19 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 ...
0
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0answers
17 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 ...
0
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0answers
33 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 ...
0
votes
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
votes
2answers
21 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 ?
1
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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
0
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1answer
18 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 ...
1
vote
0answers
34 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 ...
1
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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 ...