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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How to find polar values of complex number as quick as possible?

I need to calculate these kind of values in exams in best speedy way. Convert $1.46 + 3.17j$ to polar form ($r∠θ$) Is there is any solution to find of the values as quick as possible? By the way, ...
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39 views

Reference for studying polar coordinate

There is a theorem about justification of polar-coordinate in Folland-Real analysis p.78. I find it somewhat terse (Maybe it's just me).. I guess this kind of transform is possible even when ...
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97 views

Cartesian vector field to vector field

Ok so I have a given vector field in Cartesian coordinates, say \begin{align*} \textbf{v}(x,y)=\frac{dx}{dt}\hat{\textbf{e}}_{1}+\frac{dy}{dt}\hat{\textbf{e}}_{2} \end{align*} Where $dx/dt$ and ...
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51 views

Rectangular transformation into Polar coordinates

I was working with a simple transformation of rectangular coordinates - symmetry around the y-axis, i.e. $$f(x,y) = (x, -y)$$ I wanted to express the identical concept in polar coordinates. After ...
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53 views

Polar equation and Cartesian equation

For the polar equation, $r \sin \theta = \ln r + \ln (\cos\theta)$ Is that equivalent to $y = \ln x $ ?
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52 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
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112 views

Change to polar coordinates when evaluating limits of functions in two variables?

I have a function in two variables $f(x, y)$ and need to calculate the limit $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} .$$ If I decide to change to polar coordinates, how can I determine where $r$ ...
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299 views

Sketch the polar graph r=2+cos(theta). Find the points of intersection, if any, of this graph with the straight line y=2x-1 (use two decimal places)

I have already sketch the polar graph. and I have to find this graph's intersection point with the straight line y=2x-1 so, I try to solve it like this way: y=2x-1 Rsin(theta)=2Rcos(theta)-1 ...
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75 views

Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
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128 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
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71 views

Velocity of a particle in polar coordinates

The equations $r = 3\sin(2\theta)$ and $\frac{d\theta}{dt} = 2$ describe the motion of a particle in polar coordinates. Find the velocity of the particle in terms of the unit vectors $u_r$ and ...
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16 views

The Set of Closed Curves Representable by $r(\theta)$.

I apologize in advance if my terminology and/or notation is inaccurate; I am a little out of my depth here. If something is unclear, please point it out and I will try to explain myself better. ...
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41 views

Is it possible to change the pole and/or the polar axis in a polar coordinate system?

Citing Wikipedia's article on polar coordinates... In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance ...
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240 views

how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...
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57 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...
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34 views

Integrals in polar coordinates

Polar and spherical coordinates seem very useful for areas, however I don't understand why I can't seem to keep a direction after a spherical integral. In Cartesian coordinates, it's very easy to ...
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861 views

Area that lies inside both curves: $r=sin2\theta, r=cos2\theta$

My integral is setup as: $$A=8\int_0^\frac{\pi}8{\frac12sin^22\theta}\space d\theta - 8\int_{\frac{\pi}8}^0{\frac12cos^2 2\theta}\space d\theta$$ $$=8\int_0^\frac{\pi}8{\frac12sin^22\theta}\space ...
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416 views

Find the cartesian equation of: $r=2\cos\left(\frac {3\theta}{2}\right)$

I've managed to use identities to simplify it down to: $$r = 2\left(\cos^3\left({\theta\over2}\right)-3\sin\left({\theta\over2}\right)\cos\left({\theta\over2}\right)\right)$$ using trig identities, ...
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72 views

Hankel transform and Laplacian in cylindrical coordinates

My book solved a PDE containing the Laplacian in cylindrical coordinates. It doesn't really explain why the Hankel transform is useful in this case (symmetries etc..); just brute force math. So yeah, ...
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Derivatives of polar coordinates

I've got a problem for which I'm trying to calculate $\ddot r$. The problem is right here for the sake of reference. So far, I've got that: $$\ddot r=\frac{d}{dt}v_r=\frac{dv_r}{dr}\frac{dr}{dt}$$ ...
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95 views

Help with Polar coordinates and the length of the curve.

I have a test coming up today and I was going over our past midterms and this question came up. I tried it but its not working, please any hints or solution in how to do it will be really helpful. ...
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33 views

Polar Integral Confusion

Yet again, a friend of mine asked for help with a polar integral, we both got the same answer, the book again gave a different answer. Question Use a polar integral to find the area inside the ...
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29 views

Polar partial derivatives continuously differentiable implies holomorphic

I need to show that if $f(re^{i\vartheta}) = U(r,\vartheta) + iV(r, \vartheta)$ and $U,V$ are continuously differentiable and satisfy the Cauchy-Riemann equations, then $f$ is holomorphic. I am ...
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48 views

Can I solve for the fractional volume of a hyperboloid?

This looks like a homework problem because it is. I'm stuck at the portion where I solve for fractional volumes. Suppose you are a part of a team designing a water tank in the shape of a hyperboloid. ...
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63 views

polar coordinates, finding tangents

I've been asked to find the coordinates of the points on the curve: $r = acos2\theta, -\dfrac{\pi}{4} \leq \theta \leq \dfrac{\pi}{4} $ where the tangents are parallel to the initial line. The ...
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205 views

How to map sphere to faces of an Icosahedron

This is the mathematics behind some graphics I am trying to build in OpenGL. I believe the question belongs here. I want to represent an approximate sphere (let's say the Earth) as an icosahedron and ...
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48 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
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79 views

Polar coordinates integral

The exercise is Evaluate the double integral of the function $f(r, \phi) = r$ in the domain limited by cardioid $r = a(1 + \cos(\phi))$ and the circle $r = a$ If $T$ is the domain, I want $$\int ...
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1k views

How to convert a circle off origin to polar coordinates for integration?

