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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Why doesn't line fitting seem to work in polar coordinates

I have 2 points, $(r_1, \theta_1)$ and $(r_2, \theta_2)$. They are plotted and I'm trying to find a curve in the form of $r=\theta\beta_1+\beta_2$ to connect both of them. This is basically performing ...
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1answer
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A Conceptual Polar Curve Question

A polar curve has $r=f(\theta), 0\le \theta \le 2\pi$ has a length of $L$ and is closed by a region that has an area $A$. How can I find the area of a region closed by polar curve say $r=4f(\theta)$ ...
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34 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
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1answer
29 views

How to find the position on a circle that satisfies two constraints?

Say I'm given an point P1 at coordinates $(x_1,y_1)$, and another point $P_2$ at coordinates $(x_2,y_2)$. Then I have a point $P_0$ that needs to be at coordinates $(x,y)$ such that it is a fixed ...
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15 views

Speed and velocity in x-direction of a point in polar coordinates

I have a list of values that describe the angle (a) of the polar coordinates to a time (t). The radius is 1. I was asked to estimate the speed of the point in the x-direction at time t(3)=2.4° My ...
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68 views

Calculus - finite integration of $e^{y^3}$ in double integration

i have this problem that bugs me for 3 hours now. I searched the internet and did not find a solution to this specific problem which was asked in our final: $$\int_0^3 \;\int_{\sqrt{x/3}}^r ...
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3answers
34 views

Polar coordinates in double integral of two circles

Use polar coordinates to calculate the integral $\int\int_R(x²+y²)\,dx\,dy$ where $R$ is the region inside $x²-4x+y²=0$ and outside $x²-2x+y²=0$. This is the graphic of the region: ...
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1answer
45 views

Is it a good way to find polar equations of curves?

When I was in my first year of Prepa classes it was not at the program but we have to see it on an example and our maths teacher did it with hypocycloïd and epicycloïd too for fun, well it was very ...
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1answer
30 views

Transform the following cartesian equations in polar equations

$$4y^2-20x-25=0$$ The answer given by the textbook is $r=\frac{5}{2(1-\cos \theta)}$ and I couldn't get to this result. I have done $x=r\cos\theta$ and $y=r\sin\theta$ and it leads to ...
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Derivative of angular function by cartesian coordinates using Legendre polynomials?

I'm programing some numerical evaluation of force dependent on angle $\phi$ between vector ${\vec a}=(x,y)$ and normalized direction vector ${\hat d}$. To achive maximal performance I wan't to avoid ...
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1answer
28 views

Formula for area in a special occasion in polar coordinates.

I know that the area of a curve given in polar coordinates is $$\int_{\theta_1}^{\theta_2}\frac{r^2}{2}d\theta$$. But what is the area outside one curve and inside another, when one of them is not ...
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Unit vector of an angle in plane polar coordinates

I'm struggling to find any information, about how the tip of a unit vector of an angle in plane polar coordinates, $\hat u_{\theta}$, describes a circle - if $P$ is a moving particle - with an angle ...
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2answers
64 views

converting a differential equation to polar coordinates

I have the following family of autonomous systems, I'm having trouble with part b): $$x'=x(1-\sqrt {x^2+y^2})-y-\epsilon y$$ $$y'=y(1-\sqrt {x^2+y^2})+x+\epsilon(x+x^2+y^2)$$ a) Convert the system ...
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1answer
44 views

Calculate : $\int_{} \int_ {} e^{-(x^2+y^2)}dA$ While $R$ is the area in $x^2+y^2=1$

Hello I need to calculate this $$\iint_R e^{-(x^2+y^2)}d\mu$$ Where $R$ is the unit disk, given by $\left\{(x,y):x^2+y^2\leq 1\right\}.$ What I did : $$\int_{-1}^1 ...
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3answers
49 views

Calculate : $\int_{-\pi/2}^{\pi/2}\int_0^{asin\theta} r^2 drd\theta$

I need help with the following integral : $$\int_{-\pi/2}^{\pi/2}\int_0^{asin\theta} r^2 drd\theta$$ What I did : $$\int_{-\pi/2}^{\pi/2}\int_0^{asin\theta} r^2 ...
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Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
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Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ...
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1answer
20 views

formula for the arclength of a curve, given as polar coordinates

Let $\gamma: [a, b] \to \mathbb{C}$, $\gamma(t) = r(t)e^{i\phi(t)}$ be a continuously differentiable curve, given in it's polar coordinates, where $r, \phi: [a, b] \to \mathbb{R}$ are continuously ...
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31 views

Limit of function of 2 variables - can I use polar coordinates?

