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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Integral by polar coordinates

How to calculate the integral $$\int_0^6\int_0^y x\;dx dy$$ using polar coordinates?$$$$I know that $x=R\cos \theta$ and $y=R\sin\theta$ and that the Jacobian is $R$.
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1answer
20 views

Specific cartesian coordinates of an ellipse

I want to do the following: 1.) Ask user for the vertical and horizontal distances of the ellipse 2.) With this information calculate the circumference 3.) Divide the circumference by the closest ...
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2answers
55 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : ...
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2answers
419 views

converting improper double integrals to polar form: what do I do with infinity limits

I need to convert $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}-e^{\frac{x^2+y^2}{5}}dA$$ To polar form. I know $x^2+y^2 = r^2, $ and $dA = rdrd\theta$ But what do I do with the $\infty$ ...
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2answers
38 views

Need help converting $z = \ln(x^2 + y^2)$ to polar

The full question is this: Volume of a solid in any region R is given by: $$\int\!\!\!\int_Rf(x,y)dydx $$ where, $$f(x,y) = z = \ln(x^2+y^2)$$ and, $$x^2+y^2=r^2$$ There for, $$dydx = ...
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0answers
52 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 ...
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0answers
21 views

Double Integral Mistake with Parametric Equation

I'm trying to figure out the mass of an object bounded by $y=0$ and $y=\sqrt{1-x^2}$ the density at a given point is proportional to its distance from the origin; $\rho(x,y) = kxy$. So I set it up ...
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0answers
45 views

Convert geodetic coordinates to cartesian coordinates

I am working on some simulation software that will represent a number of entities in a defined geographic area in the world. The part of the software that I am currently working on is to implement ...
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2answers
32 views

solve polar coordinate integral

Evaluate $$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ using polar coordinates. My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct And also, when I ...
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0answers
47 views

Find arc centerpoint(x,y) with start(x,y) and end(x,y) in a conical helix

Im trying to script drawing of a conical helix in 3D software, and are stuck at the last arc when its not a full 180 degree arc. I know(calculate) the arc startpoint and endpoint, but how do I find ...
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1answer
26 views

Cartesian to Polar coordinates where alpha is real parameter

I want to convert the following equation in Cartesian form to Polar: $$-y(1+\alpha+x)+x(1-x^2-y^2)$$ so $x = r\cos(\theta)$ and $y = r\sin(\theta)$ I can get this far: ...
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1answer
28 views

Is it possible to write all of the functions in terms of polar form?

Is it possible to write all functions in terms of polar form? For example, the equation of the circle with radius one can be written like $r=1$ I'm wondering whether reform the equations of all curves ...
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2answers
66 views

Finding the Equation of a Tangent Line to a Polar Curve

Find the equation of the tangent line to the polar curve: $r=3-3\sin\theta$ at $\theta=\frac{3\pi}{4}$ I have the equation: $$\frac{dy}{dx} ...
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1answer
21 views

Tangent Line of Polar Curve

i start by changing polar coords into x and y and then find the derivatives to get the slope. $$x=(3-3\sin\theta)\cos\theta $$ $$x=3\cos\theta -3\cos\theta \sin\theta $$ and took $x'=(-3\sin\theta ...
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1answer
119 views

Transform second order partial derivatives into polar coordinates

I have the following question: Let $u(x, y)$ be a function with continuous second order partial derivatives. Use the chain rule to transform the expression: $$ x^2\frac{\partial^2u}{\partial ...
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1answer
75 views

Changing to polar form for Green's Theorem

In my text given the integral $\int_{\partial{D}} xy\,dx$, and that $$\int_{\partial{D}} xy\,dx=-\int\int_{D}x\,dx\,dy = - \int\int r\cos \theta\,r\,dr\,d\theta$$ I'm not really understanding the ...
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4answers
61 views

Limit using polar coordinates?

