Questions tagged [polar-coordinates]

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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Bound for polar equations.

When I wish to find the area of a polar equation such as $r=2-2\cos\theta$, I need to compute $$\frac12\int_\alpha^{\beta}r^2.$$ However, I am confused as to how to determine $\alpha$ and $\beta$. I ...
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Evaluating $\int_{0}^1\int_{0}^{\sqrt{4y-y^2}}x^2dxdy$ using polar coordinates

Evaluate $$\int_{0}^1\int_{0}^{\sqrt{4y-y^2}}x^2dxdy$$ using polar coordinates. My try: The region is as shown below: As i can notice $\theta$ is going from $0$ to $\frac{\pi}{2}$. But i am confused ...
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The points A $(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle.

$(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle. Derive a relation between $\alpha$ and $\beta$." /> I tried using the slope ...
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Converting simplified Gielis equation (polar) to parametric equation

I found the simplified Gielis equation, that discribes the shape of a leaf. The equation states: $ r = \frac{l} {(|cos \frac{θ}{4}| + |sin\frac{θ}{4}|)^\frac{1}{n}}$ To get to the Cartesian ...
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2 votes
2 answers
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Is it fine this change of variables? Integral in polar coordinates

$$\iint_{R} \frac{x}{y\sqrt{x^2+y^2}} dA$$ Where $R$ it's $\{x^2+y^2\leq 4,\ y\geq1\}$ I have changed to $x=r\cos{\theta}, y=r\sin{\theta}$. Seen the graph, we have $$\int_{-\pi/6}^{\pi/6} \int_{0}^2 \...
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How to formally express a function defined on unit disk?

Let $D$ be the unit disk, i.e. $D=\{(x,y): x^2+y^2\le1\}$. Let $f\colon D\to \mathbb R$ be a function. I associate to $f$ a new function $F\colon [0,1]\times [0,2\pi]\to \mathbb R$ such that $F(r,\...
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How to draw the graph of two curves look like a lemniscate which satisfy $d_1d_2=b$ where $b<a^2$ and $b>a^2$ respectively?

Actually I was drawing the graph of a lemniscate with the help of a polar ordinate equation - $r^2=2a^2cos2\theta$ And for which I had to prove that the points lie on it would satisfy the condition $...
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Hirsch, Smale, and Devaney's "Differential Equations, Dynamical Systems, and an Introduction to Chaos" - polar coordinates

To study the system $$x' = −y +\epsilon x(x^2 + y^2)\\ y' = x +\epsilon y(x^2 + y^2),$$ the authors perform a change of coordinates, getting $$r'=\epsilon r^3\\ \theta'=-1.$$ So, they state that Thus ...
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Calculating linear speed of a dough hook stand mixer.

I'm trying to calculate the effective linear speed of the dough hook arm of a stand mixer, such as a KitchenAid mixer. If I can figure out the arc length I think I can get the speed easily enough. I ...
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Iso-parametric Dido problem in polar coordinates

How to find maximum area arc between two points with fixed length, using polar coordinates? I know that I can use Lagrange multipliers, but I cannot integrate the Euler-Lagrange equations. Is there ...
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Prove that doesn't exist a Lyapunov function for and ODE in polar coordinates

Given the following ODE in polar coordinates $$ \dot{r} = r^2 sin^2(\frac{1}{r}) $$ $$ \dot{\theta} = 1 $$ Show that the origin is Lyapunov Stable Prove that doesn't exist a Lyapunov function ...
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Obtaining transfer function from polar plot

Suppose we are given the following closed loop system: We know that $G(s)$ has only one zero which is at $s = -1$. Also, when $K = 8$, the closed loop system displays sustained oscillations with ...
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Determine area bounded by $(x+y)^4 = ax^2y$

Determine the area of region $U$ bounded by the graphic of the curve: $$(x+y)^4 = ax^2y,\ a > 0 \quad \text{(loop in the first quadrant)}$$ I used polar coordinates and I arrived to: $$\iint r\,dr\,...
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Spiral enlargements on desmos

If you toggle the slider for n the point moves in a spiral. Is there a way to plot the curve for all values of n? https://www.desmos.com/calculator/kmdmfazsyy
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Rotating polar velocity vector fields

There is a great way to rotate a Cartesian vector field about the origin described in Rotating vector functions. Instead, let us suppose that we have a velocity vector field in polar coordinates i.e., ...
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Is the origin included in polar half-lines/radial lines?

In the complex plane, $\arg(z)=\alpha$ defines a half-line starting at the origin at an angle $\alpha$ from the positive real axis, however the origin itself is not included in the half-line as $\arg(...
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Column vectors in polar coordinates?

