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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211 views

Evaluate the integral by converting to polar coordinate

$$ \int^{\pi/2}_{\pi/4} \int^{\sqrt{2-y^2}}_y 3(x-y) dx dy$$ I attempted the following: $$ \int_{\pi/4}^{\pi/2} \int_{0}^{1} 3r^2 (\cos\theta - \sin\theta) dr d\theta $$ which is wrong apparently. ...
3
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30 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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69 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 ...
2
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71 views

Shifting a plot in polar coordinates

Say we have the plot of a function $r=f(\theta)$ and want to "relocate" it to $(h,k)$. Is there a general procedure for this? I have tried the following tactic to no avail on the following example: ...
2
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74 views

Wrong answer within 'Calculus Solution Manual, Michael Spivak, 3rd ed'

I have a problem with the answer provided in the solution manual of Calculus, Michael Spivak, 3rd ed, The Problem: Consider a hyperbola, where the difference of the distance between the two foci is ...
2
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74 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 ...
2
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90 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 ...
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34 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 ...
2
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646 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= ...
2
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130 views

Explain this limit of integration for radius in polar coordinates.

Use polar coordinates to find the volume of the given solid: Inside both the cylinder $x^2 + y^2 = 4$ and the ellipsoid $4x^2 + 4y^2 + z^2 = 64$ The limit of integration for the radius goes ...
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913 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. ...
2
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200 views

What is the total area enclosed by a polar curve and the x-axis?

I am aware of the formula.. but can someone give me a clear definition of what the total area enclosed by a polar curve represents. Thanks
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26 views

Double integral polar coordinate using substitution

I have to calculate: $$\iint _{R} \frac{x}{\sqrt{x^{2}+y^{2}}}dA$$ and R is this region: $$x^{2}+y^{2}=16; x^{2}+y^{2}=4; y = \sqrt{3}x; y=\frac{x}{\sqrt{3}}$$ so I used substitution: $$x = r\cos ...
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34 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
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30 views

area in polar coordinates

Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this particular question. To be completely honest I am not sure how to even approach this problem, ...
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24 views

Question concerning the domain of polar coordinate.

So in the problems I encountered, I find it confusing about the domain of $\theta$. Problems take the form: For arbitrary function $f(x,y)$, and $$\displaystyle \iint_S f(x,y)dxdy=\iint_T ...
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45 views

Algorithm for finding nearest distance from a point to a curved surface in space

I need to write an algorithm which can find the nearest distance from a point in space to a 3D curved surface which is straight in vertical direction but its projection is an arc of a circle (Similar ...
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25 views

Finding intersections points of pairs of polar curves?

Find all intersections of the curves $r=3^{(1/3)}\cos(\theta) , r=\sin(\theta)$ What I have done so far is to just put them equal to each other like this: $3^{(1/3)}\cos(\theta)=\sin(\theta)$, but ...
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25 views

Sketch the polar graphs

Any advice on how to sketch polar graphs? I have tried transforming to rectangular coordinates, but its not really much help $$ r=1+\sin(\theta) \\ r^2=4\cos(2\theta) $$ Thanks in advance :)
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42 views

Derivatives of polar coordinates

I've got a problem for which I'm trying to calculate $\ddot r$. The problem is right here for the sake of reference. So far, I've got that: $$\ddot r=\frac{d}{dt}v_r=\frac{dv_r}{dr}\frac{dr}{dt}$$ ...
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137 views

obtaining $y'$ from $r = 1 + \sin\theta$

Just want to check if this is the idea for this $$x = r\cos\theta$$ $$y = r\sin\theta$$ so now we substitute $$x = (1 + \sin\theta)\cos\theta$$ $$y = (1 + \sin\theta)\sin\theta$$ get the ...
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20 views

Simple ways to create a separating plane given two points in a polar coordinate system?

I am currently working on sensor networks, where sensors are uniformly distributed in a polar coordinate system (maximum radius $R$ is set to $1$). A few of the sensors are placed equidistantly on a ...
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127 views

Finding an area using $r = 13\cos \theta$, $r = 6 + \cos \theta$

I have these: $$r = 13\cos \theta\quad r = 6 + \cos \theta$$ I am trying to find the area. Would anyone please help?
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51 views

Regions formed by polar coordinates in double integration.

