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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2answers
47 views

Trouble finding the limits of integration for polar coordinates

Use polar coordinates to evaluate $\iint_D x \, dA $, where D is the region inside the circle $x^2 +(y-1)^2 = 1$ but outside the circle $x^2 +y^2 = 1$ as shown below. Hi all, i'm stuck on finding ...
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3answers
45 views

Trignometric Equation Solution

Question : On the interval $[0,2\Pi]$ there is one point on the curve $r = \Theta - 2cos\Theta$ whose x-coordinate is 2. Find the y-coordinate there. The solution simply states: Solving $(\Theta - ...
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2answers
46 views

Transformation of xy plane to polar coordinates. (What would be the bound of polar coordinate?)

I have a double integral $$\int_0^a \int_0^x (x^2+y^2)^{1/2} \operatorname d y \operatorname d x$$ So, I am double-integrating $r^2$ What would be the region of the polar coordinate..?
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2answers
50 views

What is the area of $[r = \frac{4}{2 - \cos \theta}]$?

It makes an ellipse, but I'm unsure where to go from here.
3
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1answer
51 views

Second order linear PDE

I have the system with the following partial differential equation. $\\ \frac{\partial u}{\partial t}=\frac{3 a}{4r^ 2}\frac{\partial^ 2 u}{\partial r^ 2}\\$ How can I solve this?
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1answer
41 views

Find the intersection between two lines in a polar notation

I've a polar chart in an application, which displays a curve: I would like to add a functionality when I click on the plot. When I click(at the point M here), I know the orientation and amplitude ...
0
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1answer
30 views

converting a circle's equation not touching axis to polar from Cartesian system for integration

I am having a really hard time figuring out how to convert this circle to polar coordinates, I am to use double integration after converting it. I know that $\theta$ has to be between $0$ and $\frac ...
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0answers
43 views

Real world Geometry - Finding the Location of a point of intersection between two known locations and one angle

This is a real world problem I'm trying to solve. If I know the locations of two points A and B have a certain latitude, longitude and elevation (GPS coordinates). I also know that a vector from ...
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1answer
25 views

I'm struggling on getting the limits right for this cartesian to polar double integral

I've been working on this homework problem for a while now and I'm just not getting it right. I'd like for some extra eyes to look at this and hint to me where I'm going wrong. The cartesian for is: ...
2
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2answers
32 views

Oblique asymptote polar equation

I have the polar equation $r(\theta)=\frac{1}{\theta-\frac{\pi}{4}}$. I can see that it has an oblique asymptote for $\theta \rightarrow\pi/4+$, but what is it in Cartesian form ?
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1answer
43 views

Name of this type of plot? Does anyone know how to produce it

Does this type of polar plot have a name? Does anyone know how to produce it in octave 3.8.1 which is compatible with matlab? Link to site
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1answer
21 views

When changing variables to polar coordinate and deriving a differential equation for r

The system is $$\dot{x}=-y+ax(x^2+y^2)$$ $$\dot{y}=x+ay(x^2+y^2)$$ and the variables are changed to $x=r\cos\theta$, $y=r\sin\theta$, and when you note $x^2+y^2=r^2$, why can you say ...
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0answers
50 views

Complex integral over sphere in polar coordinates

I have trouble evaluating the integral: $$\int_{B(0,\frac{3R}{|h|})} \frac{1}{(r e^{2i a}-e^{i a})}dr da$$ In fact I just need to estimate it from above in terms of $|h|log (\frac{1}{|h|})$, where ...
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0answers
60 views

Definite Integral of $\sqrt{(x^2+y^2)^k+B}$

I'm trying to evaluate the integral $$ \int_{-1}^1 \sqrt{(x^2+y^2)^k+B} \, \mathrm{d}y $$ WolframAlpha doesn't return a response even for simplified versions of this, but I believe it can be ...
0
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1answer
160 views

How to calculate shortest distance in polar coordinates when approaching a pole

Given a distance (generally, a large one, say of 850km), a polar coordinate on the earth, and a bearing (with respect to the north pole), I'm using the Haversine formula to calculate a second ...
3
votes
2answers
92 views

Converting an integral from polar to cartesian

Question concerning definite integrals. Lets say we have some integral in cartesian coordinates (like the integral of $$ \int^{3/4}_0 \sqrt{1+y^2} \space dy$$ I completely understand how to ...
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2answers
39 views

Need help with polar double integral problem

this is an even problem in my textbook. So this is one of the few places I can check my answer. $$\int _{-1}^1\int ...
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2answers
41 views

How to calculate the argument and its limit for the sequence $z_n=-2+i\frac{(-1)^n}{n^2}$

I am trying to show that the limit of the sequence $$z_n=-2+i\frac{(-1)^n}{n^2}$$ exists, using the polar representation. Note that $\lim_{n\rightarrow \infty }z_n=-2$. $$$$I am finding difficulty in ...
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0answers
57 views

Area under a curve with polar coordinates. Seems to be too simple?

