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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72 views

Changing operator to polar coordinates

Let $$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ be the Laplace operator on the $(x,y)$-plane. Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$. ...
0
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1answer
4k views

Velocity and acceleration of a particle in polar coordinates

I am asked to find the radial and transverse velocity and acceleration for a particle with polar coordinates $r=e^t$ and $\theta=t$ Therefore let $\underline{r}=\underline{\hat{r}}e^t$ and ...
0
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1answer
37 views

Path of an ellipse

A path is described by the position vector $\mathbf{r}$: $$\mathbf{r}=a\cos(\omega t)\mathbf{\hat{i}}+b\sin{\omega t}\mathbf{\hat{j}}$$ I am asked to show that the path is the ellipse in the form of: ...
1
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1answer
167 views

Rotation of 2D polar graph in a 3D space along some fixed axis?

Does there exist some systematic way of rotating a 2-D polar graph $r=f(\theta)$ around some axis in a 3D space? For example: $f(\theta)=cos(\theta)$ in 2-D looks like: If we want to rotate the ...
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2answers
1k views

Sketch $r=\cos(5 \theta)$? $r$ as a function of $\theta$ in cartesian coordinates

I think I have to plug in numbers into $\theta$ such as 0 and $\pi/6$. What kind of numbers should I plug in ? Sketch $r=\cos(5 \theta)$? $r$ as a function of $\theta$ in cartesian coordinates
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1answer
55 views

Heuristic approach to winding number

I'm working on problem 8.23 of Rudin's PMA, that is: Let $\gamma:[a,b]\to\mathbb C$ be a closed curve, $\gamma \in C^1([a,b])$ and $\gamma(t) \neq 0 \ \forall t\in [a,b]$. Show that $$\text ...
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1answer
611 views

Finding the centroid of a polar curve

I have absolutely no idea how to find the area centroid of this problem. I have been working at this one for ages but can't seem to get anywhere. Any first steps? How would one go about solving ...
0
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1answer
914 views

Can someone check my answer for this area between 2 polar curves question?

Find the area of the region that lies inside the circle $r = 1$ and outside the cardioid $r = 1-cos(\theta)$ I drew the graph and set it up like this: $$ \int_0^\pi \frac{1}{2} [ (1)^2 - ...
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0answers
137 views

Correct way to write the polar form of a complex number

What is the most correct way to write the polar form of a complex number? For example, given the complex number: $\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i$ I would write the polar form as follows: ...
2
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0answers
131 views

Polar Coordinates for Multivariate Limits With Three Variables

When working on limits with two variables, $f(x,y)$, I like to convert the problem to polar coordinates a lot of the time, by changing the question from $$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y)$$ to ...
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1answer
63 views

Representation of differentials in Polar Coordinates

We define polar coordinates in $\mathbb{R}^{n}$\ $\{ 0\}$ by $x=ry$, where $r=|x|>0$ and $y \in \partial B(0,1)$ is a point on the unit sphere. In the coordinates, Lebesgue measure has the ...
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0answers
44 views

what are the possible solutions to this equation?

I'm trying to find some angles for my characteristic equation , I need to know the roots or possible answers to cosine equation $$1-\cos(u)\cdot\cosh(w)=0,$$ and $$u=\sqrt{\lambda_1}\cdot L.$$ ...
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0answers
114 views

Polar equation of perimeter of half ellipse

x = Cx + a * cos(ang); y = Cy + b * sin(ang); Cx, Cy are cords of center. ...
0
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1answer
146 views

Conversion to polar equation

I would to know when asked to convert an equation to polar what it means.For example $ x^2+x+y^2-2y=0 $ My understanding so far tells me I need to derive an equation in form of: $$ r^2=x^2+y^2$$ ...
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2answers
290 views

Image of a closed curve under $w=z^2$.

I have the curve: $$r=2(1+ \cos \theta), \ \theta \in [0,2\pi)$$ in polar coordinates on the complex $z$ plane, and I want to find the image of this curve under the square function $w=f(z)=z^2$. ...
0
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1answer
690 views

Hyperbola in polar coordinates, what's wrong?

