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Questions tagged [polar-coordinates]

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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22 views

Differentiating polar functions using complex numbers

I was wondering, given some polar function $r(\theta)$ is it possible to convert it into a complex number in exponential form, then differentiate that and then convert it back and have the appropriate ...
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1answer
36 views

Mass transport equation Cartesian to polar coordinates

Can someone please advise on how to transform the following equation to polar coordinates? $$\frac{\partial \rho(x,t)}{\partial t}=v\frac{\partial \left(\rho(x,t) L(x)\right)}{\partial x}+D\frac{\...
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3answers
54 views

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
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0answers
21 views

Integral of function undefined at one point

Let us consider a plane with polar coordinates. Let us also consider the following integral over any area $A$ on the plane: $$\iint_A f(r,\theta)\ \hat{r}\ dr\ d\theta\ $$ Here the function is $\...
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1answer
38 views

Converting the polar equation $r=12-\sin\theta+2\sin3\theta+2\sin5\theta-\sin7\theta+3\cos2\theta-2\cos4\theta$ to rectangular form

How do I convert the following polar equation to rectangular equation? $$r = 12 - \sin(θ) + 2\sin(3θ) + 2\sin(5θ) - \sin(7θ) +3\cos(2θ) - 2\cos(4θ)$$
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19 views

Cross product with surface in polar coordinates

I'm struggling with a problem concerning cross products. I have a perturbation to a background magnetic field given by $\delta E \times B$ where $B(r(\phi),\phi)= \frac{B_0R_E^3}{r(\phi)^3} + \Delta ...
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1answer
47 views

velocity and acceleration in a parabola

A particle moves with constant speed along a parabola of equation $y^{2}=2px$ with $p=constant$. I want to find its velocity and acceleration vector. Since the velocity magnitude is constant I know ...
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23 views

Finding the Polar Area of two circles intersecting each other

The equations for the two circles are: $r=18\cos{\theta}$ $r=3$ $r_f= 9$ and $r_g=3$ I can see that I need to subtract the $r=3$ circle, however im not sure on how to get the boundaries of ...
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0answers
22 views

Integral of a 2-Form Over a Certain Region of Integration

This is a rather simple problem, but one that I'm struggling with nonetheless. I'm given a 2-Form, $\beta = zdx \wedge dy-x^2dy \wedge dz$ that I need to integrate over the surface S : $z=4-x^2-y^2 \...
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3answers
65 views

How to convert a straight line into polar coordinates?

I'm trying to go through a simple exercise for the Hough transform where I have a simple straight line in the form of $\;y=-x+5\;$ and I want to obtain polar coordinates $\;(\rho,\theta)\;$, I know ...
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General solution to complex number to complex power in complex form

Given the form: $(a+bi)^{(c+di)}$ Does there exist a generalized solution for the principle branch where: $(a+bi)^{(c+di)} = (e+fi)$ I ask this because addition and multiplication (with ...
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1answer
28 views

Normal vector to a polar curve

I'm struggling with working through a proof. Suppose I have a polar curve of the form $r = f(\phi)$. How do I find the $\textbf{normal vector} $ to this curve? The end result I need should be in ...
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2answers
56 views

Convert $(x-3)^2 + y^2 = 49$ to polar form.

Convert $(x-3)^2 + y^2 = 49$ to polar form. Applying $x=rcos(\theta)$ and $y=rsin(\theta)$, I get $x^2 - 6x + 9 + y^2 = 49$ $r^2-6x = 40$ $r^2-6rcos(\theta) = 40$ $r(r-6cos(\theta)) = 40$ This ...
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0answers
24 views

constructing a spiral

picture of spiral]1I am seeking to construct a spiral with the constraints shown in the picture. The problem is that I have not covered polar co-ordinates and I am quite bogged down in understanding ...
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1answer
38 views

Prove that locus of vertex is $(a+b)(x^2+y^2)+2h(x\beta + \alpha y) + (a-b)(x\alpha - y\beta)=0$

The base of a triangle passes through a fixed point $(\alpha ,\beta )$. Let the perpendicular bisectors of the sides be the lines $ax^2+2hxy+by^2=0$. It is to prove that the locus of the vertex is : $$...
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1answer
42 views

Classical mechanics problem in polar coordinates.

A smooth horizontal table has a vertical post fixed to it which has the form of a circular cylinder of radius $\displaystyle a$. A light inextensible string is wound around the base of the post (&...
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1answer
21 views

Parametrization of distance to non-unit circle/sphere with non-centered origin

I attempt to parametrize the distance $z(\theta;r,z_0)$ from the origin $(x,y)=(0,0)$ of my coordinate system to arbitrary points $(x,y)$ on a circle, as a function of the variable $\theta$ (angle), ...
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0answers
58 views

Solving an integral in polar coordinates

I am given the potential: $$\phi (x, y) = \frac{k}{2}(x^2+y^2) + axy$$ Where a is a constant. I have to compute: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \mathrm e^{-\beta\phi}dxdy$$ ...
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1answer
34 views

Simplify the complex expression…

I am asked to simplify the complex expression $$\frac{1}{2}(|{e^{i{\theta}}-1}^2|+|{e^{i{\theta}}+1}|)$$ I have gotten to $$\frac{1}{2}((2-2cos\theta)+(2+2cos\theta))$$ 1. Do I expand to get $$\...
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2answers
29 views

Two polar curves intersect problem .

