1
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2answers
33 views

Partial differentiation in transformed coordinates

Following lecture notes from MIT it says that, given some variable $A = A(x, y, z(x, y, r, t), t)$ where $r$ is a transformed vertical coordinate $\left. \frac{\partial A}{\partial x} \right|_r = ...
0
votes
1answer
22 views

How does the transformation $u=x+y$, $v=x/y$ transform the first quadrant?

How is the region $(x,y) \in [0,\infty] \times [0,\infty]$ transformed under the change of coordinates given by $$u=x+y$$ $$v=x/y$$ Would appreciate any hints on how to find the image of such ...
0
votes
1answer
73 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
1
vote
1answer
52 views

Using Cylindrical Coordinates to Compute Curl

I am given a vector field $\vec{A} = A_\rho \space \hat{e_\rho} + A_\phi \space \hat{e_\phi} + A_z \space \hat{e_z}$, and I am supposed to use the unit vectors (provided below) in cylindrical ...
1
vote
1answer
32 views

Deriving Cartesian Coordinates from Cylindrical Coordinates

The ans given was: $x = r \cos (\alpha)$ $y = h$ $z = -r \sin(\alpha)$ Could somebody explain to me how to arrive at the formula? I'm probably confused with the axes because usually, the $Z$ ...
0
votes
0answers
45 views

Derivative in non orthogonal coordinates

I am trying to transform irregular shape in common Cartesian coordinates ($x-y$) into a regular shape in a generalized coordinates(e.g.,$u-v$), in which the transform can be defined as $u=u(x,y)$ and ...
3
votes
0answers
65 views

Why does a figure look the same in every coordinate system?

After reading Maximilian M. Answer here: Gauss' Theorem - Can't understand a parameterization I'm trying to figure out why does a figure look the same in every coordinate system I choose. For ...
1
vote
1answer
38 views

Transforming from cartesian to cylindrical

Here is the question: Transform $\textbf{A} = \hat{\mathbf{x}} 2 - \hat{\mathbf{y}}5 + \hat{\mathbf{z}}3$ into cylindrical coordinates at point ($x=-2, y=3, z=1$). What I have tried is this: ...
0
votes
0answers
16 views

Parametrically defined Spheres in $R^n$

So I have 2 questions here which are closely linked: How do you parametrically define the circle $(x')^2 + (y')^2 = r^2$ using (x') and (y') as coordinates on the plane ax + by + cz = 0 that are ...
0
votes
0answers
72 views

Gradient definition of the unit vector

Given some coordinate system ($q_1,q_2,q_3$), it seems that the unit vectors $\hat{q}_i$ can be defined by $$\hat{q}_i = \frac{\nabla q_i}{||\nabla q_i||},$$ that is, the normalized gradient of $q_i$. ...
1
vote
3answers
77 views

How to find the integral by changing the coordinates?

Let R be the region in the first quadrant where $$3 \geq y-x \geq 0$$ $$5 \geq xy \geq2$$ Compute $$\int_A (x^2-y^2)\,dx\,dy.$$ I tried to use $ u= y-x, v= xy$ as my change of coordinates, but then I ...
1
vote
1answer
478 views

How to determine gradient of vector in cylindrical coordinates?

I am wondering how to actually determine the gradient of a vector in cylindrical coordinates. I have seen a lot of websites that just say what the general form is but I cannot seem to understand how ...
2
votes
1answer
171 views

Proper change of coordinates

It's a really easy one for you guys. I'm performing a simple cylindrical change of variable from Cartesian coordinates, but I want to write it out properly and I'm stuck with the differential ...
0
votes
0answers
49 views

Polar coordinates on a set T

This exercise show that f is a gradient on the set $$T= \mathbb{R}^2-\{(x,y)| y=0, x \leq 0 \}$$ consisting of all points in the xy-plane except those on the nonpositive x-axsis. If $(x,y) \in T$, ...
1
vote
1answer
26 views

Maximal region in the cylindrical space

I would like to determine a maximal region in $(r, \theta, z)$- space which maps injectively into $(x,y,z)$-space Thank you
3
votes
1answer
96 views

Unusual function format and its partial derivatives.

I came across a function of this format: $z = f(u,v)$ where $u = x^2y^2$ and $v = 5x + 1$ Because this function is not in the same format of the ones I've seen before (explicit or implicit), I don't ...
7
votes
0answers
420 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
1
vote
0answers
111 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
2
votes
0answers
252 views

Gradient when changing coordinate system

A change of variables from $\vec{r_1}$, $\vec{r_2}$ to $\vec{r}$, $\vec{R}$ is given by: $$ \vec{r} = \vec{r_1}-\vec{r_2}\text{ , }\vec{R}=c_1 \vec{r_1} + c_2 \vec{r_2} $$ I'm supposed to find ...
2
votes
1answer
59 views

gradient in polar coordinate by changing gradient in Cartesian coordinate

I'm tried to do following and I can't see what went wrong. $$\begin{bmatrix} \hat r\\ \hat \theta \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta ...
0
votes
1answer
218 views

Help needed with volume integral in Cylinder coordinates

Problem (more here and the problem XIV.6:6 here on page 976) Integrate $\int_A z dx dy dz$ in cylinder-coordinates when $$A=\{(x,y,z)\in\mathbb R^3 | x^2+y^2 \leq z \leq ...
0
votes
2answers
213 views

Help me with Cylinder -coordinates problem, back to Cartesian or not? How to do it fast?

Source of the problem, 3b here. Problem Question Electricity density in cylinder coordinates is $\bar{J}=e^{-r^2}\bar{e}_z$. Current creates magnetic field of the form ...
2
votes
1answer
3k views

Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
1
vote
1answer
343 views

Jacobian matrix normalization

I have a problem with normalization of the Jacobian matrix. There seems to be no clear method for doing it: in some literature, it has been normalized by using some characteristic length, which is ...
0
votes
1answer
335 views

transforming vector potential with a coordinate rotation

In electrodynamics, given the vector potential $\vec{A}$, the magnetic field is defined as: $\vec{B} = \nabla \times \vec{A}$ I'm having trouble figuring out how a coordinate transformation (a ...
0
votes
2answers
2k views

How to transform a Laplacian operator from (x,y) coordinate system to polar system?

for any functions in $C_2^2$, we have a $-D^2$ operator $-D^2u=-(u_{xx}+u_{yy})$ However now i need to transform this operator from $(x,y)$ to $(r,\theta)$ for some sphere boundary condition, how am ...