# 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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### How to find inverse of general curvilinear coordinates

Lets say I have a curvilinear coordinate system $A=A(x,y,z) = \frac{x^2+y^2+z^2}{2z}$, $B=B(x,y,z)= \frac{x^2+y^2+z^2}{2\sqrt{x^2+y^2}}$, $C=C(x,y,z)=\tan^{-1}(y/x)$ How do I find the inverse of ...
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### How do I derive the volume element $dV = \sqrt{g} du^1 du^2 du^3$ in a 3D curvilinear coordinate system?

I am trying to derive $dV = \sqrt{g} du^1 du^2 du^3$ for some general curvilinear coordinate $(u^1,u^2,u^3)$ system in $\mathbb{R}^3$ where $g = \mathrm{det}[g_{ij}]$. I am using the following facts:...
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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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### Find The Area of the Surface $z=xy$ for $x^2+y^2\leq 4$

This question comes under the topic of orthogonal curvilinear coordinates, but I am unsure how that topic relates to this question. How should I approach this question? This content will be covered ...
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### Finding curl$\vec F$ of the field $\vec F=f(\phi,\theta)e_\rho$

Find curl$\vec F$ in spherical coordinates for a vector field of the form $\vec F=f(\phi,\theta)e_\rho$. My query is that I have not seen a vector field in this form before. Can someone explain it to ...
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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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### Find the center of mass in 3D

Vector Calculus: [Using Integration] Find the center of mass of the "snow cone" of uniform density bounded above by the sphere $x^2+y^2+z^2=a^2$ and below by the cone $z=\sqrt{x^2+y^2}$.
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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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### 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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### 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 ...