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Questions tagged [curvilinear-coordinates]

Use this tag for questions about coordinate systems for Euclidean space for which coordinate lines may be curved.

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Name of Non-Unique Coordinate systems

Are there some examples (and a name) for non-unique coordinates (non-unique meaning may have multiple ways to represent the same point). Such as the one below.
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Proof of formula for curvilinear curl

I'm trying to prove the formula for curl in curvilinear coordinates and am following the proof in the book "Vector Analysis and Cartesuan Tensors" by Bourne and Kendall. I got this far: -rewritting $...
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Identity in vector calculus : why is it zero?

From Spiegel's "Vector Analysis", problem 7.21 a): We have general curvilinear coordinates, $u_i$, with unit vectors $\mathbf{e}_i$ for $i=1,2,3$ and a vector $\mathbf{A}=A_1\mathbf{e}_1+A_2\mathbf{e}...
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How do I generalize the dot product (bilinear form) in spherical coordinates?

In cartesian coordinates, the unit vectors $\{u_x, u_y, u_z\}$ are universal. That is, $u_x(x, y, z)$ is constant and so on for the rest of them. Because of that, the dot product $\langle v | w \...
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Scale factors in orthogonal coordinate systems

I'm trying to reduce the general tensor expression for either orthogonal or non orthogonal $$\vec{\nabla} \cdot \vec{V} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial x_i}\left(\sqrt{g}V^i \right)$$ ...
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Transform vectors from cartesian coordinates to curve coordinate system [2d]

I have an object moving in a two-dimensional space and its position is given by cartesian coordinates $(x_i, y_i)$. This object also has a velocity vector $({v_x}_i,{v_y}_i)$ and an acceleration ...
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Given $\alpha(x,y,z) = \alpha_1\hat x +\alpha_2 \hat y+\alpha_3 \hat z$, how to write in the coordinates $\hat r, \hat \theta, \hat \phi$

I'm trying to convert a parametrization of a curve $$\vec \alpha(x,y,z) = \alpha_1(x,y,z) \hat x + \alpha_2(x,y,z) \hat y + \alpha_3(x,y,z) \hat z$$ to $$\vec \alpha(r, \theta, \phi) = \alpha_r(r, \...
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Question on the derivation of Laplace operator, and its application to polar coordinate system.

Let $\mathbf r = \left[ \begin{matrix} r & \phi \end{matrix} \right]^\top \; $be some curvilinear coordinates, with corresponding unit base column vectors $\hat {\mathbf h}_r \; $and $\hat {...
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Curvilinear Coordinates Transform matrix

In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_{ij}q_j$. Equation 4.100 lists $a$ as: $$a_{ij}=\frac{h'_i}{h_j}\frac{\partial q_i'}{\...
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Is this a correct definition of a Riemannian metric tensor?

I am a total beginner in Riemannian geometry but I'm trying to teach myself the basics. So the following could contain many horrible mistakes. Suppose I describe the x,y-plane by curvilinear ...
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Curvilinear abscissa = radius * angle - Circular motion

I would like to understand why: $$ s(t) = r \, \theta(t) $$ where $s$ is the curvilinear abscissa, $r$ the radius and $\theta$ the angle in circular motion. Thank you for your time.
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Del operator apply directly to orthogonal curvilinear coordinate does not match

I understand that $\nabla$ in general orthogonal coordinate $(u_1,u_2,u_3)$ as follows: $$ \nabla=\mathbf{a}_{u_1}\frac{\partial}{h_1\partial u_1}+\mathbf{a}_{u_2}\frac{\partial}{h_2\partial u_2}+\...
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Divergence in Orthogonal Curvilinear Coordinates - Is there a mistake?

I have to prove that, $$\nabla \cdot (A_1\mathbf e_1) = \frac{1}{h_1h_2h_3}\frac{\partial (A_1h_2h_3)}{\partial u_1} $$ My approach: $$\begin{align} \nabla \cdot (A_1\mathbf e_1) &= \nabla \...
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Benjamin's Othogonal Curvilinear Coordinate System to analyze Gas Velocity

I have been trying to understand a problem given in a paper for a couple of months but cannot figure out the rationale behind the change of variables of a function. This problem is outlined below. In ...
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213 views

Inverse of a (coordinate) transformation - which method is best?

I am woring with curvilinear coordinates and got stuck on an exercise where we have a transformation from cartesian to parabolic coordinates $(x,y,z)\rightarrow (u,v,\varphi)$. My question is how I ...