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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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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65 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 ...
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1answer
302 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 ...
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2answers
134 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
40 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 _{-\sqrt{1\:-\:x^2}}^{\sqrt{1-x^2}}\:\frac{2}{\left(1\:+\:x^2\:+\:y^2\right)^2}\:dy\...
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2answers
67 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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59 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 ...
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1answer
24 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 = re^...
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1answer
63 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 $\mathbb{R}^2$ ...
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1answer
82 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
30 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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1answer
40 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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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 ...
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1answer
31 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
247 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
78 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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3answers
72 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 ...
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1answer
269 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
45 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
29 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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58 views

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

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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170 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
60 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 ...
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125 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
95 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
78 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 ...
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1answer
45 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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1answer
69 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 ...
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1answer
52 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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139 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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47 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 ...
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4answers
114 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 ...
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2answers
750 views

Area of Bernoulli Lemniscate?

Can anyone help me calculate this area? I have to use double integrals, and the question sounds like this: "Calculate the area bounded by the curve $(x^2+y^2)^2=a^2(x^2-y^2)$, where $a$ is a real ...
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2answers
221 views

formula for logarithmic spiral on a linear level

I am trying to plot the contents of a circle, which include geometric elements and spirals, on a linear graph. For example, take a circle, take the beginning and the end and make it straight. What ...
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1answer
1k views

Volume of Solid Region Between Sphere and Paraboloid

"Find the volume of the solid region above the sphere $x^2+y^2+z^2 = 6$ and below by the paraboloid $z = 4-x^2-y^2$" I am, of course, going to be solving this double integral by converting to polar ...
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2answers
46 views

Evaluate 2D integral (by change of variable)

The question asks to evaluate integral $$\iint_D \Big[3-\frac12( \frac{x^2}{a^2}+\frac{y^2}{b^2})\Big] \, dx \, dy \ $$ where D is the region $$\frac{x^2}{a^2}+\frac{y^2}{b^2} \le 4 $$ I believe ...
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153 views

Stability of dynamical system described in polar coordinates

Near a fixed point, a dynamical system $\dot{\bf{x}}=\bf{F}(\bf{x})$ can be approximated by $\dot{\bf{x}}=A\bf{x}$, where $A$ is the Jacobian matrix. From the trace and determinant of the Jacobian $A$,...
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1answer
29 views

Writing same equation in different forms

I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $ My question would be if i have a circle ...
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1answer
901 views

Find all polar coordinates of point $P$ where $P = (7, \pi/3)$.

I don't know where to go from here. Answer choices are: a) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + 2n\pi)$ b) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + (2n + 1)\,\pi)$ c) $(7, \pi/3 + (2n + 1)\,\pi)$ ...
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1answer
123 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
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4answers
1k views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting $z=r(cos\...
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1answer
190 views

How do I find the area shared by the circles $r = 2\cos(\theta)$ and $r = 1$?

I figured out the intersection points: $r=2\cos(\theta)$, $r=1$ $2\cos(\theta) = 1$ $\cos(\theta) = \frac{1}{2}$ $\arccos(1/2) = π/3$ (I), $5π/3$ (IV)
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69 views

Introducing $\mathrm π$ and polar coordinates in real analysis

From time to time, I think about how material from introductory courses like real analysis or linear algebra can be structured in a way I would have liked to see in my freshman days. So recently, I ...
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2answers
2k views

Convert the Polar Equation to Cartesian Coordinates

$$ r^2=\sec 4\theta $$ I graphed this equations using Wolfram Alpha and found it to be 2 hyperbolas. I'm having difficulty showing this using the standard equations $$ x=r\cos\theta \;, \; y=r\sin\...
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1answer
58 views

Finding the area enclosed by 4 functions using polar coordinates

I need to find the area enclosed by $x^2+y^2$ = 4x, $x^2+y^2$ = 2x, y=x and y=0. How do I use polar coordinates here? It seems to me that representing those functions using polar coordinates is too ...
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1answer
52 views

Cannot find link between trigonometric statements and reduced form

I have been trying to find a way to reduce following trigonometric statements to the reduced form below, but without succes. I haven't been able to grasp the typical train of thought I presume I would ...
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1answer
53 views

Polar coordinate double integral

I have to integrate the following integral: $$ \iint \limits_A sin({x_1}^2 + {x_2}^2) dx_1dx_2 $$ over the set: $A=\{x \in \mathbb{R}^2: 1 \leq {x_1}^2 + {x_2}^2 \leq 9,x_1 \geq -x_2\}$ I ...
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1answer
38 views

Find the area enclosed by curve with polar coordinates

I am having a little difficulty finding the area enclosed by the curve, $r(\theta) = 4 + sin\theta + cos\theta$ with $0 \le \theta \le 2\pi$. I tried integrating over $0 \le \theta \le 2\pi$ and $0 \...
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3answers
388 views

Polar Integration of $ r = 2\cos(\theta)$

$ r = 2\cos(\theta)$ has the graph I want to know why the following integral to find area does not work $$\int_0^{2 \pi } \frac{1}{2} (2 \cos (\theta ))^2 \, d\theta$$ whereas this one does: $$\...
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0answers
25 views

Polar coordinates used to evaluate a function containing a branch cut

I'm having a lot of trouble understanding how to approach these kinds of problems, if anyone could explain the approach, it would be really helpful. The problem is as follows: The function $f(z)$ is ...