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
49 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
46 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 ...
0
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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
59 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
19 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 ...
0
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1answer
54 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?
1
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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
21 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
65 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
98 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 ...
0
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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
34 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
61 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 ...
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0answers
62 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 ...
0
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1answer
36 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
52 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
35 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
21 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 ...
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1answer
53 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
40 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
81 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
18 views

Polar-Cartesian Plot Interconversion

The Question How can we interconvert any general graph of polar or cartesian function so they give the same plot in the other coordinate system. Note: I do not mean to ask about interconverting a ...
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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
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4answers
94 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
136 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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0answers
31 views

Is there anything wrong with my solution?

Finding the arc length in polar form. $r = 1 - \cos\theta$ $0<=\theta<=2$ using the formula $L =\int_a^b\sqrt{r^2+(\frac{dr}{d\theta})^2}d\theta$ $L ...
1
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2answers
88 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 ...
0
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1answer
180 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 ...
2
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2answers
39 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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0answers
39 views

Calculate the flux coming out of a surface

Let F(x,y,z)=(2xy(z-2),x^2(z-2),x^2y) be a vector field and $\Sigma $ the surface defined as the portion of cone x^2+y^2=(z-2)^2 ...
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0answers
83 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 ...
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1answer
26 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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0answers
52 views

How to numerically determine whether a curve on a polar coordinate graph is dominant over another curve?

Given 2 curves appearing on a 2-dimensional polar coordinate graph across multiple quadrants having the same range of x-axis endpoints, how do you numerically determine the dominance of 1 curve over ...
0
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1answer
236 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) $(7, \pi/3 + (2n + 1)\,\pi)$ ...
0
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1answer
79 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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0answers
24 views

Coordinates in file isn't in the range -180 to 180 respective -90 to 90?

Hello! I'm creating an android version of a PC program (I've contacted the complany who owns the PC program, so it's legal). The program is in the core a GPS, but is used to navigate pre-defined ...
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4answers
213 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 ...
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1answer
102 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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1answer
64 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 ...
2
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2answers
173 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 \;, \; ...
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1answer
53 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 ...
1
vote
1answer
49 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
42 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 ...
0
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1answer
27 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 ...
1
vote
3answers
99 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: ...