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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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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coordinate geometry passage problems [closed]

Passage $1)$ Let $A( 0, B)$ , $B ( -2 , 0)$ and $C (1 ,1)$ be vertices of a triangle then $Q1)$ Angle $A$ of triangle $ABC$ will be obtuse if $B$ lies in that is the range of $B$ is $a)$ $( -1, ...
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2answers
21 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
42 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
21 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
17 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
44 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
20 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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33 views

TI-83 gets a simple convertion to polar form wrong

I am trying to do the equation $-1-i$ converted to Polar form on the format e^i The answer (according to my own computations) should be $$ \sqrt2e^{i5\pi/4} $$ While the calculator says it is $$ ...
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2answers
60 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
22 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
12 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
41 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
214 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
23 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
48 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
24 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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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
19 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
9 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
56 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
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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
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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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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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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
20 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
47 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
40 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
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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
30 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
69 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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area of the region bounded by two circles

The two circles in polar expressions are: r=sqrt(3)sin(\alpha), and r=3cos(\alpha). I got the answer to be (1/2)integral(from 0 to pi/3) (3cos(\alpha)^2) d(\alpha) + (1/2)integral (from pi/3 to pi/2) ...
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Why aren't polar coordinates global?

Using polar coordinates, why cant we use one coordinate neighbourhood to cover $\mathbb{R}^2$? Can't every point in $\mathbb{R}^2$ can be described by its distance from the origin and its angle from ...
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1answer
96 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
51 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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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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67 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 ...
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1answer
21 views

Error with mixed derivatives during derivation of Laplacian in polar coordinates

It can be shown that, if $f:\mathbb{R}^2\to\mathbb{R}$ is a smooth function and the Cartesian and polar coordinates are related by $$ x = \rho\cos\phi\\ y = \rho\sin\phi $$ that $\partial f/\partial ...
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26 views

How to translate the polar curves up/down and right/left without referring to Cartesian equations?

If I have a polar equation $r(\theta)$, how can I translate it up/down and right/left? We can do this by converting the equations into Cartesian equations and do the translations we want and then ...
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2answers
120 views

Plotting a polar curve

The question is, to generate a polar graph using a graphing utility, and to choose parameter interval so that the complete graph is generated. $$r=\cos\frac{\theta}{5}$$ To find such an interval, we ...
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Finding the intersection of 2 coordinates in spherical coordinate system

Sorry in advance for messing up any math term or being confusing. I have the following data: lat1, lon1, alt1, v1, h1 and ...
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2answers
84 views

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ...
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1answer
29 views

why is theta in a restricted interval? (Polar coordinates)

What is the polar equation of the circle of radius 1 whose centre lies at the cartesian point (1,0)? So I got the correct answer of r=2cos(θ) But then is says theta is in the interval (-π/2)≤θ≤(π/2) ...
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3answers
39 views

Is there any way to express $\theta=c$ as some function of $r$?

I recently found this: Desmos Graphing calculator. I tried to plot the equation $\theta=45$ but it gave me an error: Sorry, you can't graph $\theta$ as a function of anything yet. So I started ...
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3answers
163 views

Is Adobe Acrobat's icon a special function?

It looks like a function in polar coordinates. Is it a special function ?
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2answers
48 views

Intersection of circle and ellipse

I'm looking for the points of intersection of a circle $x^2 + y^2 = r^2$ ($r$ is known, origin is $(0,0)$) and an ellipse $(x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 = 1$ ($a,b,x_0,y_0$ are known). ...
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Solid angle subtended in latitude-longitude maps

I need to scale a latitude-longitude map with the solid-angle each "pixel" subtend. How can I obtain the said solid angle starting from the $\phi$ and $\theta$ angles? Thank you very much
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Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
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1answer
37 views

Area of a Self-intersecting Curve

I was doing some work finding the areas of rose curves. The rose curve is a polar curve given by the equation $$ r(\theta) = \cos{k\theta} $$ When $k$ is even, the area is $\pi/2$, and when $k$ is ...
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Radians : negative and positive values

Recently I have been reading books on DSP where I came across Polar co-ordinates. I understand that on Polar graph (4 quadrants) we have 0,pi/2,pi,3/2pi and 2pi radians as we move from one quadrant to ...