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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Why $ (\cos(\theta) \frac{\partial}{\partial x} + \sin(\theta) \frac{\partial}{\partial y} ) \frac{\partial}{\partial \theta} =0$? [on hold]

Why $ (\cos(\theta) \frac{\partial}{\partial x} + \sin(\theta) \frac{\partial}{\partial y} ) \frac{\partial}{\partial \theta} =0$ ?
1
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1answer
13 views

Finding the horizontal and vertical tangents of a parametric equation.

Find the points at which the polar curve $r=2+2\sin{(\theta)}$ has a horizontal or vertical tangent line. Translate the parametric equation to Cartesian coordinates: $$ r^2=2r+2r\sin{(\theta)} ...
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1answer
18 views

Polar conversions of coordinates and parametric equations

Express the polar coordinates $P\left(6, -\dfrac{\pi}{4} \right)$ in Cartesian coordinates. $\displaystyle x=r\cos{(\theta)} ,\ y=r\sin{(\theta)} \implies x^2+y^2=r^2 \wedge \theta = ...
2
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3answers
21 views

Eliminate the parameter of a

Eliminate the parameter to find a description of the following circles or circular arcs in terms of $x$ and $y$. Give the center and radius, and indicate the positive orientation. ...
1
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1answer
59 views

Find the area using double integral and polar coordinates.

I need to find the area using double integral and polar coordinates. $$y=3-x$$ $$y^2=4x$$ This is what i figured already: $${r\cos{\theta}+r\sin{\theta}} = 3$$ $$r=0, r=3, \theta=0, \theta=\pi/2$$ ...
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0answers
22 views

Converting cartesian to polar integral

I feel like I almost have a grasp on regions of integration, I am a bit frustrated that I haven't fully gotten it but because I feel like I'm almost there. In this particular homework problem I have a ...
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2answers
16 views

Setup region of integration for polar coordinates

I've been working on a homework set for Calc III, right now we're emphasizing double integration and polar integrals. I keep having problems conceptualizing where to actually create my region of ...
1
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1answer
26 views

Equation to place points equidistantly on an Archimedian Spiral using arc-length

I am looking for a way to place points equidistantly along an Archimedes spiral according to arch-length (or an approximation) given the following parameters: Max Radius, Fixed distance between the ...
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1answer
47 views

Is this a valid example of a non-euclidean Sierpinski attractor?

I am learning the basic concepts about the Chaos Game (I did a previous question about the same topic here), the method to create fractals elaborated by professor Michael Barnsley. The basic example ...
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4answers
124 views

Adding two polar vectors

Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?
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2answers
37 views

Volume of a cube in spherical polars

Let us calculate the volume of the cube using spherical coordinates. The cube has side-length $a$, and we will centre it on the origin of the coordinates. Denote elevation angle by $\theta$, and the ...
0
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1answer
12 views

Perpendicular distance from a 3D point to a vector in spherical polar coordinates.

I have a point $(r, \theta, \phi)$ and a direction vector with angles $(\theta', \phi')$. What would be the method to calculate the shortest distance from the point to the vector?
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39 views

Vector Calculus - Polar Co-ords

I am having a lot of difficulty finding an approach to solving the following question: A dyon is a particle with both electric and magnetic charge; in suitable units $$\mathbf{E} = ...
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1answer
55 views

Express -i in polar exponential form

so express $-i$ in form $r\cdot e^{i\cdot \theta}$ $r=1$ is simple enough. As on an argand diagram $-i$ will be at $(0,-1)$ does $\theta = 3\pi/2$ here? or -$\pi/2$ to get it $-\pi < \theta ...
1
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1answer
38 views

Surface area of the circle

I was told to calculate the surface area of the following circle by the integration method (monte carlo) $x^2 + y^2 = 1$ The area of this circle is determined by the following inequalities: $-1 ≤ x ...
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1answer
22 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system (r, θ, φ), θ can range from 0 to 2pi, but φ only varies from 0 to pi. Why is that?
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3answers
51 views

Calculate the divergence of the polar coordinate vector field $\partial_\phi$ [closed]

I have to solve this problem: $v=\partial_\phi$ on $M=\mathbb{R}^2\backslash{0}$ where the components of $v$ are in polar coordinates. Calculate the divergence of $v$. Even with the help of ...
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1answer
36 views

Integration of a generic radial function in polar coordinates

I need to perform the following integral $\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$ using polar coordinates. I think the result should depend on some Bessel function, but ...
4
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1answer
47 views

Plot of $n$ concentric circles at once?

