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
42 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
48 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
votes
2answers
50 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
votes
1answer
63 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
2answers
64 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
31 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
31 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 ...
7
votes
3answers
332 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
25 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
36 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
vote
0answers
21 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 ...
3
votes
1answer
53 views

Polar to cartesian equation conversion

I have a polar equation defined as: $r = ae^{θ \tan m}$ where, $a$ and $m$ are constants, $θ$ is the angle between the horizontal axis from the origin $(x_c,y_c)$ to the coordinate. $e$ refers to ...
1
vote
0answers
27 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
48 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
36 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
votes
0answers
87 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
88 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 ...
1
vote
0answers
47 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
43 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
votes
0answers
20 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
votes
0answers
47 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. ...
0
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0answers
38 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
46 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
83 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
55 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
vote
1answer
40 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
47 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
44 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
45 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
votes
2answers
50 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
46 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
vote
1answer
34 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
votes
1answer
29 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
votes
0answers
40 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
24 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
32 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
vote
1answer
42 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
votes
1answer
20 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
46 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
vote
0answers
60 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 ...
0
votes
1answer
138 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 ...
3
votes
2answers
90 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 ...
1
vote
2answers
39 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 ...
0
votes
2answers
41 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 ...
4
votes
0answers
56 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 ...
0
votes
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 = ...
0
votes
0answers
26 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 ...
0
votes
0answers
27 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
votes
1answer
49 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 $\mathbb{R}^2$ ...
0
votes
1answer
46 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 ...