I am trying to find the surface area of $x^2+y^2+z^2=a^2$ over the region $x^2 +y^2 \leq ax$. I rewrote the region as $\left(x-\frac{a}{2}\right)^2 + y^2 \leq \frac{a^2}{4}$. This is where I am stuck. ...
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Integration of figure whose base is a quarter circle not centered at origin using polar coordinates

How do I integrate $$ \int_{1}^{2}\int_0^{\sqrt{2x-x^{2}}}\frac{1}{\sqrt{x^2+y^2}}dydx $$ using polar coordinates? The base is a quarter circle of radius 1 centered at (1,0), so my first instinct ...
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42 views

Polar-cartesian conversion

Need some help here: Transform the polar equation to rectangular coordinates and identify the curve represented by $$r = \sin t + \cos t$$ What I've done: $$r^2 = r\sin t+r\cos t $$ $$x^2+y^2 = x ...
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101 views

How can I derive differential equations for polar coordinates based on these equations?

A textbook I am using on my own to study differential equations contains a problem: given the two differential equations for $x,y$ below and a real value of $t$, derive the differential equations for ...
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229 views

Conversion of a complex number into polar form

Below is the complex number that is to be converted into Polar form. I'm facing problem in second part of this number(after + mark not the (b) itself). When I divide them(10/-5+j12) directly, by ...
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43 views

Converting polar equation into cartesian equation to obtain derivatives

If I have a polar equation such as $r=1-2\cos(\theta)$ How would I convert this into an equation for $x$ and $y$ so that I can get $dx/d\theta$ and $dy/d\theta$ ?
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Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and ...
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91 views

Changing a double integral into polar coordinates

I have the double integral $\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)} dydx$ And I am asked to evaluate this by changing to polar coordinates. I know the transformations are, $x=r ...
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256 views

Polar form of quadratic equations

I'm trying to derive a polar, general and graphing, form of a quadratic equation. Here Is what I've done so far. $$ f(x)=ax^2+bx+c $$ And $$ f(x)=a(x-h)^2+k $$ Then I substituted $$ x=r\cos(\theta) ...
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72 views

$x² + y² +6y = 7$ to polar coordinates

How do i come from $M1$ to the polar coordinates? $$M1 \qquad x^2 + y^2 +6y = 7$$ I started with: $$r^2 * (\cos^2\varphi + \sin^2\varphi)+ 6r\sin\varphi = 7$$ because: $$(\cos^2\varphi + ...
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90 views

translate coordinates on circle to percentage?

I'm coming more from a programming point of view but the question is pure math. The only strange thing, I guess, is that the coordinate system is like this: ...
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5k views

Cannabis Equation

How can an equation for the following curve be derived? $$r=(1+0.9 \cos(8 \theta)) (1+0.1 \cos(24 \theta)) (0.9+0.1 \cos(200 \theta)) (1+\sin(\theta))$$ (From WolframAlpha)
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Write function from polar to rectangular coordinates.

I need to write this functions in rectangular coordinates: $$f(r,\theta)=r^{2k+5}\cos5\theta$$ $$g(r,\theta)=r^{2k+5}\cos5\theta$$ Of course the radius is very easy to convert to $x$ and $y$. The ...
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114 views

Bounding an integral over a surface (using polar coordinates?)

Suppose $S$ is smooth $n-1$-dimensional closed and bounded (compact) hypersurface in $\mathbb{R}^n$. Suppose for simplicity that $S$ is the boundary of a Lipschitz domain, for example. Whatever makes ...
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229 views

What is the area outside of $r=1$ and inside $r=2 \cos(3 \theta)$?

What is the area inside the polar curve $r = 2 \cos(3 \theta)$ but outside of the circle given by the polar equation $r = 1$? A picture of the polar curve is at ...
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777 views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
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Find the area of the region R inside the circle $ r=2 \cos \theta$ and outside the cardioid $r=2− 2 \cos \theta$ .

I got $\theta\pi/3$ and $5\pi/3 $ and then the area I got was $-4\sqrt3-(8\pi)/3$ The area is not right, I used the area equation that takes integral of $1/2(f(\theta)^2-g(\theta)^2)$ from $\pi/3$ ...
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238 views

Changing from Cartesian coordinates to Polar coordinates

Rewrite the iterated integral $$\int_0^1 \int_0^{\sqrt{2y - y^2}} (1 - x^2 - y^2)\,dx\,dy$$ in polar coordinate form. Do not evaluate the integral. Here is my answer: ...
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123 views

Find the area enclosed by the curve $x=t-\sin t $, $y=1-\cos t$ from $0$ to $2\pi$

Find the area enclosed by the curve $x=t-\sin t $, $y=1-\cos t$ from $0$ to $2\pi$ and the x-axis Not sure how to execute. is it just that $.5\int_{0}^{2\pi}x^2+y^2$ ? and not sure how to account ...
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66 views

Problems with integrals, polar coordinates.

I am having problems parameterizing these integrals: $$\int_A{\frac{x}{1+x^2+y^2}}\mathrm{d}x\mathrm{d}y$$ for $A = \mathbb R^2 \bigcap \,\{y \ge 0\}$ and the volume of $M = \{(x, y, z) \in \mathbb ...
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62 views

Find the area inside a polar curve

I feel a bit silly asking this question as it is no doubt relatively simple, but it has been bugging me. Given the polar curve described by $r^2 = cos(2\theta)$, find the area inside the curve. My ...
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2k views

Why does $r=cos\theta$ produce a circle?

I am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta=-\pi/2\quad to\quad \theta=\pi/2\\r=0\quad to\quad ...