While we plot the Equation of $$(x^2+y^2-1)=0$$ we get: While we plot $$(x^2+y^2-4)=0$$ we get: So What will happen if we plot $$\prod\limits_{i=1}^{i=n} \Big({(x-a)^2+(y-b)^2-i^2}\Big)=0$$ ...
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0answers
41 views

How do I plot a function like this in cylindrical polar coordinates?

Considering that $R = (x^2 + y^2)^{1/2}$, $x = R\cos(\phi)$, $y = R\sin(\phi)$, plot: $$R(\phi) = \left(\dfrac{(\cos\phi)^{1/2} + (\sin\phi)^{1/2}}{ \cos\phi + \sin\phi}\right)^2$$ Have tried to ...
2
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2answers
80 views

Is Wolfram-Alpha giving me a wrong result?

I have to calculate: $$\nabla^2 \frac{e^{ikr}}{r}$$ which I know to be $\displaystyle -k^2 \frac{e^{ikr}}{r} $ (from a lecture). Doing it by hand: $$ \nabla^2 f(r) = \frac{1}{r^2} ...
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3answers
38 views

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$ The answer should be: $y = \frac{1}{12} x^2 -3$ But how to arrive at the answer? I tried replacing r with $\sqrt{x^2 + y^2}$, then ...
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1answer
30 views

Graphing function with polar coordinate

I am studying polar coordinates and I am not understanding what's the practical method for graphing this relation: $$r = \frac{1}{2} + \sin \theta, \text{for } 0 < \theta < 2\pi$$. I plotted ...
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2answers
63 views

Find the area of the entire region that lies between $r=1+\sin\theta; r=1+\cos\theta$

I have to find the area of the region that lies between the curves $r=1+\sin\theta; r=1+\cos\theta$ . The answer the book gave was $\frac {3\pi}{2}-2\sqrt{2}$ . I tried generating the curve for ...
2
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1answer
32 views

Circles limits of integration with polar coordinates

Footnote: Got caught up thinking it asked for a 'mutual region' in both functions, while the question actually asked for area of the second function not covered by the first function. I have two ...
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1answer
22 views

Integration with Polar Coordinates

I want to integrate this integral with polar coordinates: $\int \sin x \ dA$ on the region bounded by $ y=x, y=10-x^2, x=0$. So far I've got that $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} ...
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0answers
25 views

Why doesn't line fitting seem to work in polar coordinates

I have 2 points, $(r_1, \theta_1)$ and $(r_2, \theta_2)$. They are plotted and I'm trying to find a curve in the form of $r=\theta\beta_1+\beta_2$ to connect both of them. This is basically performing ...
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1answer
13 views

A Conceptual Polar Curve Question

A polar curve has $r=f(\theta), 0\le \theta \le 2\pi$ has a length of $L$ and is closed by a region that has an area $A$. How can I find the area of a region closed by polar curve say $r=4f(\theta)$ ...
2
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0answers
34 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
0
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1answer
27 views

How to find the position on a circle that satisfies two constraints?

Say I'm given an point P1 at coordinates $(x_1,y_1)$, and another point $P_2$ at coordinates $(x_2,y_2)$. Then I have a point $P_0$ that needs to be at coordinates $(x,y)$ such that it is a fixed ...
0
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0answers
14 views

Speed and velocity in x-direction of a point in polar coordinates

I have a list of values that describe the angle (a) of the polar coordinates to a time (t). The radius is 1. I was asked to estimate the speed of the point in the x-direction at time t(3)=2.4° My ...
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2answers
68 views

Calculus - finite integration of $e^{y^3}$ in double integration

i have this problem that bugs me for 3 hours now. I searched the internet and did not find a solution to this specific problem which was asked in our final: $$\int_0^3 \;\int_{\sqrt{x/3}}^r ...
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3answers
34 views

Polar coordinates in double integral of two circles

Use polar coordinates to calculate the integral $\int\int_R(x²+y²)\,dx\,dy$ where $R$ is the region inside $x²-4x+y²=0$ and outside $x²-2x+y²=0$. This is the graphic of the region: ...
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1answer
43 views

Is it a good way to find polar equations of curves?