I need to solve a limit of a f(x,y) (as a part of bigger task), but I'm bad at math. So basically here's this limit: $$\lim_{x,y\to(0,0)} \frac{y^4}{(x^2+2y^2)\sqrt{x^2+y^2}}$$ I tried to use other ...
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1answer
28 views

Give a formula for the volume of the solid under a surface $z=xy$ and a triangle?

Given is the solid with unit density lying under the surface $z = xy$ and above the triangle in the $xy$-plane with vertices $(0, 1, 0)$, $(1, 1, 0)$ and $(0, 2, 0)$. Give a formula for the ...
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2answers
76 views

Polar form to cartesian

Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$ I know this has to be converted ...
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3answers
92 views

What is $\dfrac{dr}{d\theta}$?

Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$ I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations ...
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75 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
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1answer
39 views

Calculating an integral using polar coordinates.

I want to calculate the volume enclosed by $z^2 = 1+x^2+y^2$ and the plane $z=2$. When $z = 2$, $x^2+y^2 = 3 \rightarrow r = \sqrt{3}$ So I have set up the integral in polar coordinates as: ...
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1answer
45 views

What is the advantage of describing functions in terms of polar coordinates instead of Cartesian coordinates?

Using $x = rcos(\theta)$, $y = rsin(\theta)$, we can rewrite $x^2 = y$ as $r = sin(\theta)sec^2(\theta)$ This seemed very unnecessary while I was learning calculus. Does anyone know if there are ...
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1answer
32 views

Evaluate the given double integral by converting to polar coordinates

Evaluate the given double integral by converting to polar coordinates $$\int_0^2 \int_0^{\sqrt{4-x^2}} e^{x^2+y^2} \, dy\,dx$$ my work is that $$x^2 +y^2 =r^2$$ $$x= r\cos(\theta)$$ ...
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2answers
86 views

Random Uniformly Distributed Points in a Circle

I know that by just using a random angle and a random radius within the bounds of your circle, you will end up with points near the center of a circle. Whereas if you do ...
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How to make this change of variable?

I have the following system of equations:$$\begin{cases}\dot{r}(\tau)=H_w\\\dot{w}(\tau)=-H_r\end{cases}$$ where is a given function of the form $H(w(\tau),r(\tau),t,\tau)$ and $\tau=\varepsilon t$. I ...
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30 views

How to calculate the first and second order derivatives of the curve given in polar coordinates?

How do I calculate the first and second order derivatives, $dy/dx$ and $d^2y/d^2x$, of the following curve given in polar coordinates, $r=cos(\theta)$? I really have no idea where to start on this ...
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2answers
32 views

How to write the equation in polar coordinates

I have the equation $(x-3)^2+y^2=9$ which is a circle centered at 3 and I need to find the polar coordinates for this equation, but I am not sure where to start because the $(x-3)$ is throwing me off. ...
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1answer
32 views

weird conversion of polar coordinates into rectangular coordinates.

Given $r=\frac{4}{1-\cos(\theta)}$, convert into rectangular (cartesian) coordinates. My solution: Square both sides: $r^2=\frac{16}{\sin^2(\theta)}$ Multiply the denominator by $r^2$ to get: ...
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3answers
58 views

Evaluating $\int_{-\infty}^{\infty}e^{-x^2}dx$ using polar coordinates. [duplicate]

I need to express the following improper integral as a double integral of $x$ and $y$ and then, using polar coordinates, evaluate it. $$I=\int_{-\infty}^{\infty}e^{-x^2}dx$$ Plotting it, we find a ...
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What type of Hopf bifurcation takes place here?