So, we can represent the Cartesian vector $r= x\hat{x}+ y\hat{y}$ as the column vector where the entries are: $$\begin{bmatrix} x\\ y\\ \end{bmatrix} $$ How would the polar coordinates position ...
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Slope of Tangent line of Polar Curve at Point

Question We had an examination today in college and the question was this: Find the slope of the tangent line of polar curve $r = 3(1-\cos\theta)$ at point $B(\pi/3, 3/2 )$. Answers are: $2\pi$, $\pi/...
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How to write the region in simple Cartesian form

The polar equation of the annular is $$1 \leq r \leq 2, \pi/4 \leq \theta \leq 5\pi/4$$. How to write the region in simple Cartesian form. Clearly, it is like: $$D=\{(x,y)| 1\leq x^2+y^2\leq 4, ......
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Line integral over a lemniscate verification

The task is to find $\int_L {x\sqrt{x^2-y^2}}$ds, where $L: (x^2+y^2)^2 = a^2 (x^2 - y^2),\ x\ge0$. The curve is a right loop of a lemniscate. By tranistioning to polar coordinates I found: $$r^2 = a^...
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Computing the length of the polar curve $r = \frac{2}{\theta}$ using a hyperbolic substitution

$$\begin{align} L&:=\int_{\theta_1}^{\theta_2}\sqrt{\left({\mathrm{d}x\over\mathrm{d}\theta}\right)^2+\left({\mathrm{d}y\over\mathrm{d}\theta}\right)^2}\,\mathrm{d}\theta \\\\r&:={2\over\theta}...
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Polar coordinate differentials

We all know that when switching to polar coordinates we have: $$ x = r\cos\theta,\qquad y = r\sin\theta $$ and either by a geometric argument or using the Jacobian we have: $$ dx\,dy = r\,dr\,d\theta $...
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Find $\lim\limits_{(x,y)\to(0,0)}\frac{x^6+y^6}{x^3+y^3}$ using the $\epsilon-\delta$ definition?

My textbook asks the question $$f(x,y) = \frac{x^6+y^6}{x^3+y^3}$$ Does $f(x,y)$ have a limit as $(x,y) \rightarrow (0,0)$? I used polar coordinates instead of solving explicitly in $\mathbb R^2 $, ...
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Area inside the cardioid $r=2+2\sin\theta$ and outside the circle $r=1$

Simply I saw a friend asking about the area inside the cardoid $r=2+2\sin \theta$ and outside the circle $r=1$ and I couldn't help. I know that the area is equal to $$ \int_{a}^{b} \frac{1}{2}((2+2\...
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A curious limit: area between a regular n-gon with side length 1 and an inscribed curve with the same perimeter

A curve $S$ with polar equation $r=a\cos{(n\theta)}+b$ is inscribed in a regular n-gon with side length $1$. $S$ touches the midpoint of each side of the n-gon. $S$ has the same perimeter as the n-gon....
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Integral of $\frac1{|x|^3}$ on a circular segment

How can I evaluate $$\int \int _C |(x,y)|^{-3} dx dy$$ Where $C$ is the part between the chord $AB$ and the arc $AB$, and $|(x,y)| = \sqrt{x^2 + y^2}$? The radius of the circle is $R$. I tried using ...
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2 answers
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What is the cartesian equivalent of $r=0$?

The question I have asks to convert $$\int_{\pi/4}^{\pi/2} \int_{0}^{2/\sin{\theta}} r^{3/2} dr d\theta$$ I think I understand how to do most of it: $r=\frac{2}{\sin{\theta}}\implies r\sin\theta=2 \...
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Converting a double integral from cartesian to polar coordinates

Question: The integral $$\int\limits_0^3 \int\limits_{|x|}^{\sqrt{18-x^2}} \sqrt{x^2+y^2} dy\ dx$$ has the following form in polar coordinates My work so far: $y=\sqrt{18-x^2} \implies x^2+y^2=18 \...
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Show that $ \frac{x^2}{(x^2+y^2)^{3/4}} $ is continuous on $\mathbb{R}^2$.

I tried with polar coordinates, I found the limit is infinite (which is not true the graph show it's 0). I tried to majorate with something who has for limit 0 but still impossible. with polar ...
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14 votes
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Why does it seem more natural to think of $r$ as a function of $\theta$, rather than the other way around?

When teaching functions in polar coordinates, the nearly universal practice is to consider functions of the form $r = f(\theta)$. I think I have never seen any examples in which $\theta$ is expressed ...
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Limit of 2 variables: two similar cases with different outcomes

thanks for your time and excuse me if the english is bad, it's not my first language. Practicing for calculus 2 I found these two similar limits which have a very different result and I can't figure ...
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6 votes
3 answers
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Why does this cycle of 44 show up in the Collatz Conjecture?

Consider this function: $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{...
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How can I tell which function of two variables is larger?