$$\lim_{(x,y) \to (0,0)} \frac{x y^2}{(3x^2 + 4x^2)}$$ How would one calculate above "using polar coordinates"? It was mentioned during class shortly, but we won't be introduced to this until next ...
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1answer
301 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
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2answers
40 views

Divergence of vector in spherical coordinates

How should I calculate the divergence for $$\vec{V}=\frac {\vec{r}}{r^2}$$ Is it possible to convert it from spherical coordinates to cartesian?
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1answer
50 views

Function in Polar Coordinates

Let $f,g:I\to\mathbb{R}$ be two function in $C^{k}(I)$, with the property that $f^2(t)+g^2(t)=1, \ \forall\ t\in I$. Is there a function $\theta: I\to\mathbb{R}$, $\theta\in C^{k}(I)$, such that: ...
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1answer
31 views

What is the cartesian equation of $r = 4 + \frac{\sin(\theta)}{2}?$

This is extremely similar to this question, but as there is no r next to the constant 1, when I multiply everything by r I'm going to end up with: $r^2 = 4r + r\frac{\sin(\theta)}{2}$ And I don't ...
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1answer
21 views

Double integral in cylindrical coordinates

I'm having trouble with a double integral problem in cylindrical coordinates. I'm sure the answer is staring me in the face, but I'm missing something. In the multivariable version of the Community ...
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1answer
57 views

Double Integral to Polar Coordinates

Evaluate $$\int_{0}^{2}\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2}dydx$$ by converting to polar coordinates. I sketch the region which is a half circle from $0$ to $2$ on the $x$-axis and $0$ to $1$ ...
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1answer
24 views

Graphing A Polar Equation

So, I encountered a question r = -|sinø|. So, I thought the polar graph would look like (2) but it actually looks like (3) and I don't understand why. Can someone explain it to me? I've attached a ...
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2answers
173 views

In polar coordinates, can r be negative?

I'm getting different answers for this. Many websites say that when you get a negative value of r, you flip the coordinate 180 degrees across the pole. However my teacher says that you cannot have a ...
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1answer
29 views

Arc length of a polar curve in terms of theta

Is there an equation for such? I know that there is an equation for such in terms of r, but I must calculate the length of $tan(\theta) = 3/5$ (cartesian equivalent: $y = 3/5x$) from r = 0 to 1.457. ...
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2answers
21 views

Points of intersection for two polar equations question

Why is it that when I try to find the points of intersection for $r=2$ and $r=4*\cos(2\theta)$, I only get the $\theta$ where the reference angle is $\pi/6$? There is clearly another solution between ...
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2answers
47 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
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3answers
233 views

Polar coordinates confusion

This seems to be very easy, however I cannot understand, where I am mistaking. Here's the integral to be computed: $$\iint_Dx^2+y^2dydx$$ with $D:=\left\{(x,y)\in \mathbb{R}^2:x \ge0, \; ...
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2answers
28 views

Transforming a cartesian equation to a polar one when it has different x and y denominators?

$$\frac{x^2}{9}+\frac{y^2}{16}=1$$ Needs to be replaced with an equivalent polar equation. I'm sure the identity I'll have to use will be $$x^2+y^2=r^2$$ though other options are: $x=rcos\theta$ ...
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1answer
58 views

How may I use this C loop to solve that integral?

Let C be the curve of polar equation $r = 2cos^2(\theta)$ and D the area bounded by the loop C which is situated in the half-plane $x \ge 0$ region. How may I calculate the D's area and use it to ...
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1answer
100 views

How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$ How may I describe E in cylindrical and spherical coordinates? And how may I ...
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0answers
40 views

Fourier inversion of an infinitely divisible multivariate gamma measure represented in polar form.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
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1answer
22 views

Area of region in polar coordinates

I have to verify a point: I'm supposed to find the area of the region given in polar coordinates $$\sec{\theta}\le r\le 2\cos{\theta}$$ I plotted the curves of $\sec{\theta}$ and $2\cos{\theta}$ ...
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0answers
14 views

