Questions tagged [curvilinear-coordinates]

Question 1

I am trying to follow this paper:https://accelconf.web.cern.ch/p03/papers/WPAB088.pdf and I would like to derive the Laplacian of a divergence-free vector field where the space curve has zero torsion ...

Question 2

In the theory of generalized (curvilinear) coordinates, usually one defines coordinates first, e.g. $(u^1,u^2,u^3)$, then a position vector $\mathbf{r}(u^1,u^2,u^3)$, and finally the basis vectors, $$ ...

Question 3

In the following, $U$ is a scalar field while $\mathbf A$ is a vector field. The symbol $\partial_r$ stands for $\partial/ \partial q_r$ where $(q_1,q_2,q_3)$ are curvilinear coordinates. Some well ...

Question 4

I’m working with Reynolds‐stress production tensor and I need to be sure whether my projection from Cartesian coordinates into a curved Frenet frame already accounts for the extra Christoffel terms (i....

Question 5

Let $\mathcal{M}$ be a Riemannian manifold. Let $(U,x)$ be the Riemann normal coordinates about some chart $U$. Let $(U,y)$ be some arbitrary coordinates. Define $D := x(U) \cap y(U) \subset \mathbb{R}...

Question 6

Consider relaxation of a vector field $\textbf{T}$ to the minimum of a free energy in a curvilinear coordinate with orthogonal basis and a metric $g_{ij}$. The free energy is defined as $F= \int dv f,$...

Question 7

I am trying to use the following formula for spherical coordinate to find the divergence of a velocity field. The formula reads $$\nabla_k u^k = (\frac{\partial u^k}{\partial x_k}+ u^j \Gamma_{kj}^i) ....

Question 8

Could some one post the formula (or a reference) on laplacian of a rank two tensor in curvilinear coordinate? I have spent hours looking for it with no success. in particular, I found two ...

Question 9

Given a (sufficiently nice) function $f(x,y) : \mathbb{R}^2 \to \mathbb{R}$, how would one find a function $g(x,y)$ such that $(f,g)$ form a 2-D orthogonal curvilinear basis (i.e. if the contour lines ...

Question 10

I am trying to understand the derivation of divergence in curvilinear coordinate systems. I stumbled upon this equation and it seems enigmatic to me. I know that $\mathbf{e}_{u_i}$ are the unit ...

Question 11

I am having trouble finding the Lamé coefficients for the following curves. I have specified all the necessary theory, however I am not sure how to practically compute it. $$ \begin{cases} x = \frac{...

Question 12

I'm interested in constructing a curvilinear coordinate system with coordinate axes being these two curves: $$x=a, \qquad y^2+x=b,$$ where $a, b$ are parameters that specify the families of the curve, ...

Question 13

In orthonormal curvilinear coordinate system, we define unit basis vector as $$\mathbf{\hat{e}}_u = \frac{1}{h_u} \frac{d\mathbf{r}}{du},$$ where $h_u = |\frac{d\mathbf{r}}{du}|$. Suppose we only know ...

Question 14

In a typical ODE course, we learn that families of orthogonal curves to parabola $y=Ax^2$ are given by families of ellipses, given by $x^2/2+y^2=c^2$. However, there is this thing called parabolic ...

Question 15

There are curvilinear coordinates with coordinate curves that look like rays and circles, i.e. polar coordinates, and there are curvilinear coordinates with coordinate curves that look like parabolas, ...

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Questions tagged [curvilinear-coordinates]

Vector Laplacian in Frenet–Serret coordinates with zero torsion

Calculating coordinates out of basis vectors

General formula for $\nabla^2 \mathbf A$ in curvilinear coordinates

Does Cartesian projection onto a Frenet frame include curvature corrections?

Is this a possible definition for the Levi-Civita connection coefficients?

inconsistency in upper and lower index dynamics in differential geometry

using christoffel symbol gives a wrong divergence in spherical coordinate

Laplacian of a rank two tensor in curvilinear coordinate

Algorithm to extend function $f(x,y)$ to orthogonal curvilinear coordinate system

Understanding the Equation for Unit Vectors in Curvilinear Coordinate Systems

Finding Lamé coefficients (scale factors) for the curves

Can the two curve $x=a$, $y^2 + x=b$ form a coordinate system?

Is knowing unit basis vector enough to specify a coordinate system?

Families of orthogonal curves to parabola.

Curvilinear Coordinate with coordinate curves that look like cosine in 2D

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