I read that the equation of a conic in polar coordinates is $$r=\frac{l}{1+e\cos \theta}.$$ But when I try to reduce the hyperbola $$x^2 - y^2 =1$$ to that form by setting $x=r\cos \theta $, $y=r ...
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3answers
883 views

Pushforward of a vector field

Can someone help me with that ? We define $\phi:=(\phi^1,\phi^2):\Omega\subset\mathbb{R}^2\to\phi(\Omega)$ with $\Omega$ such that $\phi$ is a diffeomorphism by ...
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2answers
261 views

Evaluating the area in the polar coordinates

So the problem asked me to find the area of the region that lies inside both of the circles $$r=2sin\theta, \quad r=sin\theta +cos\theta $$ I know that $r=2sin\theta$ is $x^2+(y-1)^2=1,$but ...
0
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1answer
52 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
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1answer
90 views

How to use $dz=d[r(t)(\cos t + i\sin t)]$ as a change of coordinates?

This notation comes in handy for some path integrals, but I don't know yet how this is calculated. Is it simply a change of coordinates? Is this correct: $$z=r(t)(\cos t+i\sin t) \quad ...
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2answers
411 views

Equation of circular sine waves in the water

I have to write the equation of a sine wave expanding circularly from a point $P_0=(x_0,y_0)$. The wave has the form $\eta(\rho)=A\sin(\omega\rho)$ where $\rho$ is the distance from the point $P_0$. ...
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2answers
156 views

need help finding the coordinates of AB

Find AB if the coordinate of A is -5 and the coordinate of B is 17. i have been out of school for over 20 years and have little to no memory of this process. i examined my daughter's book and there is ...
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1answer
178 views

Software for drawing two-variable functions in polar coordinates

I am in difficulty of finding a software for drawing two-variable functions in polar coordinates. Could someone introduce useful software for me? Thanks in advance. For example $$ f(r, ...
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3answers
151 views

Obtain polar form of a line from two points

I need to work with the lines in polar form, but i only have two points in cartesian form for each line. I tried this: From the points, i got the slope-intercept form: $$y = mx + b$$From this url: ...
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1answer
52 views

Geometry finding area problem

A regular 2N -sided polygon of perimeter L has its vertices lying on a circle. Find the radius of the circle and the area of the polygon.
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77 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
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2answers
138 views

Double integrals transforming into Polars

This is my first post here. I'm reading about double integrals and can't catch how to get the new limits of integration when converting to polar form. $$\left(\int_{-\infty}^{\infty} ...
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3answers
191 views

Polar to rectangular $r = 7$

I don't follow this at all. I have $r = 7$ and the formula states $x = r \cos\theta$ $y = r\sin\theta$ but my book gives $x^2 + y^2 = 49$ this is impossible. It doens't follow the formula at all. ...
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3answers
5k views

Polar curve $r = 2\cos \theta -1$

$$r = 2\cos \theta -1$$ I am suppose to find the polar curve of the inner loop. Here is its graph, courtesy of Wolfram|Alpha, I am having trouble working out this polar function on a cartesian ...
0
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1answer
185 views

Defining a spiral in polar coordinates

I'm trying to find a general form for a spiral that fits the following criteria: the inner radius is $N$, and for any point $q$ on the spiral, the arc length from the start of the spiral to $q$ is ...
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2answers
44 views

Find the image of a ring

I'm working on the following problem: Find the image of the ring defined by $4 \lt x^2 + y^2 \lt 16 $ under the mapping $$F(x,y) = \left(\frac{x}{x^2+y^2} , \frac{y}{x^2+y^2}\right)$$ It looks to ...
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1answer
126 views

Plotting an angle on a graph

So I know, my origin "(0,0)", my angle "theta" degrees, and the distance from the origin, "d" Now I think I can work this out with polar coordinates, but really have no idea how to go about it. My ...
2
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1answer
345 views

How do I define the limits of a double integral in polar coordinates over an annulus?

Evaluate the double integral by re-writing them in polar coordinates: $\displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA$, where $R$ is part of the annulus (ring) $9\leq x^2+y^2\leq 25$ lying ...
2
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1answer
54 views

What's the name of each pseudo-rectangle in a spherical surface?

