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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1answer
32 views

convert equation from polar coordinate to cartesian coordinate

I have the following equation $$r= \frac{A}{\log\left[B\tan\left(\frac{\theta}{2N}\right)\right]}$$ For using an optimization program, I would like to have this equation in cartesian coordinate ...
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0answers
19 views

Polar coordinates for vector difference in $\mathbb{R}^2$

I have a function $F(\boldsymbol X)=\tilde F(x,y)$ of $x$ and $y$ in the plane, and I can transform it in a function of $r$ and $\theta$, say $f=f(r,\theta)$, through the change of coordinate $$x=r ...
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1answer
48 views

Making sense of polar coordinates transformation on the derivatives

I would like to make sense of the transformation of the differentials in polar coordinates (to fix the ideas). To be more precise, the "right" way to find the transform for the differential and the ...
3
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1answer
40 views

$\int_{\mathbb{R}^{n-1}} \frac{1}{(y_1^2 + y_2^2 +…y_{n-1}^2 + C^2)^\frac{n}{2}} dy = \frac{n\alpha(n)}{2C}$

$$\int_{\mathbb{R}^{n-1}} \frac{1}{(y_1^2 + y_2^2 +...y_{n-1}^2 + C^2)^\frac{n}{2}} dy = \frac{n\alpha(n)}{2C}$$ where $\alpha(n)$ is the volume of the unit ball in $\mathbb{R}^n$. Could anyone help ...
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0answers
75 views

Approximate Laplace Operator with Central Difference in Polar Coordinates

I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel ...
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0answers
18 views

Polar-Cartesian Plot Interconversion

The Question How can we interconvert any general graph of polar or cartesian function so they give the same plot in the other coordinate system. Note: I do not mean to ask about interconverting a ...
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0answers
19 views

Expressing Curvature of a Polar Function in Terms of its Derivatives

I could use a little guidance with this question: Consider the curve $r=f(\theta)$, where $f$ is any twice differentiable function. Determine an explicit formula for the curvature $\kappa$ in terms ...
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0answers
33 views

How to convert this polar equation to cartesian?

$r=\cos \left(\frac{13}{7}\theta \right)$ When I try I get this: $\left(x^2+y^2\right)^{.5}=\space \cos \left(\frac{13\space }{7\space }\arctan \left(\frac{y}{x}\right)\right)$ But that doesn't seem ...
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4answers
91 views

Conversion from Polar to Rectangular

Can someone please explain to me how to convert the following equation from polar to rectangular? r=$2^\theta$ Thus far I got: $4^{\arctan(y/x)}$=$x^2$+ $y^2$ by squaring both sides and ...
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2answers
107 views

Area of Bernoulli Lemniscate?

Can anyone help me calculate this area? I have to use double integrals, and the question sounds like this: "Calculate the area bounded by the curve $(x^2+y^2)^2=a^2(x^2-y^2)$, where $a$ is a real ...
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0answers
31 views

Is there anything wrong with my solution?

Finding the arc length in polar form. $r = 1 - \cos\theta$ $0<=\theta<=2$ using the formula $L =\int_a^b\sqrt{r^2+(\frac{dr}{d\theta})^2}d\theta$ $L ...
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2answers
75 views

formula for logarithmic spiral on a linear level

I am trying to plot the contents of a circle, which include geometric elements and spirals, on a linear graph. For example, take a circle, take the beginning and the end and make it straight. What ...
0
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1answer
164 views

Volume of Solid Region Between Sphere and Paraboloid

"Find the volume of the solid region above the sphere $x^2+y^2+z^2 = 6$ and below by the paraboloid $z = 4-x^2-y^2$" I am, of course, going to be solving this double integral by converting to polar ...
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2answers
39 views

Evaluate 2D integral (by change of variable)

The question asks to evaluate integral $$\iint_D \Big[3-\frac12( \frac{x^2}{a^2}+\frac{y^2}{b^2})\Big] \, dx \, dy \ $$ where D is the region $$\frac{x^2}{a^2}+\frac{y^2}{b^2} \le 4 $$ I believe ...
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0answers
39 views

Calculate the flux coming out of a surface

Let F(x,y,z)=(2xy(z-2),x^2(z-2),x^2y) be a vector field and $\Sigma $ the surface defined as the portion of cone x^2+y^2=(z-2)^2 ...
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0answers
79 views

Stability of dynamical system described in polar coordinates

Near a fixed point, a dynamical system $\dot{\bf{x}}=\bf{F}(\bf{x})$ can be approximated by $\dot{\bf{x}}=A\bf{x}$, where $A$ is the Jacobian matrix. From the trace and determinant of the Jacobian ...
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1answer
26 views

Writing same equation in different forms

I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $ My question would be if i have a circle ...
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0answers
51 views

How to numerically determine whether a curve on a polar coordinate graph is dominant over another curve?