I need to sketch the region of integration of the following double integral in the $xy$ plane: $$\int_0^{\pi/2}\int_0^{1/\cos\theta} f(r,\theta) \ dr \ d\theta,$$ where $f(r,\theta)= ...
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36 views

Polar coordinates: fixing ratio between arc and radius

When drawing a coordinate system with fixed step size, the standard polar coordinates $$ x=r\cos(\theta), y=r\sin(\theta) $$ exhibits stretched pixels for large $r$. Ignoring the singularity in ...
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154 views

How to plot a stream function

This question relates to fluid mechanics and I have the components in polar coordinates. The components of the velocity field are; $$v_r= \frac{-kr}{z}$$ $$v_z= kz$$ $$v_\theta= 0$$ and I have ...
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105 views

Correct way to write the polar form of a complex number

What is the most correct way to write the polar form of a complex number? For example, given the complex number: $\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i$ I would write the polar form as follows: ...
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41 views

what are the possible solutions to this equation?

I'm trying to find some angles for my characteristic equation , I need to know the roots or possible answers to cosine equation $$1-\cos(u)\cdot\cosh(w)=0,$$ and $$u=\sqrt{\lambda_1}\cdot L.$$ ...
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180 views

Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
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36 views

From cartesian to polar, on a 'wavy' sphere surface

For a hobby project I'm trying to transform a wavy halfsphere surface into smaller segments. For this I need to be able to go from cartesian coordinates to polar coordinates. One of the formulas for ...
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90 views

Curl in cylindrical coordinates

I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating: ...
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181 views

gradient of an axis symmetric vector field in cylindical coordiantes

I am trying to calculate with a general approach the gradient of an axis symmetric vector field in cylindrical coordinates and then express it in cartesian coordinates. First I write my vector ...
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825 views

Mass and center of mass of lamina in polar coordinates

I need some help with the following problem which is question number 15.5.4 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the mass and center of mass of the ...
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220 views

(Calculus 3) Having trouble finding the polar equation of a hyperbola.

Eccentricity e=sqrt(2), and one vertex is located at (2,0). I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...
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65 views

Knowing coordinates of a point having two coordinates and the distance.

I have the two geographic coordinates of the lower corners of a wall. So, for example, i want to know what is the coordinate that is for example 15cm on the right of the lower corner left. Is that ...
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212 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: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, ...
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111 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
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822 views

Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?

Say I have the following equation of motion in the Cartesian coordinate system for a typical mass spring damper system: $$M \; \ddot{x} + C \; \dot{x} + K \; x = ...
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56 views

Minimization of matrix of vectors in polar field

The problem I am facing is the reduction of vibrations of a rotating object. I have a series of vibration measurements taken at 5 different states with magnitude and phase components, and a set of ...
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17 views

How many kinds of simple coordinates are there in a 2D space?

The question comes form an idea to solve a motion-with-potential problem in 1D space by finding a mathematically equivalent uniform-motion problem in 2D space. ...
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12 views

Heat equation on a circular plate

I'm in trouble with the following problem: assuming a circular plate of radius $R$, the heat equation on it, is: $$\partial_t ...
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28 views

Differential Operators in different coordinates

How does one show this identity? $$\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}=\frac{\partial^2}{\partial r^2}+{1\over r}\frac{\partial}{\partial r}+{1\over ...
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27 views

polar co ordinates integration

integrate the polar co ordinates $$ \int^{r=\infty}_{r=0} \int^{z=\infty}_{z=-\infty} \delta(r) \delta(z-z_s) dz dr$$ => I want to integrate the above equation. integral of $ \int^ {z=\infty} _ {z= ...
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21 views

Transformation in polar coordinate system

I have a point $P \in \mathbb{R}^2$. $P$ in a local Cartesian coordinate is given as $(x,y)$ or alternatively the polar representation is given as $(\rho, \theta)$. This local coordinate is located ...
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18 views

Polar coordinates: Slope of tangent

Would anyone mind telling me how to solve this problem? It seems strange as my answer is $-1$. Do I have to apply this formula, $(r'sinθ +rcosθ)/(r'cosθ -rsinθ)$ ?
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34 views

How to solve for r: rsin( θ) = sin(rcos( θ))

Please help me out. I am trying to figure out how to put y=sinx in polar form and this is as far as I can get. If this is as far as possible, then how are you supposed to graph this when you need r ...
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25 views

Simplifying Inequalities Before Converting Cartesian Coords. to Polar

I have a 3 dimensional region defined by Cartesian coordinates and I have to convert them to cylindrical coordinates. That is the easy part, but what I don't understand is how to treat the ...
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18 views

Using spirals to draw log scale?

Taking inspiration from the fact every math textbook in existence puts a picture of a nautilus shell on its logarithms chapter, I did some research, and found that many different spirals (most easily ...
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205 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
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56 views

Finding an arc-length between 2 points in 3 dimensions

I know how to find an arc-length between two points with coordinates, say $X=(a,b)$ and $Y=(c,d)$. But how do I find the same thing but for, say $X=(a,b,c)$ and $Y=(d,e,f)$? Thanks!