Consider the common surface of a spherical segment crossed with a spherical wedge. This produces a pseudo-rectangle in the sphere surface, and a perfect rectangle in a mercator projection. What's the ...
2
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1answer
251 views

How to integrate over polar coordinates

Evaluate the following double integral by rewriting it in polar coordinates: $\displaystyle\iint\limits_Dxy\,dA$, where $D$ is the disc with center at the origin and radius 5 I have very little ...
2
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1answer
341 views

Test for symmetry for polar graphs

From a calculus book I'm reading: "Unlike the graphs of an equation in $x$ and $y$, the graph of an equation $r=f(\theta)$ can be symmetric with respect to the polar axis, the line $\theta = \pi/2$, ...
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3answers
3k views

Square root of complex number in polar or rectangular form

I am trying to find how to simplify: $$\sqrt{\frac{A+jb}{C+jd}}$$ My calculator errors out, giving a math error, and I don't know how else to solve this.
2
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1answer
96 views

Need a hint on what's wrong - polar coordinates

I'm asked to solve the following $$ \int^2_0 \int^\sqrt{4-y²}_0 \sqrt{4-x^2-y^2} dxdy $$ I thought about using polar coordinates: (1) $0 \le x \le \sqrt{4-y^2}$ is the upper half of a circumference ...
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2answers
392 views

How to verify a conversion to spherical coordinates?

Is it possible to verify if a conversion of an integral in Cartesian coordinates to spherical coordinates was done correctly other than revising it looking for mistakes? I mean, is there some kind of ...
2
votes
2answers
94 views

How to find the number of intersection for $ \rho =\frac{\theta} {2\pi+1} $ and $\rho =\frac {1} {2-\cos\theta} $

How to Find the number of intersection for curve $ \rho =\frac{\theta} {2\pi+1} $ and curve $\rho =\frac {1} {2-\cos\theta} $ .
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2answers
392 views

Kepler's First Law in 3D

Kepler's First Law in 2D polar is $$ r = \frac{p}{1 + \varepsilon\cos(\nu)}. $$ How can this be written to consider ellipses in ...
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0answers
362 views

Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
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0answers
49 views

From cartesian to polar, on a 'wavy' sphere surface

For a hobby project I'm trying to transform a wavy halfsphere surface into smaller segments. For this I need to be able to go from cartesian coordinates to polar coordinates. One of the formulas for ...
4
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2answers
364 views

question about continuity: using polar coordinates

Given a function $f\colon\mathbb R^2\rightarrow \mathbb R$ I want to study continuity. So I know the $\varepsilon-\delta$ and sequence criterion. Now we had polar coordinates in lectures: set ...
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1answer
179 views

Converting from polar to Cartesian coordinates.

I'm looking at some notes that I was given for my Calculus II class on converting from Cartesian to polar coordinates. Now I understand how to solve for r and $\theta $ but I'm looking at how she ...
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1answer
80 views

Polar parametrization surface intersection

here is my problem: I need some help, i need the parametrization of the intersection of this two surfaces: $\ z^2= x^2+y^2 $ $\ (x-1)^2+y^2=1 $ Well, i can do it with cartesian equations $\ ...
4
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2answers
366 views

limits of Surface area of revolution in polar co-ordinates.

My Question is Find the area of the surface generated by revolving the right-hand loop of the lemniscate $\;r^2=\cos2\theta\;$ about the vertical line through the origin (y-axis). I know the formula ...
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1answer
73 views

Another polar integral bounds question.

A plane region $R$ is determined by the inequalities $y\ge0$, $y\ge-x$, $x^2+y^2\le3\sqrt{x^2+y^2}-3x$. Sketch the region and find it's area. I have foregone sketching the area and tried to use ...
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1answer
171 views

area between two polar curves

I am trying to find the area between the following two curves given by the following polar equations: $r=\sqrt{3}\cos\theta$ and $r=1+\sin\theta$. I did the following: First, I found the points of ...
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74 views

Obtaining the cardioid by mirroring the square root function in a line

In what line of the plane $C_{W}$ is the cardioid $$p= 2 (1 + \cos\theta)$$ mirrored, from the branch of the function $$w=\sqrt{Z}$$ which takes positive values in $X>0$ and $Y=0$. Seriously this ...