Given 2 curves appearing on a 2-dimensional polar coordinate graph across multiple quadrants having the same range of x-axis endpoints, how do you numerically determine the dominance of 1 curve over ...
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1answer
198 views

Find all polar coordinates of point $P$ where $P = (7, \pi/3)$.

I don't know where to go from here. Answer choices are: a) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + 2n\pi)$ b) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + (2n + 1)\,\pi)$ c) $(7, \pi/3 + (2n + 1)\,\pi)$ ...
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1answer
74 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
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0answers
24 views

Coordinates in file isn't in the range -180 to 180 respective -90 to 90?

Hello! I'm creating an android version of a PC program (I've contacted the complany who owns the PC program, so it's legal). The program is in the core a GPS, but is used to navigate pre-defined ...
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4answers
189 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
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1answer
95 views

How do I find the area shared by the circles $r = 2\cos(\theta)$ and $r = 1$?

I figured out the intersection points: $r=2\cos(\theta)$, $r=1$ $2\cos(\theta) = 1$ $\cos(\theta) = \frac{1}{2}$ $\arccos(1/2) = π/3$ (I), $5π/3$ (IV)
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1answer
63 views

Introducing $\mathrm π$ and polar coordinates in real analysis

From time to time, I think about how material from introductory courses like real analysis or linear algebra can be structured in a way I would have liked to see in my freshman days. So recently, I ...
2
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2answers
128 views

Convert the Polar Equation to Cartesian Coordinates

$$ r^2=\sec 4\theta $$ I graphed this equations using Wolfram Alpha and found it to be 2 hyperbolas. I'm having difficulty showing this using the standard equations $$ x=r\cos\theta \;, \; ...
0
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1answer
52 views

Finding the area enclosed by 4 functions using polar coordinates

I need to find the area enclosed by $x^2+y^2$ = 4x, $x^2+y^2$ = 2x, y=x and y=0. How do I use polar coordinates here? It seems to me that representing those functions using polar coordinates is too ...
1
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1answer
49 views

Cannot find link between trigonometric statements and reduced form

I have been trying to find a way to reduce following trigonometric statements to the reduced form below, but without succes. I haven't been able to grasp the typical train of thought I presume I would ...
1
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1answer
41 views

Polar coordinate double integral

I have to integrate the following integral: $$ \iint \limits_A sin({x_1}^2 + {x_2}^2) dx_1dx_2 $$ over the set: $A=\{x \in \mathbb{R}^2: 1 \leq {x_1}^2 + {x_2}^2 \leq 9,x_1 \geq -x_2\}$ I ...
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1answer
26 views

Find the area enclosed by curve with polar coordinates

I am having a little difficulty finding the area enclosed by the curve, $r(\theta) = 4 + sin\theta + cos\theta$ with $0 \le \theta \le 2\pi$. I tried integrating over $0 \le \theta \le 2\pi$ and $0 ...
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3answers
86 views

Polar Integration of $ r = 2\cos(\theta)$

$ r = 2\cos(\theta)$ has the graph I want to know why the following integral to find area does not work $$\int_0^{2 \pi } \frac{1}{2} (2 \cos (\theta ))^2 \, d\theta$$ whereas this one does: ...
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0answers
17 views

Polar coordinates used to evaluate a function containing a branch cut

I'm having a lot of trouble understanding how to approach these kinds of problems, if anyone could explain the approach, it would be really helpful. The problem is as follows: The function $f(z)$ is ...
1
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1answer
65 views

Two-dimensional limit, is my approach correct?

The limit is $$\lim_{(x,y)\to(0,0)}\frac{x^3y}{x^4+y^2}$$ As usual, I tried checking along particular paths, namely the axes and the curves $y=mx^n$ for various values of $n$, but to no avail; all the ...
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0answers
21 views

From what source should I learn about analytic functions given in polar coordinates?

In the Calculus 1 course that I am currently taking, we only discussed functions given in polar coordinates as some sort of side note, but I am eager to explore them more thoroughly. Namely, what I am ...
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1answer
52 views

Polar coordinate system : Is radial coordinate is a function of angular coordinate?

