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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84 views

Polar equation of perimeter of half ellipse

x = Cx + a * cos(ang); y = Cy + b * sin(ang); Cx, Cy are cords of center. ...
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1answer
93 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
127 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 $w=f(z)=z^2$. ...
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1answer
201 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
230 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
236 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 ...
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1answer
47 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
63 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
205 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
105 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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88 views

Compute $ \int_{-1}^{1} \int_{\left| x\right| }^{ \sqrt{2-x^2} } \frac{1}{\left( x^2 + y^2\right)^{1000} } \mbox{d}y \mbox{d}x $

Compute $$ \int_{-1}^{1} \int_{\left| x\right| }^{ \sqrt{2-x^2} } \frac{1}{\left( x^2 + y^2\right)^{1000} } \mbox{d}y \mbox{d}x $$ We have (by using polar system): $$ \int_{-1}^{1} ...
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1answer
104 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
76 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
45 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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60 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
103 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
88 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
2k 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 ...
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1answer
142 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
37 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
84 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 ...
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1answer
124 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 ...
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1answer
43 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
127 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 ...
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1answer
92 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
1k 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.
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1answer
78 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
259 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
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2answers
85 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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189 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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144 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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19 views

Superquadrics with inverse functions?

I'm looking for any website that has a catalog of superquadrics / polar surfaces AND their inverse functions (such that I can get phi and theta if I know a cartesian point on the surface). It doesn't ...
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0answers
30 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 ...
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2answers
141 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
112 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
69 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 $\ ...
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2answers
169 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
67 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
106 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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0answers
64 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 ...
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1answer
60 views

Area of $\left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2$

I've used the modified polar coordinates: $x = 3r \cos \theta$, $y =5r \sin \theta$, which got me to $$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$ What now?
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1answer
79 views

Find polar equation from 4 polar points

Given $4$ polar coordinates $(3, -\pi/6)$, $(1, \pi/3)$, $(3, 5\pi/6)$, $(-3, 4\pi/3)$, graph and find the polar equation. I know that the general polar equation is $r = ep / 1+- e \cos (\theta)$. ...
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3answers
97 views

Finding a length of arc, what's wrong?

Find: $$ \int \sqrt{x^{2}+y^{2}}dl$$ $$L: x^{2}+y^{2}= Rx$$ (at image $p' = -R\cdot \sin(\phi)$ )
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4answers
139 views

Find the maximum value of $r$ when $r=\cos\alpha \sin2\alpha$

Find the maximum value of $r$ when $$r=\cos\alpha \sin2\alpha$$ $$\frac{\rm dr}{\rm d\alpha}=(2\cos2\alpha )(\cos\alpha)-(\sin2\alpha)(\sin\alpha)=0 \tag {at maximum}$$ How do I now find alpha? ...
8
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3answers
299 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
2
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1answer
347 views

Heat equation in polar co-ordinates

I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge $r=a$ of a circular plate is kept at temperature $f(\theta)$. The plate is insulted so that there is ...
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1answer
79 views

Sketch the polar graph $r=e^{-2\phi}$

How are you supposed to sketch this type of polar graph? Are you supposed to somehow relate this to $\cos\phi+i\sin\phi$ but can polar graphs even have an imaginary axis?! I am thinking that you ...
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3answers
732 views

Why is the formula for the area of a cardioid $ \int_a^b \frac{1}{2} r^2 d \theta$

I've seen this expression in many places :$\int_a^b \frac{1}{2} r^2 d \theta$ and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction ...
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2answers
64 views

How do you find the maximum value of $r$ in a polar function?

I have $\, r=\cos\alpha +\sin2\alpha,\quad 0\le\alpha\le\frac{\pi}{2}.$ Do you then find $\dfrac{dr}{d\alpha}$ and let that $=0$ ? I am after just a few set of instructions.
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1answer
436 views

Find the area of the shaded region between $r=e^{\theta/2}$ and $r=θ$ .

That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ...