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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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 ...
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0answers
50 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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9 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 ...
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1answer
29 views

Is this polar equation correct? [on hold]

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?
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2answers
40 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 ...
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0answers
23 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 ...
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1answer
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Find the Area Bounded by a Polar Curve Over a Given Interval [closed]

Find the area of the region bounded by the given curve: $r=7e^{\theta}$ on the interval $6\pi/7 \leq θ \leq 2\pi$
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22 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 ...
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1answer
17 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 ...
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1answer
35 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). ...
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2answers
97 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 ...
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2answers
23 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 ...
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2answers
15 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 ...
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1answer
13 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 ...
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4answers
46 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 ...
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0answers
67 views

Show that there is a limit cycle in the dynamical system

I have the dynamical system \begin{align} \dot{x}_1 & = -x_2+x_1(1-x_1^2-x_2^2), \\ \dot{x}_2 & = x_1 + x_2(1-x_1^2-x_2^2) \end{align} With the initial conditions $x_1(0)=x_{10}$ and ...
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1answer
22 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 ...
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2answers
36 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 ...
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1answer
11 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 ...
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2answers
24 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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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 ...
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0answers
26 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 ...
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1answer
23 views

Convert to Cartesian (rectangular) form

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 ...
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1answer
31 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 ...
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0answers
45 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 ...
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0answers
5 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 ...
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21 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 ...
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2answers
35 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, ...
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62 views

Changing polar coordinates: Calculating $\iint_R\dfrac{dxdy}{\sqrt{x^2+y^2}}$ where $R=\{(x,y):1\leq x^2+y^2\leq 2, x\leq0, y\geq0\} $

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $d$ and graded ...
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2answers
44 views

Area of a quadrilateral on cartesian plane A(0,0), B(4,0), C(3,${\pi \over 8 }$), D(1, ${3\pi\over 8}$)

I'm having trouble on this question. Could anyone find a solution and answer for this? What is the area of quadrilateral ABCD whose vertices have polar coordinates A(0,0), B(4,0), ...
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15 views

Express function in terms of polar coordinates and find residues of poles

The function $f(z)$ is given by $$f(z) = (z + \sqrt{3})^{1/2}ln(z-1).$$ The branch of this function is such that $$-\frac{4\pi}{3}<arg(z-1)\le\frac{2\pi}{3} and ...
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1answer
22 views

Width of a spiral

I'm attempting to generate a Archimedes spiral (defined as $r = a\theta$) from a given width $w$ and spacing $a$ between 'arms'. I have plotted Cartesian coordinates generated from my workings, but ...
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1answer
40 views

Polar coordinates of an egg-shaped curve

For a simple 3D-rendering project, I need to get the shape of an egg. (actually, a prism with an egg-shaped based). The idea behind it is to explain how a camshaft in an engine is working. From ...
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2answers
38 views

Evaluate the double integral by changing to polar coordinates for $x^2+y^2\leq4$

Change the double integral $\int_{D}\int \sqrt{4-x^2-y^2}dxdy$ where D={$(x,y):x^2+y^2\leq4,y\geq0$} by changing to polar coordinates $r, \phi$ So am I right in thinking the limits would be 0 and ...
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22 views

Verify my calculation of the surface integral without divergence theorem

I have $F=xyi-y^2j+zk$ Over surface $z=0$, $s \le1 $, $x^2+y^2 \le s$ My approach to calculate $ \iint F.ds$ was the outward normal is $k$ the dot product of this with F gives z so integral becomes ...
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29 views

Conversion of polar equations when you change the position of the origin

I'm working on a physics problem that is described as follows: "I am standing on the ground beside a perfectly flat horizontal turntable, rotating with constant angular velocity w. I lean over and ...
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1answer
30 views

How do you find the angle of intersection between two given polar curves?

How does one find the angle of intersection between two given polar curves? For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
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1answer
22 views

Is this a viable way of trisecting an angle in polar coordinates, using an Archimidean spiral?

Say you plot $ r = \theta $ from [0, 2pi] Consider an arbitrary angle $\alpha$ The length $r(\alpha)$ can be trisected using a ruler and compass. Arcs can be "swept out" from the points of ...
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9 views

Time for finite beam to cross a point in circular region

I'm trying to find the time a finite width beam takes to cross a point in circular region. Assuming the beam width at distance $r$ from the center is some constant times $r$, $kr$. I have calculated ...
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20 views

Calculate flux of vector field

I want to calculate the flux of the vector field $$X(x,y)=y\partial_x-x\partial_y$$ in $\mathbb R^2$ written in polar coordinates ($\partial_x:=\frac{\partial}{\partial x}$ and so on). Step 1: ...
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1answer
39 views

Transformation matrix in polar coordinates

I'm trying to write a software widget that allows the user to resize the component, so I can write a transformation matrix $\mathbf T_\text{xy}$ that will map $(x,y)$ to a transformed $(x',y')$, that ...
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1answer
29 views

Transforming ODE into polar form

Let $z=\rho e^{i\phi}$ be a complex number and $\alpha$ some parameter. I determined the following ODE $$ \dot{\rho}e^{i\phi}+i\rho\dot{\phi}e^{i\phi}=\rho e^{i\phi}(\alpha+i-\rho^2). $$ How to get ...
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2answers
23 views

find coordinates from known angles and length in 3d

Suppose I have 3 vectors with length a,b,and c. They are oriented in 3D space such that the angles between the three vectors are $\alpha$, $\beta$, and $\gamma$ (suppose all less than 90 degrees). If ...
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4answers
33 views

Polar equation of an ellipse given the origin coordinates and major and minor axis lengths?

I've been trying to create a polar equation that will give me all points on an ellipse with the independent variable being theta and the dependent variable being the radius, but I'm having a great ...
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15 views

find the equations of the tangents at the pole.

For the graph with polar equation $r = 1 + sin 3\theta$, find the equations of the tangents at the pole. My attempt, When $r=0$, $\sin3 \theta=-1$ $\theta=\frac{\pi}{2}, ...
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1answer
24 views

What is the polar coordinate equation for an Archimedean spiral with arc length known relative to theta?

What is the equation for the radius of a polar coordinate for an Archimedean spiral with the arc length known relative to theta? arc length: L = ...
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3answers
52 views

Prove a function is harmonic

This problem is from Conformal Mapping by Zeev Nehari: If $u(x,y)$ is harmonic and $r=(x^2+y^2)^{1/2}$, prove $u(xr^{-2}, yr^{-2})$ is harmonic. The hint is obvious: "Use polar coordinates." I ...
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2answers
65 views

Interval for area bounded by $r = 1 + 3 \sin \theta$

I'm trying to calculate the area of the region bounded by one loop of the graph for the equation $$ r = 1 + 3 \sin \theta $$ I first plot the graph as a limaçon with a maximum outer loop at $(4, ...
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2answers
31 views

Question about integrals in polar coordinates

I've just made on question where is asked the area of a region enclosed by one loop of rose $r=\cos3\theta$ and had one uncertained. In this case, the figure is the following: Suppose if it is ...
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0answers
17 views

Polar co-ordinates dr/dtheta

How can you visualise what is the curve doing by calculating Dr/dtheta in polar co-ordinates form. Also, what will it mean for Dr/dtheta to be zero? Thank you.