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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1answer
901 views

How to find the shortest distance from a line given in polar coordinates and a point given in Cartesian coordinates?

How do I find the shortest distance from a line given in polar coordinates and a point given in Cartesian coordinates? For example, say that the line is given by the polar coordinates rho = 2 and ...
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0answers
186 views

How to find the formula of a spiral using least-squares(regression)?

Assume data from a plane which are roughly showing a spiral. I want employ the rationale of regression to find the parameters for the best fit by some spiral. That means, I have to estimate the ...
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1answer
734 views

Cardioid given by the polar equation $r = 1 − \cos(\theta)$

Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$. (a) Find the intersection points of the curve with the line $\theta = \pi/4$. (b) Find the ...
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3answers
74 views

Put x in terms of u and v

I have these two equations $$u = \frac{2x}{x^2 + y^2} \\ v = \frac{-2y}{x^2 + y^2}$$ And I need to put $x$ in terms of $u$ and $v$. If I take polar co-ordinates and plug them in I get(in the case of ...
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0answers
235 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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1answer
137 views

Find the area of polar coordinates

The questions is to find the area in the bounded region in polar coordinates $r = \sqrt{\theta}$ from $3\pi/2$ to $2\pi$ Here is what I did: I got the integral of $\cfrac{1}{2}\theta d\theta$ from ...
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1answer
284 views

Calculating the gradient of a tangent of a radial function

I'm trying to determine the gradient of a tangent to a curve defined by a radial function $r = f(\theta)$. It's a programming application and the actual function is gigantic but lets say that $r = ...
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1answer
248 views

map on a unit sphere with polar coordinates

My brother, who is in hospital atm and cannot verify by himself asked me to post the following question, thank you in advance, and sorry if the topic has already been covered, i do not have the math ...
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1answer
177 views

Probability functions with 2 random variables

I really dont understand this problem, I was hoping someone could help me out with this and explain how to do this. It goes like this: Suppose you throw a dart at a circular target with a 10 cm ...
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3answers
1k views

Need hint: Prove that $r = a (\sin t) + b (\cos t)$ is a circle, where $ab \neq 0$

"Show that the polar equation $r = a \sin(t) + b \cos(t)$, where $ab \neq 0$, represents a circle, and find its center and radius." I don't need the answer - I just need a hint to nudge me on. I've ...
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2answers
133 views

Help? I cannot do the integration by parts correctly??

Can someone please show me how to integrate $$\int_0^\infty\frac4{\pi b^2}x^2e^{-x^2/b^2}dx\;?$$ please show steps how to integrate this problem. This is what i have so far. $$\frac4{\pi ...
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1answer
10k views

Magnitude of average velocity..

A particle is at the position with Cartesian components $(5.0 m, 4.0 m)$ and $5.0 s$ later it is at the position with components $(8.0 m, 0.0 m)$. What is the magnitude of its average velocity? I ...
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2answers
364 views

How do I calculate numerically a tensor in polar coordinates?

You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid? a) fit a (spline) surface, ...
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1answer
925 views

How to convert the equation of a line from polar to standard form?

How do you convert a polar line to a line in standard form? That being, change a line with parameters rho and theta in a polar ...
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1answer
81 views

Getting coordinate between two coordinates knowing the distance and latitude

That is my wall: http://imageshack.us/f/266/wall2z.png/ I know the coordinates of the lower points (left and right). (X1,Y1,Z) and (X2,Y2,Z) where X es the latitude, Y longitude and Z the altitude. ...
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0answers
68 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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1answer
353 views

The point C has coordinates (2,k) and the distance from A to C is 5. Find the 2 possible values of the constant k

The coordinates of A are (6, -2) I have managed to do every other question in this section but this one, my brother tried to help but just drew a graph and the answers without explaining it, i dont ...
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6answers
362 views

Simple partial differentiation $x = r\cos\theta$ and $y = r\sin\theta$

If \begin{align} x &= r\cos\theta,\\ y &= r\sin\theta, \end{align} find $$\dfrac{\partial^2\theta}{\partial{x}\partial{y}}.$$ How can I find this partial derivative? I need to prove ...
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4answers
350 views

Find the length of a curve

Find the length of the following curve: $r(t) = e^{-t} \sin(t)+e^{-t}\sin(t) i$ for $0 \leq t\leq 1$. Any ideas?
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0answers
228 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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0answers
122 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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1answer
965 views

Numerical Integration of a Gaussian Distribution in Polar Coordinates

I want to numerically evaluate a 2D-integral of a specific probability distribution over some given area (I use MATLAB so all the code below is MATLAB code). I broke down the problem so that it ...
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4answers
5k views

Find the area of the region inside the limaçon

I'm struggling to figure out the answer to this: Find the area of the region inside the limaçon, $r=3 + \sin(\theta)$ Could someone please help me out?
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1answer
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Area Bounded by Polar Curves

I am answering sample exams for my Calculus class and my attention was caught by the following item. Set-up the definite integral or sum of definite integrals equal to the area of the region above ...
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1answer
872 views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
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2answers
2k views

