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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Calculating volume of a subset of $\mathbb{R}^n$

Suppose $A\subset \mathbb{R}^n$ is a compact, convex and centrally symmetric set such that $(x_1,\ldots,x_n)\in A$ if $$ |x_1|+\ldots+|x_r|+2\left(\sqrt{x_{r+1}^2 + x_{r+2}^2} + \ldots + \sqrt{x_{n-1}^...
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Choice of the limits for multivariable integral

Let $A \subseteq \mathbb{R}^2$ a limited set bordered through $x=0, x=1, y=-1+x, y=1-x^2$. Rotate A around the y-axis and define this set with $B$. Calculate the integral $$\int_B y\,\mathrm{d}x\...
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Cylinder defined on 3d coordinate plane

This is the first time, I have seen a problem like this: I feel as though if i knew where to start i would be able to do this problem easily. In other words, question 1-4 make sence to me and i ...
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20 views

The direction of the angle in polar coordinate system? [on hold]

Polar coordinate is $(r,\theta)$.In which direction $\theta$ will be counted;clockwise or anti clockwise?
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Diagram of a multivaribale function

I have to draw the diagram of the function: $$(x^2+y^2)^{\frac{3}{2}}=x^2-y^2$$ I transformed it with polar coordinates to: $$r=\cos^2(\varphi)-\sin^2(\varphi)$$ with $r \ge 0$ and $\cos^2 \ge sin^2$....
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Equation of a circle in polar coordinates under a linear transformation

Let's say we translate a circle with origin $(0,0)$ on the x axis by some constant $c$. What would the new equation of the circle be in polar coordinates? I have tried subbing in the equation of the ...
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2answers
40 views

Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
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26 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
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2answers
26 views

Area between two polar curves method

The question is not too hard. I sketched them and they were correct which was not too bad. I then did the second part by finding the intersection points between the two curves which are $\frac{\pi}{...
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1answer
144 views

Laplace Challenge in One Examples, Is there any help?

this question is taken from 2014 exam on CE Entrance Exam, Question $32$ on the end of page $6$. Consider the Laplace equation of following polar coordination, $$\frac{1}{r}\frac{\partial}{\partial ...
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61 views

Calculus 3 Integrals

Find the volume of the solid bounded by $z=4x^2+4y^2, z=0, x^2+y^2=1$ and $x^2+y^2=2$. What I know: I know that when I draw the graph I will get two paraboloids giving me a radius of $1$ and $2$,$\...
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36 views

Triple integral using Cylindrical Coordinates

Evaluate the iterated integral $x= 0$ to $x=1$, $y= -\sqrt{1-x^2}$ to $y=\sqrt{1-x^2}$ and $z= 0$ to $z=2-x^2-y^2$ and $f(x,y,z)= \sqrt{x^2+y^2}$ What I know: I know that I have to use cylindrical ...
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Triple integral of $e^{{(x^2+y^2+z^2)}^\frac32}$

I'm trying to find the result of $$ \iiint_{R} e^{(x^2+y^2+z^2)^{3/2}} \,dz\,dy\,dx $$ with $$ R = \left\lbrace (x,y,z) \in \mathbb{R}^3 \mid -1 \le x \le 1, 0\le y \le \sqrt{1-x^2}, 0\le z \...
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1answer
23 views

Why Circle is traced counterclockwise and ellipse is traced clock wise?

In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise. "Which was this one here. And first we noted that this does parameterize, as we ...
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Area of the graph $z=\sqrt{x^2+y^2}$ over the ring $r^2 <(x^2+y^2)<4$ with $0<r<2$

Could someone please help me calculate the surface area of the graph $z=\sqrt{x^2+y^2}$ over the ring $r^2 <(x^2+y^2)<4$ with $0<r<2$ Thanks!
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Help writing a parametric equation from this complex polar one

A particle is moving along the curve $r=4-2\sin(\theta)$ at the moment when $\theta = t^2$. I need to write a x(t) and y(t) function that will model the particle behavior with its x position and y ...
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Difference of Entropy of two-dimensional Gaussians

I encountered a putative contradiction. Assume we have two 2-dim. Gaussian variables $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ with all components being independent, normal distributed variables: $x_1,...
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21 views

Is there any way that a polar coordinate function can be shifted inward or outward without changing the derivative where it exists?

question Let f(x) be a polar coordinate function defined from 0 to pi Is there a translation that pulls f outward (like in the sense f(x) + c does) without altering the polar coordinate derivative? ...
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Is there a way to shift a portion of a polar coordinate graph without disrupting its shape?

A polar coordinate function f(x) can be rotated around the axis by h with the shift f(x - h). However, this rotates the graph. Is there a way to shift a polar coordinate function by x and y ...
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33 views

Calculate the area bounded by the polar curve $ r(\theta)=2+\sin(\theta)$

How can I calculate the area bounded by the following curve: $$ r(\theta)=2+\sin(\theta) $$
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Converting a higher order operator to polar coordinates?

The operator: $$\frac{\partial^4}{\partial x^4}+2\frac{\partial^4}{\partial x^2 \, \partial y^2}+\frac{\partial^4}{\partial y^4}$$ ... appears in the equation of motion of an oscillating rigid ...
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21 views

polar cordinates, finding teta

the integral,and a picture of the graph of D in the left corner you have the integral and below it the range D. the right side has the ranges i found. Hi im having a hard time understanding ...
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Finding flux using polar coordinates.

