Tagged Questions

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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 ...
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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 ...
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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 ...
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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 $Z$ ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 $q_i$. ...
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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 ...
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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 ...
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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 ...
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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 $(x,y) \in T$, ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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} + ...
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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 ...
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 ...
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 ...