Consider the system: $\dot{x} = \mu x-y-xy^2-x^3$ $\dot{y} = x+\mu y - x^2y-y^3$ I have shown that a Hopf bifurcaiton takes place at the origin $(0,0)$ as a stable spiral becomes an unstable spiral ...
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A family of curves has polar equations

A family of curves has polar equations r = (1 -􏰃 a cos(theta) 􏰁/ (1 +􏰂 a cos(theta)) 􏰁 Investigate how the graph changes as the number a changes. In particular, you should identify the ...
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1answer
20 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
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How to force prime numbers into a line?

Inspired by an article on Prime Spiral and Hough transform I tried to analyze patterns created by plotting numbers on spiral (Archimedean?). $$x = \cos( angle ) * radius$$ $$y = \sin( angle ) * ...
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1answer
110 views

Help determining angle

Let $R$ be the triangle defined by $−x\tan(\theta) \le y \le x\tan(\theta)$ and $x \le 1$ where theta is an acute angle. Sketch the triangle and calculate \begin{equation*} \iint_R(x^2+y^2)\mathrm ...
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1answer
29 views

polar coordinates question

I was tasked with writing $\iint_D f(x,y) \,dx \,dy$ for $ [ D:{4\leq x^2 + y^2 \leq}9]$ through ''reoccurring integrals'' in polar and Cartesian systems? what are ''reoccurring integrals''? and how ...
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polar system in a plane?

what is a polar system in a plane and how it helps in calculating integrals in certain areas? I'm looking for a good explanation/a fair/ readable source on the matter.
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Geometric interpretation of $\frac{dr}{d\theta}$ in Cartesian Coordinates

You'll have to excuse me if this questions is extremely trivial; it's been years since I went back to elementary calculus and I humbly accept that I haven't really gotten deep into polar ...
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2answers
87 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
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1answer
26 views

Derivative of a polar coordinate equation

I was trying to plot the polar curve: $r=\cos(2n\theta)$ ($0\leq\theta\leq 2\pi$) and tried differentiating with respect to $\theta$ to get some information about where the petals would be. My ...
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1answer
41 views

Intersection of polar curve with line

I apologize for the horrible title. I came across this in an exam question: You're given $C_1$ as $$r = 1 + cos 2\theta $$ For $\frac{\pi}{2} \leq \theta \leq -\frac{\pi}{2}$. The only symmetry $C_1$ ...
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132 views

Help me out with my assignment question? Area of triangle using double integral in terms of polar coordinates.

Let $R$ be the triangle defined by $-x \tan t \leq y \leq x \tan t$ and $x \leq 1$ where $t$ is an acute angle. Sketch the triangle and calculate double integral $(x^2 + y^2) dA$ using polar ...
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2answers
31 views

How to convert from cartesian to polar equation

I am trying to convert the equation $y=4/x$ into a polar equation. I have done this work but I am not sure if it is right. I just subsituted $r\sin(\theta)$ for $y$ and $r\cos(\theta)$ for $x$ and ...
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2answers
73 views

Find exact length of polar curve $ r= \frac{6}{1 + \cos \theta}$

I find myself frustrated with the solution of this problem since profit not find it, I'm stuck in the middle of the problem I can not solve the integral, I'm stuck in the solution of the integral is ...
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1answer
66 views

Surface area generated by revolving $r = \sqrt {\cos 2\theta}$

I've been giving a good time trying to solve this problem, I do not find a clear way to solve appreciate your help. \begin{array}{rcl} r& =& \sqrt{\cos 2\theta } \end{array} This Around to ...
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1answer
24 views

Area shape calculating

Can't find the area of the figure bounded by the curve in polar coordinates $$\phi=r\arctan(r), \phi=0, \phi=\frac{\pi}{\sqrt 3}.$$ I tried use the formula $$S=\frac 12\int_{0}^{\frac{\pi}{\sqrt ...
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1answer
37 views

How to prove this ODE is stable but not asymptotically stable?

Consider the ODE in polar coordinates: $$ r'=f(r),\theta ' =1 $$ where $$ f(r)=r\sin (1/r^2), r\neq 0, f(0)=0. $$ show that the origin is stable but not asymptotically stable.
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Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 ...