Curve is given by equation: $$r^2 = 2a^2|\cos \phi|$$ I would like to use the formula: $$A = \frac{1}{2}\int_a^b (f(\phi))^2 \, d\phi$$ So, since equation is already squared, i can put the right ...
0
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1answer
21 views

Polar Represantation of Shifted Disk

How to represent a shifted circle or disk (I mean the center of the circle is not at origin) in polar coordinate? For example I have a circle/disk in z-Domain like this: I thought this: $z = ...
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0answers
31 views

Tangential and normal components of acceleration of a point moving along a curve

If a point is moving along a curve in polar coordinates, is the tangential component of its acceleration given by $r\left(d^2\theta \over dt^2\right)$ and the normal component by $r\left(d\theta \over ...
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0answers
27 views

Single integral of 2 variable function in polar coordinates

I have a function $f(x,y)$ and I want to integrate it in polar coordinates, but only along one variable. What is this integral equal to? $$\int f(x,y) dx \overset{?}{=}\int f(r\cos\theta,r\sin\theta) ...
2
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1answer
50 views

Vanishing of the Riemann tensor

The Riemann tensor in a coordinate basis is $$R^{i}_{\,jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - \Gamma^m_{jk}\Gamma^i_{ml}$$ Consider $\mathbb{R}^2$ ...
0
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1answer
49 views

Is this function continuous? Polar coordinates “identity”

Is the function $f:\mathbb D\to S^1\times I$ given in polar coordinates by $f(r,θ)=(θ,r)$ (or to be precise: $f(r\cos\theta,r\sin\theta)=((\cos\theta,\sin\theta),r)$) continuous? How would one prove ...
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1answer
25 views

Polar form of equation of line in $xy$-plane

Urgent help requested!! Anything I can do to get an answer faster, in terms of my question?? The question, diagram, and my work are attached. Any help or suggestions or hints are extremely welcome ...
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0answers
60 views

complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
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1answer
36 views

Computing $\iint \limits_R \frac{xy}{x^2 + y^2} \mathrm{d}x \, \mathrm{d}y$ where $R=\{ (x,y) \in \mathbb{R} : y \geq x, 1 \leq x^2 + y^2 \leq 2 \}$

Homework question, so just hints please Sketch the region $$ R=\{ (x,y) \in \mathbb{R} : y \geq x, 1 \leq x^2 + y^2 \leq 2 \} $$ and, by changing to polar coordinates, compute $$ \iint ...
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0answers
20 views

Question about rewriting polar to rectangluar coordinates

I'm asked to rewrite the function $$ f(\alpha,r) = \left\{\begin{aligned} &\frac{1}{2}\sin(2\alpha) &&: r\not=0 \\ &0 &&: r=0 \end{aligned} \right.$$ to rectangluar ...
2
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1answer
28 views

Need help converting this to Polar integral and evaluating it

I have to convert this to polar integral and evaluate it. $$\int _{-1}^0\int _{-\sqrt{1-x^2}}^0\:\frac{2}{1\:+\:\sqrt{x^2\:+\:y^2}}\:dy\:dx$$ I attempted the conversion and ended up with this ...
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1answer
62 views

Conversion of the polar equation $ r=\sin(4\theta) + 2$ into Cartesian.

Can some one give me a hand converting $r= \sin(4\theta) +2$ into an x,y equation?
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1answer
52 views

Is this function continuous? Polar coordinates

Is the function $f:\mathbb R^2\to \mathbb R^2$ given in polar coordinates by $f(r,\theta)=(1,\theta)$ continuous? How would one prove it? My guess would be yes, since geometricly it simply change ...
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0answers
22 views

Continuous function generating continuous angle function

Let $f,g:I\to\mathbb{R}$, where $I\subseteq\mathbb{R}$ is an open interval, be two continuous functions. Show that there is a continuous function $\theta: I\times I\to\mathbb{R}$ such that: ...
2
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3answers
66 views

evaluating Polar Integrals. Cartesian to Polar? [duplicate]

I can't for the life of me figure out this problem. There's not example in my textbook. I'm suppose to convert this into a polar integral and evaluate it $$\int_0^6 \int_0^y x \;dx \;dy$$ I have my ...
3
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1answer
104 views

Complex number times conjugate equals square of modulus (proof check)

My textbook asked me to prove that a complex number $r\operatorname{cis}(x)$, denoted by $z$, when multiplied by its conjugate is equal to its modulus squared. I realise that the second half of my ...
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2answers
39 views

Converting an integral into polar form

$$\int_{0}^{1} \int_{0}^{\sqrt{2 - x^2}} \frac{x}{\sqrt{x^2 +y^2}} \ dy\ dx$$ How to convert this into polar form as there would be 2 parts? What is the use of limits x=0 to x=1 as i am finding no ...
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2answers
28 views

Region in Polar Coords

Hi I am intersted in the following question regarding polar corodinates: Can anyone see how the region inside the circle $$(x-1)^{2} + (y-1)^{2} = 1$$ is described in polar coordinates? Thanks for ...
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0answers
35 views

Tangent undefined for polar curves ($r^2=a^2sin(s\theta)$)?

