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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2
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1answer
64 views

Work out the area enclosed?

I am doing a simple exercice and I think that either the book's solution is wrong or I misunderstood the problem. Here is the problem, 平面上で次の曲線又は直線で囲まれる図形の面積を求めよ。 極座標系について、曲線 $r = a(1+2\cos ...
2
votes
1answer
66 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
0
votes
1answer
32 views

Converting from parts of a circle to polar coordinates

I have the area defined by $$A = \sqrt{x/4-(x/2)^2} < y < \sqrt{1-x^2)} \text{ and }0 < x < 1$$ and I'm supposed to find the integral of the function bound by these limits; $$I = ...
3
votes
2answers
85 views

Area of the region inside $r=\cos{\theta}$ but outside of $r=4\cos{3\theta}$.

I have been crazy finding the area of the region inside $r=\cos{\theta}$ but outside of $r=4\cos{3\theta}$. I can't decide the integral bounds
1
vote
1answer
21 views

Asymptotic behaviour of the riemannian metric in polar coordinates

I'm studying the section 7 ("Local Geometry in Constant Curvature) of chapter 5 of "Riemannian Geometry" written by Petersen. At the beginning there is a Lemma which says how behaves the metric $g$ ...
0
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1answer
12 views

What is the equation of a sphere of radius R centred at the origin in cylindrical coordinates?

I said that $r = R, -R \leq z \leq R$, and $0 \leq \theta \leq 2\pi$. Saying that $r = R$ is incorrect, however, but I don't understand why because clearly, at all points of the sphere the radius ...
0
votes
1answer
54 views

Can't get right solution for this polar problem…

Given: $$\frac{d\theta}{dt}=2$$ $$y = r(\sin\theta)=(3 \theta+\sin \theta)(\sin \theta)$$ Find $\dfrac{dy}{dt} = \dfrac{\left(\dfrac{dy}{d \theta}\right)}{\left( \dfrac{dt}{d \theta}\right)} = ...
3
votes
1answer
33 views

Confusion on polar coordinates of an ellipse

The polar coordinates of an ellipse are given by: $$x=\frac{abcos(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$ $$y=\frac{absin(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$ However, ...
0
votes
1answer
30 views

New limits when changing to polar coordinates for calculating a double integral

Calculate $$\iint_D {1 \over {(x^2+y^2)^2}} dxdy$$ when: $$D= \{\space (x,y) \in \mathbb R ^2 \space |\space {1 \over 2} \le x \space,\space x \le y \le \sqrt 3 x\space,\space x^2+y^2 \le 1\space ...
2
votes
1answer
47 views

When is ${{x^2y} \over {(x^2+y^2)^\alpha}}$ continuous, using polar-coordinates

Given $$f ({x,y})= \begin{cases} {{x^2y} \over {(x^2+y^2)^\alpha}},&(x,y) \ne {(0,0)}\\ 0,&(x,y)={(0,0)} \end{cases}$$ For what values of $\alpha$, $f$ is continuous in ${(0,0)}$? I set ...
2
votes
1answer
61 views

Pullback metric, coordinate vector fields..

I'm doing this computation on $\mathbb{R}^3$ with cylindrical coordinates $(r, \theta, z)$, (which aren't defined on the whole of $\mathbb{R}^3$, but I don't care about that) and I seem to get a ...
0
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0answers
18 views

Find polar coordinates parameters given a cartesian point

I'm stuck with a rather simple problem: I have the following polar coordinates equation where I know the values for the $x$ and $y$ terms and I want to find $a$, $b$ and $\Phi$. $$x = a\cos(\Phi)$$ ...
1
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0answers
27 views

Direction angle of Line segment in polar coordinates

I have a line segment given by two points $A$ and $B$, that are $(r_1,\theta_1)$ and $(r_2,\theta_2)$ in Polar coordinates. I know that the direction angle of the line segment is given by: ...
2
votes
2answers
71 views

Expressing $(-8)^{\frac13}$ in polar form

I want to express $(-8)^{\frac{1}{3}}$ in polar and cartesian coordinates. What I did was to solve the equation $-8 = r^3e^{3i\theta}= r^3(\cos(3\theta)+i\sin(3\theta))$ which implies that I must ...
1
vote
2answers
12 views

Expressing a value in polar and cartesian coordinates

I have to express the value of $\sqrt{i}$ in polar and cartesian form. I really don't know where to start this problem, any hint could help me please!
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0answers
25 views

Relationship between Normal coordinates and Spherical Coordinates

I am using the following coordinates on $S^3: (\psi, \theta, \phi)$ where $$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi ...
0
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3answers
25 views

ellipse polar co-ordinate conversion

I have a somewhat trivial question out of interest. Given the equation of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ why is the substitution $x = \sqrt{a}\cos t$ and $y = \sqrt{b}\sin t$ ...
0
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0answers
24 views

Divergence of outer product in polar coordinates

Right now I am trying to solve Euler's conservation equations for circular domain. Due to several factors, I am restricted to polar coordinates. I can't manage to correctly calculate divergence of ...
0
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4answers
51 views

How would I convert $2x^2 + y^2 + 3y = 0$ into polar form?