Del operator in 2D polar coordinates

I need to show that the del operator in 2D polar coordinates is $\nabla=e_r\partial_r+\frac{1}{r}e_r+\frac{1}{r}e_{\phi}\partial_{\phi}$. I try the following approach: ...
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2answers
220 views

Length of a plane curve in polar coordinate

Consider the plane curve $\gamma$ in polar coordinates: $$ r=r_0+e^{\lambda\theta}, \quad \theta_1 \le \theta \le \theta_2, $$ where $r_0,\lambda,\theta_1>0$. Is it possible to compute explicitly ...
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2answers
7k views

area between polar equation $r = \sin\theta$ and $r = \cos\theta$

Below is the exact question and answer from my textbook: Find the area of the region enclosed between the two curves $C_{1}$ and $C_{2}$ where $C_{1}$ has the polar equation $r = \sin\theta$ and ...
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4answers
4k views

find all points for intersection between 2 polar equations

I stumped at one of the exercise in my multivariable calculus textbook. I try to search online but I can't seem to search on how answer no 3 and 4 below is derived. I also plot both of polar ...
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1answer
5k views

Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3).

Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$. Here's my steps: 1.Write the two points in cartesian coordinates: the ...
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2answers
854 views

Find the area enclosed by the loop $r=2(1-\sin\theta)\sqrt{\cos\theta}$

The diagram shows a sketch of the loop whose polar equation is $$r=2(1-\sin\theta),\qquad -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$$ a)Show that the area enclosed by the loop is 16/3. ...
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1answer
2k views

Polar equation of cartesian $y = 1 + 3x$

I have no idea at all what to do on this I got $$\cos^{-1} \left(\frac{r\sin \theta+1}{3} \right) = \theta$$ Which can be $$\cos^{-1} \left(\frac{r\sin \left(\cos^{-1} \left(\frac{r\sin ...
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2answers
817 views

Polar equation of $y = 2$

Maybe I do not understand what is going on here but I cannot get the right answer. $$y = 2 $$ $$y^2 + x^2 = r^2$$ $$4 + 0^2 = r^2$$ $$ r = 2$$ $$y = r \sin \theta$$ $$1 = \sin \theta$$ $$\theta ...
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1answer
72 views

gradient in polar coordinate by changing gradient in Cartesian coordinate

I'm tried to do following and I can't see what went wrong. $$\begin{bmatrix} \hat r\\ \hat \theta \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta ...
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2answers
902 views

Single variable integral to polar coordinates?

I took calculus about 2 semester ago, and I'm trying to brush up on polar coordinates. I integrated $-x^2+3$ from $x = -\sqrt{3}$ to $\sqrt{3}$ and I got $6.93$ Now I tried to convert it to polar ...
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1answer
318 views

Maximum point of a polar function

I have a curve C with polar equation $$r^2 = a^2\cos{2\theta} $$ And I am looking to find the length $x$ when $r=max$ Judging from the equation: $$r = \sqrt{a^2\cos{2\theta}} $$ R will be maximum ...
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1answer
371 views

Find the area of the region determined by two curves

Find the area of the region $R$ given by two curves. So the region $R$ describes the area that is common between the two curves: $$\begin{align*} \text{Function 1: } r&= 2\sin(\theta)\\ ...
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2answers
125 views

Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?

I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are ...
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2answers
1k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
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1answer
2k views

How to “rotate” a polar equation?

Take a simple polar equation like r = θ/2 that graphs out to: But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is ...
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1answer
353 views

Fitting of Closed Curve in the Polar Coordinate.

I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I'm thinking of deducing a ...
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4answers
2k views

Polar equation of a circle

A very long time ago in algebra/trig class we did polar equation of a circle where $r = 2a\cos\theta + 2b\sin\theta$ Now I forgot how to derive this. So I tried using the standard form of a circle. ...
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2answers
5k views

Set up double integral of ellipse in polar coordinates?

How do you set up a double integral for an ellipse in polar coordinates without using Jacobian or Greens Theorem? I can't seem to figure out what (or if) the limits of r can possible be. $x = ...
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3answers
2k views

Argand Diagram - Quadrants help

I forgot the rules of adding angles when it comes to argand diagrams. In the first quadrant, you add 90 degrees to whatever angle you get, what about Q2 Q3 Q4 ? This picture will explain what i mean ...
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1answer
1k views

Find the Cartesian equation corresponding to $r = \frac{5}{3-2\cos(\theta)}$

Find the Cartesian equation corresponding to $r = \frac{5}{3-2\cos(\theta)}$ I got it into the form: $(x^2 + y^2)(3-2x)^2 = 25$ and can see that maybe the equation of a circle will appear, but ...
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4answers
4k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
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1answer
262 views

Find volume of a revolved solid by integrating wedges.

So, lets say that I wanted to find the volume of the solid formed by rotating the area between $f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a ...
2
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2answers
142 views

Integral variable substitution using Hausdorff measure

Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral: $$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$ where $B \subset ...
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3answers
471 views

Expressing $e^z$ where $z=a+bi$ in polar form.

I am reading a passage of text that states: "We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
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1answer
218 views

Complex Numbers and polar form

I am given the following information: $$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$ where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general. I was wondering how the following is ...