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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29 views

polar system in a plane?

what is a polar system in a plane and how it helps in calculating integrals in certain areas? I'm looking for a good explanation/a fair/ readable source on the matter.
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1answer
74 views

Geometric interpretation of $\frac{dr}{d\theta}$ in Cartesian Coordinates

You'll have to excuse me if this questions is extremely trivial; it's been years since I went back to elementary calculus and I humbly accept that I haven't really gotten deep into polar equations/...
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2answers
155 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
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1answer
32 views

Derivative of a polar coordinate equation

I was trying to plot the polar curve: $r=\cos(2n\theta)$ ($0\leq\theta\leq 2\pi$) and tried differentiating with respect to $\theta$ to get some information about where the petals would be. My ...
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1answer
50 views

Intersection of polar curve with line

I apologize for the horrible title. I came across this in an exam question: You're given $C_1$ as $$r = 1 + cos 2\theta $$ For $\frac{\pi}{2} \leq \theta \leq -\frac{\pi}{2}$. The only symmetry $C_1$ ...
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2answers
42 views

How to convert from cartesian to polar equation

I am trying to convert the equation $y=4/x$ into a polar equation. I have done this work but I am not sure if it is right. I just subsituted $r\sin(\theta)$ for $y$ and $r\cos(\theta)$ for $x$ and ...
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2answers
155 views

Find exact length of polar curve $ r= \frac{6}{1 + \cos \theta}$

I find myself frustrated with the solution of this problem since profit not find it, I'm stuck in the middle of the problem I can not solve the integral, I'm stuck in the solution of the integral is ...
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1answer
287 views

Surface area generated by revolving $r = \sqrt {\cos 2\theta}$

I've been giving a good time trying to solve this problem, I do not find a clear way to solve appreciate your help. \begin{array}{rcl} r& =& \sqrt{\cos 2\theta } \end{array} This Around to ...
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1answer
24 views

Area shape calculating

Can't find the area of the figure bounded by the curve in polar coordinates $$\phi=r\arctan(r), \phi=0, \phi=\frac{\pi}{\sqrt 3}.$$ I tried use the formula $$S=\frac 12\int_{0}^{\frac{\pi}{\sqrt 3}}(r^...
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1answer
67 views

How to prove this ODE is stable but not asymptotically stable?

Consider the ODE in polar coordinates: $$ r'=f(r),\theta ' =1 $$ where $$ f(r)=r\sin (1/r^2), r\neq 0, f(0)=0. $$ show that the origin is stable but not asymptotically stable.
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2answers
63 views

Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 +y^...
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1answer
30 views

Finding limits of integration for double integral

Given a region where the $x$ limits are $-1< x<1$ and $0< y<\sqrt{4-x^2}$, with the option of converting into polar coordinates, i.e. the function $(x,y)$ can be replaced by $r^2$. I'm ...
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1answer
43 views

Evaluating area using an integral in polar coordinates

I am trying to find the area of a circle which is given by the polar parameterization $$r(\phi) = \cos\phi + \sin\phi.$$ I can evaluate it in 2 ways and don't know why I get different answers. First ...
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0answers
54 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 $$\lim\limits_{a\rightarrow0}\int\limits_a^\infty\...
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2answers
61 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? ...
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0answers
32 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 $$(x,y) = (r,\phi) \;.$$ ...
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1answer
105 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 ...
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3answers
99 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 = \...
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2answers
47 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 ...
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2answers
538 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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1answer
96 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
3k 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 ...
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1answer
25 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
16 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 semi-random,...
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2answers
153 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
67 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 ...
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1answer
45 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: http://jwilson.coe.uga.edu/...
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1answer
107 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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3answers
116 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 )}{\sqrt{r^2}}rd\theta{}dr=...
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2answers
86 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 ...
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0answers
33 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 ...
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1answer
177 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
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$Arg(z+1) = \frac{π}{6}$ 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!!!!
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1answer
55 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 ...
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2answers
56 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 ...
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1answer
447 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 ...
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2answers
95 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
33 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)+ \frac{1}{\rho^...
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2answers
46 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 ...
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3answers
358 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
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2answers
31 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 already....
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1answer
38 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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35 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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1answer
82 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 ...
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0answers
31 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
88 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
41 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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1answer
1k 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 $\frac{...
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0answers
64 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 + \...
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3answers
45 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: \begin{align}\cos\left(\theta\right):\sin\left(\theta\right):1:...