We are currently working with rectangular and polar equations. How would I convert $$2x^2 + y^2 + 3y = 0$$ into polar form? So far, I have tried to make the equation into rectangular and back ...
2
votes
2answers
26 views

How can I calculate, how the volume element transforms under change of co-ordinates?

Suppose I transform an integral $$I=\int f(x,y) \, dx \, dy$$ using polar coordinates, setting $x=r\cos\theta$ and $y=r\sin\theta$. We get $$ \begin{split} dx &= \cos\theta \, dr - r\sin\theta \, ...
-1
votes
2answers
60 views

How would one convert the cartesian expression y=1/x to polar form?

How would one convert the cartesian expression y=1/x to polar form? I'd really appreciate a step-by-step solution so I can apply the same principle to other problems. Thanks!
0
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2answers
25 views

Equations of motion into Binet's equation

I have been given these two equations in polar coordinates: $m(r''− rθ'^2) =−f,$ $m(r''θ + 2r'θ') = 0$ And have been told that I need to differentiate to show the angular momentum $L=mr^2θ'$ is ...
2
votes
2answers
50 views

How to reverse the integration order of the double integral $\int_{\theta=0}^{2\pi}\int_{r=0}^{1+\cos\theta}r^2(\sin\theta+\cos\theta)drd\theta$.

I am given the integral $$ \iint\limits_H \, (x+y) \mathrm{d} A $$ where $H$ is the area of the cardioid $r=cos(\theta)+1$. I have translated the double integral to polar coordinates in order to solve ...
0
votes
1answer
21 views

Convert $\int_0^1dx \int_0^{x^2}f(x,y) dy $ to polar integration

Converting $\int_0^1dx \int_0^{x^2}f(x,y) dy $ to polar coordinates: $(r \cos\theta)^2 = r \sin\theta$, so $ r= tg\theta\sec\theta$, then the result is, $$\int_0^{\frac\pi4} d\theta ...
1
vote
2answers
154 views

Arc length of Archimedes Spiral $ r = \theta $ from $ 0 \le \theta \le 2\pi$

The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ...
1
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0answers
75 views

Polar System with Short Answers, How $U(0, \theta)=\pi$ will be calculated?

I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial ...
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0answers
10 views

limits of a sphere away from centre in polar coordinates

I have a sphere whose centre is d distance from origin.I need to find r the distance from centre of origin to any point on that sphere and the limits of that sphere in polar coordinates. I have tried ...
1
vote
1answer
31 views

Is this polar equation correct? [closed]

Find the area of the region bounded by: $$r=5\cos(10\theta),~~~~~ 0 \leq \theta \leq 2\pi$$ When I did this, I got $\frac{1}{2\sin(20\pi)}-\frac{1}{2\sin(0)}$ getting $0$, is this correct?
1
vote
2answers
48 views

Area inside polar curve

Find the area of the region inside $r=7\sin\theta$ but outside $r=1$. I have tried finding the area of both using $A=\frac{1}{2}\int_\alpha^\beta r^2 dr$, tried arc length...but i cant seem to find ...
0
votes
0answers
25 views

Polar Equations (Complex)

So I'm trying to figure out what the angle $θ$ would equal at $x=-2$ for the polar equation $r=θ+sin(2θ)$. All I know is that $θ$ has a domain of $0\leθ\le \pi$ and $y < 1$ (pretty sure). I ...
0
votes
2answers
36 views

Converting $y = -\sqrt{1 - x^2} + 2$ to polar coordinates

Question: Convert $y = -\sqrt{1 - x^2} + 2$ to polar coordinates: What I have done $$ y = -\sqrt{1 - x^2} + 2 $$ $$ 2-y = \sqrt{1 - x^2} $$ $$ (2-y)^2 = (1-x^2) $$ $$ x^2 + y^2 -4y ...
1
vote
1answer
27 views

How to express an angle in terms of pi

I have the complex number $z = 5 + 6i$ in polar form $$z = \sqrt{61} (\cos \theta + i\sin \theta)$$ and $$\theta = \tan^{-1}\left(\frac{6}{5}\right) = 0.87605805059 \text{ rad}$$ But I need that ...
0
votes
1answer
44 views

why is $\dfrac{dr}{r~d\theta} = \cot \psi$?

why is $\dfrac{dr}{r~d\theta} = \cot \psi$ ? Extracted from Ordinary Differential Equations, Garrett Birkhoff, in the chapter of Linear Fractional Equations (First order Differential Equations). ...
5
votes
3answers
209 views