Suppose we have two curves given by: $$r=20sin2\theta $$ $$r= 20 cos2\theta$$ Now by solving the equations, we get the solution as $\theta = \frac{\pi}{8}$. However, on graphing the equations, I ...
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1answer
35 views

Volume inside sphere bounded by plane in double integrals

I am trying to solve for volume below a plane bounded by a sphere given by $$x^2+y^2+z^2 = 9$$ below a plane z $\in$ [-3,3] using a double integral with polar coordinates. If the plane is given by z=...
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2answers
31 views

Equation for arc with decaying radius

Hoping for some insight into the equation and mechanics of an arc with a decaying radius. Say at 0 degrees / 0 rad, the radius ...
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1answer
20 views

Simplifying addition of polar complex conjugate exponents in the denominator

From Schuam’s Outlines, Digital Signal Processing, Second Edition, 2012, page 44: Book claims that solving this system of equations: $$\left[\begin{matrix}1&1\\e^{i\ \pi/3}&e^{-i\ \pi/3}\\\...
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0answers
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Confusion about polar equation of hyperbola

Suppose I have a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. I know that when choosing the right-hand side focus as the pole and the polar axis has the same direction as x-axis, the equation of the ...
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3answers
48 views

Why is $\sqrt{(\cos^2 \phi + \sin^2\phi)} = 1$?

A rather short question: Why is $$\sqrt{(\cos^2 \phi + \sin^2\phi)} = 1$$ I have seen that in
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0answers
15 views

Gaussian integral and polar change of variable

I would like to compute the Gaussian integral $\int_{-\infty}^{+\infty}e^{-x^2/2} dx$ using a polar substitution. I know the usual proof using Fubini's theorem and the polar substitution of variables....
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0answers
38 views

Integration by substitution to find the arc length of an ellipse in polar form.

I have that $l/r = 1+e.\cos(x)$, for $l = a(1-e^2)$ (constant). The question asks for the mean distance over angle of the planet from the sun, where the planet moves on an elliptical orbit with the ...
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4answers
60 views

Equating Coefficients of Cos and Sin

I've got a nonlinear system \begin{align} x'&=\frac{1}{2}x-y-\frac{1}{2}(x^3+y^2x)\\ y'&=x+\frac{1}{2}y-\frac{1}{2}(y^3+x^2y) \end{align} I am to analyse the system when the system is changed ...
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1answer
24 views

Integration in Polar Coordinates Finding Bounds

In polar coordinates we have to find the area enclosed by a certain function. I am confused on how to find the limits of integration without sketching a graph. I have set the function equal to zero ...
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1answer
59 views

Integral and polar coordinate transformation

I want to calculate $$\int \limits_{\mathbb{R^2}} e^{i \ x \cdot \xi} \varphi(\xi) \ \ d\xi$$ using polar coordinate transformation. Where $x,\xi \in \mathbb{R}^2$ , $\varphi \in \mathcal{S}(\mathbb{...
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2answers
66 views

Evaluating $\iint_D\frac{y}{\sqrt{x^2+y^2}}\ln(x^2+y^2)\,dx\,dy$ using polar transformation

Evaluate $$\iint_D \frac{y}{\sqrt{x^2+y^2}}\ln(x^2+y^2) \, dx \, dy$$, where $$D = \left\{(x,y):\frac{1}{4}\leq x^2+y^2\leq1,x^2+y^2\leq2x,y\geq0\right\}$$ I tried solving it by changing to polar ...
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3answers
44 views

Solving a double integration $ \int_{0}^1\int_0^{\sqrt{2y-y^2}}\ dxdy$ using polar coordinates

Using polar coordinates find the value of the double integral: $$ \int_{0}^1\int_0^{\sqrt{2y-y^2}}\ dxdy. $$ My attempt was as follows : For the limits, $\theta$ will vary from $0$ to $\pi/2$ and $...
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3answers
35 views

Evaluate an integral in polar form

This problem is driving me crazy : $$\int_{0}^{1}\int_0^{\sqrt{2y-y^2}}{dx.dy}$$ Someone please solve this problem in details by sketching the required boundary and how did he calculate it. The ...
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1answer
34 views

Proving arc length polar coordinate formula

I take the definition of arc length of a smooth curve between $x=a$ and $x=b$ to be: $$\int^a_b \sqrt{1+f'(x)^2}dx$$ Then how do I derive the formula for polar coordinates? And please don't use ...
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0answers
28 views

Folland 2.49 Polar Coordinants

I apologize if the picture is hard to read but it seemed easier than typing this all out. I am confused about the first line of theorem 2.49 "Proof. Equation (2.50), when f is a characteristic ..." ...
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0answers
10 views

Multivariable limits, using polar coordinates. Have I done it correcty?

