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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2
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43 views

How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$

want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$. I did this using polar coordinate and got ...
3
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2answers
27 views

Dynamical System transformation

How can the system $$\frac{dx}{dt}=-y+\epsilon x(x^2+y^2)$$$$\frac{dy}{ dt}=x+\epsilon y(x^2+y^2)$$ be transformed into $$\frac{dr}{dt}=\epsilon r^3$$ $$\frac{d\theta}{dt}=1$$ via polar coordinates? ...
2
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0answers
24 views

Repeatedly interpreting polar coordinates as Cartesian

Start with a Cartesian point $(x,y) \in \mathbb{R}^2$, convert it to polar coordinates $(r,\phi)$ ($\phi$ in radians), and then reinterpret $(r,\phi)$ as $(x,y)$, i.e., set $$(x,y) = (r,\phi) \;.$$ ...
0
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1answer
13 views

Finding orthogonal angles / polar coordinate of an n-dimensional vector

Orthogonal angles are the angles used when converting a vector to polar coordinates. So for vector $(1, 1)$, the orthogonal angle is $45$ degrees. Given a vector $(x_1, x_2, x_3, ..., x_n)$, what is ...
3
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3answers
34 views

Expressing $ r = \cot(\theta) $ as an equation in terms of Cartesian coordinates $ (x,y) $.

I need to show this equation $r = \cot(\theta)$ as $x$,$y$ using the following laws: $x=r\cos(\theta)$, $y=r\sin(\theta)$ $r^2=x^2+y^2$, $\tan(\theta)=\frac{y}{x}$ This is what I've done : $$r = ...
1
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2answers
23 views

How would you convert this particular polar equation to cartesian?

How would you go about converting the polar equation $r^2 = 4cos(2\theta)$ into a cartesian equation in terms of y? I have just started working on polar-cartesian equations, but do not yet have ...
1
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2answers
18 views

Help with finding degrees/radians when converting rectangular to polar coordinates?

I was given the problem "determine two pairs of polar coordinates for (-3,0) when theta is greater than 0 degrees and less than 360 degrees" and I know the radii are 3 and -3. When I use arctan 0/-3, ...
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0answers
43 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? ...
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2answers
25 views

Caclulate X,Y coordinates of point after rotation around another point of given degrees

There are Two Points A and B. The linear distance between the points is R. I have the ...
1
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1answer
15 views

Difference of roots of unity in polar form

I want to write the difference between $n$-th roots of unity in the form $re^{i \theta}.$ It is enough to find the polar form of $1 - \zeta^k$. By thinking geometrically, I can guess the formula $$1 ...
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1answer
11 views

Find the length of a curve specified by a series of polar co-ordinates.

I have a curve defined by a series of polar co-ordinates, $P_a(r_a,\theta_a)$ through $P_b(r_b,\theta_b)$. I would like to determine the length of this curve. Because the points are from ...
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0answers
12 views

Normal form of plane

I propose a simple equation of plane generalized from 2D Normal form of straight line as follows, starting from Straight line Normal form: $$ x\, \cos \alpha + y \sin \alpha = p. \tag{1}$$ ...
3
votes
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. ...
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0answers
63 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
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1answer
37 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 ...
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18 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 ...
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3answers
22 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 ...
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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
56 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 ...
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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
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1answer
38 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
41 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
24 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
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1answer
44 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 ...
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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
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2answers
27 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
292 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
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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
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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
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
25 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
36 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
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2answers
32 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
38 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
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1answer
38 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: ...
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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 ...
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0answers
32 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
30 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
45 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
66 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
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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 ...
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1answer
39 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
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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 - ...
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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
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.