In this case, $z = 1$ and $z = \sqrt{x^2 + y^2}$. How can I tell which function is bigger to choose the upper and lower bound?
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2 answers
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$\iint_Sz^2d\sigma$ where $S$ is an area of the cone $z=\sqrt{x^2+y^2}$ between planes $z=0$ and $z=1.$

What is the value of $\iint_Sz^2d\sigma$ where $S$ is an area of the cone $z=\sqrt{x^2+y^2}$ between planes $z=0$ and $z=1.$ To solve this by using polar integration I think the integral transforms ...
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Area which is enclosed by a function of $~r^2=a^2\sin(2\theta)~~(a>0)~$

Find the area enclosed by the following curves. $$r^2=a^2\sin(2\theta)~~~(a>0)$$ To hold the above identity, we want to determine the range of $~ \theta ~$ $$0\le2\theta\le\pi\iff \underbrace{0\le\...
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(-3,4) to polar coordinates

I tried to convert $(-3,4)$ to polar coordinates and did the following to calculate it. \begin{align*} r^2 & =x^2+y^2\\ r^2 & =9+16\\ r& =5 \end{align*} I got the value of $r$ correct, but ...
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3 votes
3 answers
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Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{x^2}{\sqrt{x^2 +y^2}} = 0$ without substituting

I found this possible solution: Let $r^2=x^2 + y^2, x = r \cos(\theta)$ and $y=r\sin(\theta)$. Then: $$ \begin{split} \lim_{(x,y) \to (0,0)} \frac{x^2}{\sqrt{x^2 +y^2}} &= \lim_{r \to 0} \frac{(r ...
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What is the unit vector $\hat{\theta}$ in polar coordinates?

Vector addition is defined algebraically as the sum of vector components, and it's usually taught geometrically by drawing two little arrows, placing them head to tail, with the second arrow's tail at ...
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Find the area of the region that lies inside $r=\sin(2\theta)$ and outside of $r=\cos(2\theta)$ [closed]

Find the area of the region that lies inside $r= \sin(2\theta)$ and outside of $r=\cos(2\theta)$ using polar coordinates. Generally, I could use help setting up the integral in order to solve for the ...
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double integral showing equal to 1

The question asked is as follows: Given that the nonnegative function $g(z)$ has the property that $\int_{-\infty}^\infty g(z)dz=1$ show that $f(x,y)=\frac{g(\sqrt{x^2+y^2})}{\pi\sqrt{x^2+y^2}}$ for ...
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Volume above a cone and within a sphere, using triple integrals and cylindrical polar coordinates

Consider the region above the cone $z = \sqrt{x^2+y^2}$ and inside the sphere $x^2+y^2+z^2 = 16$. Use cylindrical polar coordinates to show that the volume of region $R$ is $\frac{64\pi}{3}(2-\sqrt{2}...
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Contravariant Vector Component Transformation from Polar to Cartesian

I am new to tensors and I just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention) $$A^{'i}=\frac {\partial x^{'i}}{\partial x^j}...
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1 answer
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Show the system has one equilibrium point

I was wondering how we would show that the system: $$\frac{dx}{dt}=-x^3+2x-4y \\ \frac{dy}{dt}=-y^3+2y+4x$$ has only one equilibrium point. I have seen cases where the system is, for example: $$\...
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How does the ellipsis $x^2+2y^2=2$ gets represented to $x=\sqrt{2}\cos\theta; y= \sin(\theta)$ in polar coordinates?

Just like the title says, How does the ellipsis with equation $$x^2+2y^2=2$$ becomes represented as $$x=\sqrt{2}\cos\theta; y= \sin(\theta)$$ in polar coordinates? can someone help me to understand ...
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Express Laplacian in polar coordinates

Part of this problem is asking to express $u$ in polar coordinates and express the domain and BCs to those coordinates. The PDE is the Laplacian on disc with BC $u=0$: $\Delta u+\lambda u=0, \quad$ ...
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Length of a line to the surface of a sphere when line does not originate from center

Finding Length of a line to the surface of a sphere when line does not originate from center. Here is what I know: I have a 3D sphere of radius $7$ I have a line (...
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Oddity in the change of volume due to a change of coordinates

I was thinking about coordinates change and, using polar coordinates on a 2D plane, i found an incoherence in my equations $$x=r cos \theta \\ y=r sin \theta$$ from wich $$dx=cos\theta dr-rsin \theta ...
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Finding roots of a polar trigonometric equation

I have an equation which I am using to describe the squared distance from a polar point $(r_1, \theta_1)$ to a function $g(\theta)$ which is $C_1$ smooth over the period $0-2\pi$. $$r = r_{1}^{2}+g\...
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Why does Spivak claim there's a definable tangent line through $(0,0)$ for the graph of the polar coordinates described by $f(\theta)=|\cos(2\theta)|$

There is a problem in the Chapter 12 Appendix of Spivak's 4th Ed. Calculus (Problem 6b) that asks the reader to consider the tangent lines of different graphs of polar coordinates. For the polar ...
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Double and triple integral in spherical polar coordinates, $\iint\sqrt{4-x^2-y^2}\mathrm{d}A$

Consider the double integral $$ I=\iint\limits_{\mathcal D} \sqrt{4-x^2-y^2}\mathrm{d}A $$ where $\mathcal D=\{(x,Y): x^2 + y^2 \leq 4\}$ is the disc on the $xy$ plane (source) A.) Use polar ...
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