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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125 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 ...
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59 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 ...
4
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658 views

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: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
3
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35 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 \leq ...
3
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90 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 {\tan(\theta)}a}\sec(...
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100 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
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38 views

Pure differential equation whose solution is a siluroid?

I am trying to find a differential equation for the siluroid that DOES NOT contain explicitly $\theta$, $\sin\theta$, or $\cos\theta$, but only $\rho$, $\dot\rho$, $\ddot\rho$. The siluroid equation ...
3
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481 views

Drum membrane wave equation general solution (non-symmetrical)

I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, r&...
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33 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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50 views

Solving exp integral in closed form?

I am trying to solve the following integrals: 1) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 m^2})} dxdy $ 2) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 n^2})} dxdy $ 3) $\int \int ...
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47 views

Christoffel symbols in polar coordinates calculation

I'm currently studying Riemannian Geometry and I would like to get familiar with the basic concepts. I considered the simple Riemannian manifold $(\mathbb{R}^2, can)$ with its Levi-Civita connection $...
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48 views

What does it mean for a polar coordinate system to have basis vectors?

So I understand that every element of a vector space can be represented uniquely by a linear combination of the basis vectors: $v=\alpha_1v_1+\cdots+\alpha_nv_n$ Then coordinates to those basis ...
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28 views

Polar form of generalized superellipse

I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have ...
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34 views

Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
2
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90 views

Scale factors in cylindrical coordinates - geometrical meaning

I am trying to make sense of the scale factors in cylindrical coordinates and their geometrical meaning. To start with something simpler, begin with Cartesian coordinates: $$h_x=h_y=h_z=1$$ One can ...
2
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27 views

How to calculate the integral of a function defined by polar variables?

Let be $f(r,\theta)$ a function defined over a circle of radius $R$ where $0\leq r\leq R$ and $\theta$ is defined as the angle being $0\leq \theta\leq 2\pi$. My question is how to calculate the double ...
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45 views

Diffusion equation in polar coordinates with non-zero boundary conditions (BC)

I'm trying to solve the diffusion equation in polar coordinates: $$c_t = \frac{D}{r^2}[2r\,c_r + r^2\,c_{rr}] = \frac{D}{r}[2\,c_r + r\,c_{rr}] \tag{1}$$ with the following BC: $$c(0,t)=0, \quad c(...
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38 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 $...
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44 views

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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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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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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116 views

What function has a 3D graph that will look like a spiral into a singularity?

I am trying to draw text spiraling into a black hole, from a more interesting slightly off-orthogonal viewpoint. I think a function that defines a black hole/singularity surface might look something ...
2
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130 views

Integration of figure whose base is a quarter circle not centered at origin using polar coordinates

How do I integrate $$ \int_{1}^{2}\int_0^{\sqrt{2x-x^{2}}}\frac{1}{\sqrt{x^2+y^2}}dydx $$ using polar coordinates? The base is a quarter circle of radius 1 centered at (1,0), so my first instinct ...
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185 views

Polar Coordinates for Multivariate Limits With Three Variables

When working on limits with two variables, $f(x,y)$, I like to convert the problem to polar coordinates a lot of the time, by changing the question from $$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y)$$ to ...
2
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0answers
1k views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= -\...
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2k views

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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15 views

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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20 views

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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31 views

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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58 views

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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56 views

Find imaginary part of complex expression

Given the system of ODEs, $$x'=x^3-3xy^2$$ $$y'=3x^2 y-y^3,$$ it can be shown that the system may be written as $z'=z^3$, where $z=x+iy$. However, I don't seem to get how to show that $\Im m\{\frac{1}{...
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19 views

Area of infinitesimal polar rectangle

In taking a double integral in polar coordinates, I'm learning that we can break up the surface into little polar rectangles with $\Delta r$ and $\Delta \theta$. Therefore, the Riemann sum of a ...
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29 views

Direction angle of Line segment in polar coordinates

I have a line segment given by two points $A$ and $B$, that are $(r_1,\theta_1)$ and $(r_2,\theta_2)$ in Polar coordinates. I know that the direction angle of the line segment is given by: $$\...
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76 views

Polar System with Short Answers, How $U(0, \theta)=\pi$ will be calculated?

I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial r})+\...
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74 views

How to Trace a Real-Life Flower Using Polar Equations?

Here is the flower I'm trying to trace: $\hskip2cm$ How can I trace this flower using polar equations? I currently have the formulas \begin{align} r_{1}&=1.75\sin(10\,\theta + 18) +3\\ r_{2}&...
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14 views

Hint for setting up this surface integral

\begin{equation} \iint_S z+x^2y \,\, dS \end{equation} Where S is the part of the cylinder $y^2+z^2=1$ that lies between the planes $x=0$ and $x=3$ in the first octant. I tried to convert to Polar ...
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23 views

Integration problem in polar

How to integrate double integral $$\int_{0}^{\infty}\!\int_{0}^{2\pi}\ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right)g_m \bar{g_n} , d\theta dr$$ where $$g_a=(x+...
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509 views

How to solve this integral to find the exact length of an equation in the polar plane?

I hope it is only because it's late and I've been studying for a calculus exam for several hours, but I cannot see how to solve this integral. The problem states: Find the exact length of the ...
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37 views

Rewrite the equation of a conic in cartesian coordinates

Consider the equation for a conic in polar co-ordinates $(r,\theta)$ $$r = \frac{k}{1 - e\cos(\theta)} \qquad \qquad (1)$$ in the case where $k > 0$ and $e > 1$. Show that ...
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43 views

Polar coordinates for double integral for $\theta$

To evaluate the integral $$\iint_D \sqrt{x^2+y^2}dA$$ $$D=\{(x,y)\mid0\leq(x-1)^2+y^2\leq1 \}$$ it should be best to change variables into polar coordinates to get $$\int_{\frac{-\pi}{2}}^{\frac{\pi}...
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25 views

Evaluating $\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz$ between balls in $\Bbb R^3$

I am working on the following problem: Evaluate: $$\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz,$$ where $R = \big\{(x, y, z) : 1 \leq x^2 + y^2 + z^2 \leq 2^2 \big\}$ is the ...
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45 views

Volume Generated by the revolution of plane figure about the polar axis, with Boundaries formed by Two Polar Curves

The plane figure bounded by the cardioid $r_1=2α(1+cos\ θ)$ and the parabola $r_2=\frac{2α}{1+cos\ θ}$, rotates around the polar axis. Show that the volume generated is $18πa^3$. So the plane i have ...
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122 views

Calculating X & Y coordinates of a point offset from an ellipse at a given polar angle and intersecting point

I am using the following equations to identify the x and y coordinates of a point on an ellipse at polar angle θ. $x=\pm\cfrac{ab}{\sqrt{b^2+a^2 (\tan^2\theta)}}$ $y=\pm\cfrac{ab}{\sqrt{a^2+b^2 (\...
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24 views

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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30 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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0answers
28 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 ...
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67 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 ...
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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.