In polar coordinate system: The polar coordinates $r$ is called the radial coordinate and $\theta$ is called the angular coordinate, often called the polar angle. I am confused when answering the ...
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2answers
69 views

Write ODE in Polar Coordinates [closed]

I want to write this ODE system in polar coordinates (r,$\theta$). $$\dot x =x-y-x^3 $$ $$\dot y = x+y-y^3$$
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3answers
55 views

Real and imaginary part of $ (1-i\sqrt{3})^6$

i am a bit stuck here. As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$ I already found out that the ...
0
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1answer
37 views

When looking at motion in a circle, why do they say that $ r \dot{\theta}$ is transverse velocity when it doesn't look like it is a vector?

In my lecture notes it says that $r \dot{\theta}$ is called the transverse velocity of a particle if it is travelling in a circle. What I don't understand is why this is called a velocity when neither ...
0
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2answers
34 views

Two variable limit

Suppose I have a function which is defined in different parts, for example: $$f(x,y)=y\cos\left(\frac{x}{y}\right)\ \ \ y\neq0$$ $$f(x,0)=0$$ and I have to calculate the limit when $(x,y)\rightarrow ...
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0answers
46 views

Inversion of Rose Curve in Unit Circle

The inversion of a polar curve r(t) in the unit circle is given by 1/r(t). A rose is a polar curve defined (eup to similarity) by an equation of the form: r(t) = cos(nt) or r(t) = cos(p/q t) Does ...
3
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2answers
56 views

Why does the radius come before the angle?

Based on my understanding, when delineating two variables (for a coordinate system or otherwise) convention is to label the 'independent variable' first, then the 'dependent variable'. So for a ...
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0answers
52 views

Find the flux through a closed volume with the divergence theorem and using the definition

Given the vector field F(x,y,z)=(xy,xy,z) and $D= \{(x,y,z) \in R^3 : x^2 + y^2 + z^2 \le 4, x^2 + y^2 \le 1, z\ge 0 \}$ Find the flux through ...
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1answer
56 views

arc length of the polar curve $r^2= \sin2\theta$

given curve is $r^2 = \sin2\theta $ I got $L= \int_0^{2\pi} \sqrt{r^2+ ({\dfrac{dr}{d\theta}})^2}\ d\theta$ = $\int_0^{2\pi} \sqrt{\dfrac{1}{r^2}} d\theta = \int_0^{2\pi} ...
0
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1answer
426 views

Area of the region inside $r = 1 - \cos(\theta)$ and also inside $r = \cos(\theta)$

Pretty simple polar integration question that I've been having trouble with... The question says it all. I identified the limits of integration by setting $1 - \cos(\theta) = \cos(\theta)$ so that ...
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2answers
41 views

Integrating exponential function with elliptic bounds

I am trying to integrate the following: $$\iint_R\exp\left(\frac{x^2}{4}+\frac{y^2}{16}\right)\:\mathrm{d}A$$ With the region $R$ having the bounds: $$\frac{x^2}{4}+\frac{y^2}{16}=3$$ ...
2
votes
2answers
85 views

Does the inverse function theorem fail for $\frac {\partial r}{\partial x}$

This is a follow-up to a question that I answered (though, not well enough). Why is it that $\frac {\partial r}{\partial x} = \cos(\theta) = \frac {\partial x}{\partial r} = \frac {\partial}{\partial ...
2
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2answers
42 views

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly.

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly. I know that we are looking for the smallest positive value ...
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4answers
39 views

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of y. Then indicate which one is the principle square root.

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of $y$. Then indicate which one is the principle square root. Use the polar form of complex numbers to accomplish this task. I'm not ...
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0answers
43 views

Polar coordinate for complicated curves

In general polar representation of a closed curve is done by coordinate $(\theta,r(\theta))$, $\theta\in (0,360)$. When working with real data, I got a closed curves whose graph looks like the below ...
2
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1answer
36 views

Convert $\frac{1+ \sqrt{3i} }{1- \sqrt{3i} }$ to polar form

How do I convert $\frac{1+ \sqrt{3i} }{1- \sqrt{3i} }$ to polar form? I came across it in this question but I don't know much about complex numbers and have no idea how to figure out $\theta$.
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1answer
45 views

Domain of a Bounded Archimedian Spiral???

So I have a question about a bounded Archimedian Spiral: In one context I get that an Archimedian Spiral's domain and range are all Reals. Thus if I'm looking at what appears to be a bounded spiral: ...