Rigorous proof that $dx dy=r\ dr\ d\theta$

I get the graphic explanation, i.e. that the area $dA$ of the sector's increment can be looked upon as a polar "rectangle" as $dr$ and $d\theta$ are infinitesimal, but how do you prove this ...
0
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2answers
27 views

using polar integration to solve $\iint\limits_D \sqrt{R^2-x^2 - y^2} dxdy, \ D: x^2 + y^2 \le Rx$

$$I = \iint\limits_D \sqrt{R^2-x^2 - y^2} dxdy, given\ D: x^2 + y^2 \le Rx$$ solving this using polar integration, $$(r\cos\theta)^2 + (r\sin\theta)^2 = R \cdot r\cos \theta$$ $$ -\frac\pi2 \le \theta ...
1
vote
2answers
22 views

Angle vector in polar system represented by Cartesian vector

$x=r\cos\theta,\,y=r\sin\theta\implies r^2=x^2+y^2,\,\theta=\arctan(y/x)$ I can show that $\hat{r}=\cos\theta\hat i+\sin\theta\hat j$, where the hat vectors are ...
0
votes
1answer
19 views

Convert polar coordinate to Cartesian coordinate

$$x=r\cos\theta,\,y=r\sin\theta,\;r^2=x^2+y^2,\,\theta=\arctan(y/x)$$ I was told that $\frac{\partial r}{\partial x}=\cos\theta,\,\frac{\partial r}{\partial ...
1
vote
4answers
52 views

What is the polar formula for $y=x$?

$y=x$ is a basic cartesian equation, but I'm at a loss as to what it is in polar form. It seems the only way I've found to express it is with $r$ on both side of the equation, but is there a way of ...
0
votes
1answer
24 views

Equation for making an oval based on r=cos(th)

I have a polar graph (on paper) with a curve similar to 7.8*cos(theta). Plotting the data estimate from a nice big print-out of the graph gives a fairly close fit, but I seems that the circle needs to ...
0
votes
2answers
45 views

Using polar coordinates to find area of a circle

Since the area of a polar curve is defined as: $$ \int_a^b \frac 12 r^2 d\theta $$ and since $r$ is constant, independent of $\theta$, can this be re-written as? $$ \frac 12 r^2 \int_a^b d\theta ...
0
votes
1answer
19 views

Rectangular to polar conversion angle error

I am trying to determine the polar form of the following rectangular vector: -105 + 140j The polar form is $\sqrt(-105)^2+140^2$ = 175 and the angle is ...
0
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2answers
32 views

If $x=r\cos(\theta)$ then in $?=r\cos(\theta+a)$ what is $?$ equal to?

What I mean by $r\cos(\theta+a)$ is that it's the same function $r\cos(\theta)$ but it's translated by $a$ units, if this makes any sense. I just want to know what it means in terms of $x$.
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2answers
41 views

When should I add $\pi i$ to the exponent when computing the polar form of complex nubmers?

This is maybe math $101$ question: Let $z_1=1+i$. I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$ But now if I take a look at $z_2=-1-i$, I ...
2
votes
0answers
37 views

What does it mean for a polar coordinate system to have basis vectors?

So I understand that every element of a vector space can be represented uniquely by a linear combination of the basis vectors: $v=\alpha_1v_1+\cdots+\alpha_nv_n$ Then coordinates to those basis ...
0
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1answer
33 views

Convert to Cartesian (rectangular) form [duplicate]

Convert the following to Cartesian (rectangular) form and provide a graph. $$e^{i7\pi /2}$$ The problem comes after a long series of similar problems. However, the noticeable difference with this ...
1
vote
1answer
42 views

Polar to Cartesian: r = 3 + sin(theta/2)

I am asked to convert the following polar function to cartesian: $$r = 3 + sin(\theta/2)$$ I would be able to do it if it weren't for the fraction. I have already tried substituting the identity ...
0
votes
0answers
47 views

How do I solve for the volume of a hyperboloid using a double integral in polar coordinates?

Here is the problem text, with my attempts at solving it at the bottom: Suppose you are part of a team designing a water tank in the shape of a hyperboloid. The tank is to have a top radius a of 2 ...
0
votes
0answers
18 views

Understanding unit normal curvilinear vectors to the surface of an octant of a sphere

I'm supposed to test divergence theorem on an octant of a sphere for a given vector field. The triple integral part was easy. However, I'm stuck with the double integral part. Now, there are four ...
0
votes
0answers
24 views

Integrating dot product in polar coordinates in the vicinity of pole

I am trying to build finite difference scheme for energy equation for compressible gas, in polar coordinates. Right now i am stuck with integrating the equation over the central cell of the ...
0
votes
2answers
39 views

Finding the angle of $-2i$.

Given $z = -2i$, I am to find the exponential form. Now, the radius $= 2$. The angle is derived as $\tan^{-1} \frac{y}{x}= \theta $ . $y$ and $x$follow the form $z = x + yi$. Now, given all this, ...