When I was in my first year of Prepa classes it was not at the program but we have to see it on an example and our maths teacher did it with hypocycloïd and epicycloïd too for fun, well it was very ...
2
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1answer
29 views

Transform the following cartesian equations in polar equations

$$4y^2-20x-25=0$$ The answer given by the textbook is $r=\frac{5}{2(1-\cos \theta)}$ and I couldn't get to this result. I have done $x=r\cos\theta$ and $y=r\sin\theta$ and it leads to ...
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0answers
8 views

Derivative of angular function by cartesian coordinates using Legendre polynomials?

I'm programing some numerical evaluation of force dependent on angle $\phi$ between vector ${\vec a}=(x,y)$ and normalized direction vector ${\hat d}$. To achive maximal performance I wan't to avoid ...
1
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1answer
28 views

Formula for area in a special occasion in polar coordinates.

I know that the area of a curve given in polar coordinates is $$\int_{\theta_1}^{\theta_2}\frac{r^2}{2}d\theta$$. But what is the area outside one curve and inside another, when one of them is not ...
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0answers
28 views

Unit vector of an angle in plane polar coordinates

I'm struggling to find any information, about how the tip of a unit vector of an angle in plane polar coordinates, $\hat u_{\theta}$, describes a circle - if $P$ is a moving particle - with an angle ...
2
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2answers
51 views

converting a differential equation to polar coordinates

I have the following family of autonomous systems, I'm having trouble with part b): $$x'=x(1-\sqrt {x^2+y^2})-y-\epsilon y$$ $$y'=y(1-\sqrt {x^2+y^2})+x+\epsilon(x+x^2+y^2)$$ a) Convert the system ...
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1answer
44 views

Calculate : $\int_{} \int_ {} e^{-(x^2+y^2)}dA$ While $R$ is the area in $x^2+y^2=1$

Hello I need to calculate this $$\iint_R e^{-(x^2+y^2)}d\mu$$ Where $R$ is the unit disk, given by $\left\{(x,y):x^2+y^2\leq 1\right\}.$ What I did : $$\int_{-1}^1 ...
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3answers
47 views

Calculate : $\int_{-\pi/2}^{\pi/2}\int_0^{asin\theta} r^2 drd\theta$

I need help with the following integral : $$\int_{-\pi/2}^{\pi/2}\int_0^{asin\theta} r^2 drd\theta$$ What I did : $$\int_{-\pi/2}^{\pi/2}\int_0^{asin\theta} r^2 ...
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2answers
47 views

Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
3
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0answers
36 views

Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ...
0
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1answer
20 views

formula for the arclength of a curve, given as polar coordinates

Let $\gamma: [a, b] \to \mathbb{C}$, $\gamma(t) = r(t)e^{i\phi(t)}$ be a continuously differentiable curve, given in it's polar coordinates, where $r, \phi: [a, b] \to \mathbb{R}$ are continuously ...
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3answers
29 views

Limit of function of 2 variables - can I use polar coordinates?

I need to solve a limit of a f(x,y) (as a part of bigger task), but I'm bad at math. So basically here's this limit: $$\lim_{x,y\to(0,0)} \frac{y^4}{(x^2+2y^2)\sqrt{x^2+y^2}}$$ I tried to use other ...
2
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1answer
27 views

Give a formula for the volume of the solid under a surface $z=xy$ and a triangle?

Given is the solid with unit density lying under the surface $z = xy$ and above the triangle in the $xy$-plane with vertices $(0, 1, 0)$, $(1, 1, 0)$ and $(0, 2, 0)$. Give a formula for the ...
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2answers
60 views

Polar form to cartesian

Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$ I know this has to be converted ...
6
votes
3answers
91 views

What is $\dfrac{dr}{d\theta}$?

Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$ I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations ...
5
votes
1answer
66 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
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1answer
39 views

Calculating an integral using polar coordinates.

I want to calculate the volume enclosed by $z^2 = 1+x^2+y^2$ and the plane $z=2$. When $z = 2$, $x^2+y^2 = 3 \rightarrow r = \sqrt{3}$ So I have set up the integral in polar coordinates as: ...