Finding Polar Components by Raising/Lowering Indices

This is (I think) a simple question—I'm just making sure everything's correct: I'm given a vector field, $v^a$, which has constant Cartesian components $v^x = 0$ and $v^y = 1$. I'd like to find its ...
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1answer
243 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
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2answers
79 views

for $z,w\in\mathbb{C}$, $\sqrt{zw} = \sqrt{z}\sqrt{w}$?

for $z,w\in\mathbb{C}$, $\sqrt{zw} = \sqrt{z}\sqrt{w}$ I started by writing $z$ and $w$ in polar coordinates, and writing it out, giving another form for the question: $$ \begin{align} y &= ...
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1answer
23 views

Product rule trig

This was given as a solution to a question and I've tried working it out but can never get the same answer. Here $x=rcosϕ$ and $y=rsinϕ$ It's mostly the first 2 lines I don't understand. Wouldn't ...
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0answers
30 views

How to visualize a vector from its components (in spherical coordinates)

Let $$\mathbf{v} = A (1 + \cos \theta) \cos \phi \mathbf{\hat{u}}_{\theta} + A (1 + \cos \theta) \sin \phi \mathbf{\hat{u}}_{\phi}$$ be a vector; $\mathbf{\hat{u}}_{\theta}$ and ...
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31 views

Lagrangian transformed to polar-coordinates

I recently came across the following variational question: Transform the Lagrangian... $ L(t,x,\dot{x}) = \sqrt{(t^2+x^2)(1+\dot{x}^2)} $ ... to polar coordinates so as to show that the extremals ...
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1answer
21 views

Show a polar function's diffrentiability

I need to show that $f(r,\theta)=r\sin(2\theta)\ r>0$ is differentiable at each point in its domain, and also decide whether it's $C^1$ or not. How should I approach this?
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3answers
57 views

Are we really ever plotting in polar coordinates?

Is it true that when we plot in polar coordinates, we are still actually plotting in the x-y coordinate system? Wouldn't plotting in the polar coordinate system really be plotting with $\theta $ and $ ...
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1answer
90 views

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?
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1answer
39 views

Symmetry of polar equations

In your opinion how to show symmetry in polar equations without graphing. i thought of these methods :- converting to cartesian then test . check the period of the function . please help me any ...
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1answer
65 views

Set up triple integral for volume (cylindrical coordinates)

I am given the following question Let $D$ be the region in $\mathbb{R}^3$ that lies within $x^2 + y^2 =4$, underneath the surface $z= 4- x^2 - y^2$ and above the surface $z=- \sqrt{9-x^2 - y^2}$ ...
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3answers
73 views

How to integrate $\frac{y^2-x^2}{(y^2+x^2)^2}$ with respect to $y$?

In dealing with the integration, $$\int\frac{y^2-x^2}{(y^2+x^2)^2}dy$$ I have tried to transform it to polar form, which yields $$\int\frac{\sin^2\theta-\cos^2\theta}{r^2}d(r\cos\theta)$$ But, what ...
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1answer
104 views

Two ways to evaluate $\int (\Delta u) v d\Omega$, two different results

I would like to evaluate the integral $\int (\Delta u) v d\Omega$, where the domain $\Omega$ is a cylinder. On the boundaries, either the normal derivative $\partial_n u$ is zero or $v$ is zero. An ...
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2answers
57 views

seemingly simple question about polar coordinates

I was recently looking at a problem that looked like this: Let $x = rcos(t)$ and $y = rsin(t)$. Assuming that x is held constant, what is $\frac{\partial t}{\partial r}$? Apparently the correct ...
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0answers
20 views

Comparison of Parametric and Polar Equations

Having been introduced to parametric equations, I cannot help but question the similarities between parametrized functions and polar functions. A parametric circle is defined by the following: ...
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2answers
93 views

Evans 's PDE proof

Again, I got stuck. Please help me to understand the following: What is the meaning when you change from integration over the Ball B(x,r) to the surface integration dB(x,s), with another integral ...