Have I done it correctly? $\lim_{(x,y)\to(-1,1)} \frac{4xy}{x^2+y^2} = \frac{-4}{0} = \lim_{r\to0} \frac{4 \ \cdot \ r\cos\alpha \ \cdot \ r\sin\alpha}{r^2cos^2\alpha\ + \ r^2\sin s^2\alpha} = \lim_{...
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0answers
9 views

Let b, d > 0. The locus of all points p(r,θ) for which the line OP(where 0 is the origin) cuts the line r sin θ = b in Q such that PQ = d is

I came across a question in the challenge section of the textbook which threw me in for a loop and that was this options for it. (a) (r-d) sin θ = b (b) (r-+d) sin θ = b (c) (r-d) cos θ = b (d) (r-+d)...
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2answers
46 views

Why in polar coordinate, $(r,\theta )$ doesn't mean $re_r+\theta e_\theta $?

1) I asked a question here to compute length of set written in polar form. I'm very confuse about something : In cartesian, when I write $(x,y)$ this mean $xe_x+ye_y$. In $\{e_u,e_v\}$, when I right $(...
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0answers
39 views

Integral equation in polar coodinate system

I need an inversion formula with the form $f(r)=\cdots$, from this integral relation: $$g(r)=\frac{1}{2\pi}\int_0^{2\pi}d\theta\,f\left(\sqrt{r^2+r_0^2-2rr_0\cos\theta}\right)$$ where $r_0\geq0$ is a ...
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1answer
32 views

Finding a limit using polar co-ordinates.

Let us suppose that we have to find the following limit: $$\lim_{(x,y)\to (0,0)} f(x,y).$$ Can we solve such a limit using polar co-ordinates? I have seen the following method somewhere on the ...
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1answer
41 views

Integal of ds ==> Integal Integal of r dθ dr [duplicate]

From Page 177 of differential equations demystified (2005): We must evaluate the integral I: (1) $$ I = \int_{\infty}^{0} e^{- s^{2}} $$ observe that: (2) $$I \cdot I = \int_{0}^{\infty} e^{- s^{...
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1answer
62 views

Why exactly limit in polar coordinates isn't sufficient to find the limit in two variables?

I'm currently facing the following problem, my math teacher told us that the following statement is true: $\lim_{x\to(a,b)} f(x)=L \iff \forall \theta \lim_{r\to0} f(a+r\cos\theta, b+r\sin\theta)=L$ ...
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1answer
25 views

How to express the given disk $D$ in polar coordinates.

Given $D=\{(x,y)|0\le x\le2,0\le y\le\sqrt{4-x^2}\}$ Express $D$ in polar coordinates. I have the answer as $D=\{(r,\theta)|0\le\theta\le\dfrac{\pi}{2},0\le r\le2\}$ One more example is $D=\{...
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2answers
40 views

Vector geometry proofs

Prove that all the roots of the equation $$z^n \cos(n \alpha)+z^{n-1} \cos((n-1) \alpha)+ \cdots +z \cos(\alpha)=1$$, where alpha is real, lie outside the circle $|z|=1/2$. How do I approach this ...
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1answer
51 views

Continuity of $\begin{cases}(xy+y^2)/(x^4+y^2)&\text{if }(x,y)\neq(0,0),\\0&\text{if }(x,y)=(0,0)\end{cases}$ at origin using polar coordinates

Study the continuity of $$f(x,y)=\begin{cases}\dfrac{xy+y^2}{x^4+y^2}&\text{if }(x,y)\neq(0,0),\\0&\text{if }(x,y)=(0,0),\end{cases}$$ at $(x,y)=(0,0)$ using polar coordinates. I know that $f(...
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0answers
38 views

Laplace's Equation - Existence and Uniqueness with Robin conditions on an annulus

Let $k$ be a non-zero real number. Consider the problem $$ \nabla^2 \phi = 0 \ \ \ \mbox{for} \ \ \ 1 \leq r \leq 2, \ \ \ \ \alpha\phi + \frac{\partial\phi}{\partial r} = k\cos\theta \ \ \ \mbox{...
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0answers
40 views

Understanding the Jacobian Determinant in polar coordinates

I am trying to derive $$\mathrm dx\ \mathrm dy = r\,\mathrm dr\ \mathrm d\phi.$$ I start with the following ansatz: $$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r\cos\phi \\ r\sin\phi \...
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2answers
39 views

Calc 3 triple integral bounds

I was given a region below a hemisphere of $z=\sqrt{25-x^2-y^2}$ and $z=3$ in the order of $dr\cdot dz \cdot d \theta$ for Cartesian and $dp\cdot d \phi\cdot d \theta$ for spherical. I tried finding ...
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1answer
30 views

Conversion of polar equations

How would you say have some equation in the polar coordinate system as: $$r=3\sin3\theta $$ I know how to find the are of one petal of this shape using polar coordinate integration, but say if I ...
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0answers
34 views

Is this a natural metric on the space of all unoriented lines in a 2D place with positive slope?

Let's parameterize a 2D (unoriented) line by the slope $m$ and intercept $b$, and let the slope be positive. Thus, we are looking at all lines in the first quadrant. I want to know if there's a ...