I solved exercise using polar coordinates. I thought that $x=rcos(\alpha)$ and $y=rsin(\alpha)$. In my exercise given. Given vector field and I solved it using divergence theorem. Result is: $\iiint$...
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66 views

Find the area of the region that is bounded by $r = \cos(\theta)$ from $0$ to $\pi/6$

Find the area of the region that is bounded by $r = \cos(\theta)$ from $0$ to $\pi/6$. I'm pretty confident that I know what I am doing for this problem but I just want to make sure I have it down. ...
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2answers
39 views

Find the area of the region that is enclosed by:

$r = 2\cos(\theta)$ and $ r = 1$ I went ahead and tried it and my answer was just $2\pi$. I was wondering if someone could check if I got it right, and if I didn't, tell me what I did wrong? ...
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26 views

Eliminate the parameter

Given the parametric equations: $x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$ Eliminate the parameter. I am completely lost. Please help.
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3answers
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Find polar coordinates $(r, \theta)$ of the point, where $r > 0$ and $0 \leq \theta < 2\pi$

Given these Cartesian coordinates: $(2,-3)$ This is my fourth problem of this type, I solved the other 3, but this one has weird numbers and I don't know what to do. $\tan\theta = -\frac{3}{2}$ ...
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line integral problem, stuck in the curve

Problem: A force field is given in polar coordinates by the equation $$F(r,\theta)= (-4\sin (\theta), 4\sin (\theta)).$$ Compute the work done in moving a particle from the point $(1,0)$ to the ...
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Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?

Is it possible to convert the polaer equation $$\ r = k \cos (\theta n) + 2$$ into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$. The ...
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39 views

Changing the order of integration on a rectangular and polar region

change the order of integration for the following integral from dydx to dxdy, and from dydx to polar coordinates. $$ \int \int f(x,y) dydx$$ where $$ 0≤y≤(-x^2)+2 $$ $$ 0≤x≤1$$ From dydx to dxdy $...
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Eliminate the parameter to find a cartesian equation for the curves

For the first part I am just unsure as to how the book has a different answer than mine. The book has the answer $y = \frac{3}{4} x - \frac{1}{4}$ but given the functions $x(t) = 3 - 4t$ and $y(t) = ...
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How do I find the Integral of $\sqrt{r^2-x^2}$?

How can I find the integral of the following function using polar coordinates ? $$f(x)=\sqrt{r^2-x^2}$$ Thanks!
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1answer
82 views

Solve differential equation by using polar coordinates

For $\alpha, \beta>0$ the differential equation, I am trying to solve, is given by $$\begin{pmatrix}\dot x_1\\\dot x_2\end{pmatrix}=\alpha\sin(x_1^2+x_2^2)\begin{pmatrix}x_2\\-x_1\end{pmatrix}+\...
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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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Cauchy rieman equation. What is u?

Using cauchy rieman equation, i want to show the function is analytic. So i want to decompose from f(z) to two term (real part and imaginary part) With rectangular form or polar form. But it is so ...
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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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450 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
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Calculating a likely position in 0.5 seconds with only the knowledge of the last second (x,y coordinates)

I have x,y coordinates for football players (22) at a rate of 10 records per second - for an entire football match. I have created an animation of the match with the player locations updates at a ...
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75 views

Solve a Laplace in Poolar Coordinate under two minutes?

I trouble with calculating the following example from previous exam with short solution on this Link. OP says there is a Laplace ٍPoolar Coordinate: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\...
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Solve $z^5=-32$ and draw its solutions in complex space, then describe their characteristic geometrical property.

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
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Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?

Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?
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51 views

How to think in terms of Polar Coordinates?

I am currently studying Polar Coordinates and many times I've noticed that one converts polar equations in cartesian form to do further analysis. Is there a way to think in terms of Polar Coordinates? ...
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Polar angle of a given point

How to find out polar angle of a given point $A(x_1,y_1)$ relative to another point $B(x_2,y_2)$ in a 2D space?
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Convert Polar Equation to Rectangular Equation

I have been trying to convert the polar equation to rectangular. $U(1,\theta) = 10+3*sin(\theta)-10*cos(2*\theta)$ I started off my multiplying everything by r and using the trig identity for cos($...
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Intersection between roses with given polar equations

$ r_1 \ = \ 4 \sin(3 \theta) \ $ and $ \ r_2 \ = \ 3 \cos(3\theta) \ $ a) find the solutions to the system using polar coordinates I was able to solve this by setting $ \ r_1 \ $ and $ \ r_2 \ $ ...
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Converting polar equation to cartesian coordinates

I have been trying everything to convert the polar equation $r=\frac{2}{1-\cos(\theta)}$ to cartesian coordinates but I simply didn't manage to know the right answer. Please help me..
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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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How to multiply a number in polar form by $-1$

Given a number in the form of $a\angle b$, what happens if I multiply it by $-1$? Is it $-a\angle -b$ or $a\angle −b$? Does a negative modulus even make sense? Thanks!
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$r=a\sin \theta$ $r=a\cos \theta$ intersect at right angles

In Stewart's Calculus book, he asks: Show that the curves $r=a\sin \theta$, $r=a\cos \theta$ at right angles. I do not understand the question. It is clear that they meet for example when $\...
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Trig Identities when converting to polar form

Been practicing conversions from rectangular form to polar form but got stuck on one question. rectangular form: $\frac{1}{1+j\omega}$ polar form: $r = \left|\frac{1}{1+j\omega}\right|$, $\phi = -\...