# Tagged Questions

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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### Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?

Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name? It's very similar to hippopede, also known as lemniscate of Booth, ...
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### 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 ...
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### 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 ...
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### What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$x=r \cos(t) \\ y=r \sin(t)$$ ...
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### What type of Hopf bifurcation takes place here?

Consider the system: $\dot{x} = \mu x-y-xy^2-x^3$ $\dot{y} = x+\mu y - x^2y-y^3$ I have shown that a Hopf bifurcaiton takes place at the origin $(0,0)$ as a stable spiral becomes an unstable spiral ...
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### Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
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### Polar Coordinates Volume of Solid, Angle for integration?

I'm trying to understand how to find the angle for the integration in polar coordinate form for a solid. Here's an example of what I'm trying to solve: Find the volume of the solid bounded by the ...
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### Cartesian and Polar Coordinates, proving the same number-pair

The question states: Prove that a necessary and sufficient condition for a point to be represented by the same number-pair (a,b) both cartesian and polar coordinates is that it lies on the initial ...
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### Evaluating an integral in polar coordinates with exponential and sec^2

I'm stuck trying to find the closed form expression of an integral. I was able to upper bound it, but if anybody can help me find a way to determine the exact answer, it would be appreciated. Thanks ...
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### Can I switch to polar coordinates if my function has complex poles?

You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta ...
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### Perimeter of Overlap of $r_1 = 3+2\cos(\theta)$ and $r_2 = 8\cos(\theta)$

I'm trying to find the perimeter of the overlap of the 2 curves. I started off by finding the points of intersection of the two graphs, getting $(4, \pi/3)$ and $(4, 5\pi/3)$. Here's my integral setup:...
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### 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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### 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 ...
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 ...