I am considering the function $r^2=a^2\sin(2\theta)$ and am trying to find tangents perpendicular to the initial line, so $\frac{dx}{d\theta}=0.$ However, when I take the derivative by implicit ...
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0answers
67 views

Tangents perpendicular to the initial line for cardioid? Polar coordinates…

For the polar curve $r=a(1+cos\theta)$, I am trying to find the equations of the tangents perpendicular to the initial line by setting $\frac{dx}{d\theta}$ equal to zero. I am able to factorise a sine ...
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2answers
54 views

use polar coordinates to evaluate the integral $\int^2_0 \int^{\sqrt{1-(1-x)^2}}_0 \frac{y}{y^2 + x^2} dydx$

use polar coordinates to evaluate the integral $\int^2_0 \int^{\sqrt{1-(1-x)^2}}_0 \frac{y}{y^2 + x^2} dydx$ I have no problem evaluating the integral but the limits of integraation is what I am ...
0
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0answers
66 views

Find coordinates of a point 30degrees from another point

I need to find the coordinates of a line that is 30degrees away from another point. (If you look on the attached image it should explain, I want the coordinates of the top of all the blue lines.) I ...
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1answer
38 views

How can I prove non-geometrically that there is a bijective correspondence between polar and cartesian representations of coordinates?

We have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ as $f(x,y) = (\sqrt{x^2 + y^2}$, $\tan^{-1}\left(\frac{y}{x}\right))$ which takes a Cartesian pair $(x,y)$ to its polar form, and a ...
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1answer
54 views

$\delta$ in spherical coordinates: $\int_0^R\int_0^{2\pi}\int_0^{\pi}\delta(\theta-\pi/2)(r^2\sin(\theta)\,d\theta \,d\phi \,dr)$

Suppose you have a disc of radius $R$, we can find its area in polar coordinates by: $$\int_0^R\int_0^{2\pi}(r\,d\phi \,dr)=\pi r^2$$ Naively, I also expect to be able to integrate in spherical ...
0
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1answer
37 views

convert equation from polar coordinate to cartesian coordinate

I have the following equation $$r= \frac{A}{\log\left[B\tan\left(\frac{\theta}{2N}\right)\right]}$$ For using an optimization program, I would like to have this equation in cartesian coordinate ...
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0answers
22 views

Polar coordinates for vector difference in $\mathbb{R}^2$

I have a function $F(\boldsymbol X)=\tilde F(x,y)$ of $x$ and $y$ in the plane, and I can transform it in a function of $r$ and $\theta$, say $f=f(r,\theta)$, through the change of coordinate $$x=r ...
0
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1answer
55 views

Making sense of polar coordinates transformation on the derivatives

I would like to make sense of the transformation of the differentials in polar coordinates (to fix the ideas). To be more precise, the "right" way to find the transform for the differential and the ...
3
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1answer
43 views

$\int_{\mathbb{R}^{n-1}} \frac{1}{(y_1^2 + y_2^2 +…y_{n-1}^2 + C^2)^\frac{n}{2}} dy = \frac{n\alpha(n)}{2C}$

$$\int_{\mathbb{R}^{n-1}} \frac{1}{(y_1^2 + y_2^2 +...y_{n-1}^2 + C^2)^\frac{n}{2}} dy = \frac{n\alpha(n)}{2C}$$ where $\alpha(n)$ is the volume of the unit ball in $\mathbb{R}^n$. Could anyone help ...
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0answers
84 views

Approximate Laplace Operator with Central Difference in Polar Coordinates

I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel ...
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0answers
19 views

Expressing Curvature of a Polar Function in Terms of its Derivatives

I could use a little guidance with this question: Consider the curve $r=f(\theta)$, where $f$ is any twice differentiable function. Determine an explicit formula for the curvature $\kappa$ in terms ...
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0answers
35 views

How to convert this polar equation to cartesian?

$r=\cos \left(\frac{13}{7}\theta \right)$ When I try I get this: $\left(x^2+y^2\right)^{.5}=\space \cos \left(\frac{13\space }{7\space }\arctan \left(\frac{y}{x}\right)\right)$ But that doesn't seem ...
4
votes
4answers
97 views

Conversion from Polar to Rectangular

Can someone please explain to me how to convert the following equation from polar to rectangular? r=$2^\theta$ Thus far I got: $4^{\arctan(y/x)}$=$x^2$+ $y^2$ by squaring both sides and ...