03. Coordinate Transformation

Orthogonal Coordinate Systems

The distance between two points in one system is:

And in another coordinate system, where each of our coordinates is a function of some of its coordinates :

$dx_i=÷{∂}{∂x̄_k}(x_i(x̄))⋅dx̄_k$

$ds²=÷{∂}{∂x̄_k}(x_i(x̄))⋅dx̄_k⋅÷{∂}{∂x̄_l}(x_i(x̄))⋅dx̄_l$

$ds²=÷{∂}{∂x̄_k}(x_i(x̄))÷{∂}{∂x̄_l}(x_i(x̄))⋅dx̄_k⋅dx̄_l$

We introduce $g_{k,l}(x̄):=÷{∂}{∂x̄_k}(x_i(x̄))⋅÷{∂}{∂x̄_l}(x_i(x̄))$ . g is called the metric tensor.

$ds²=g_{k,l}⋅dx̄_k⋅dx̄_l$

Iff the metric tensor is diagonal, then the new coordinate system is orthogonal.

$g=\diag(U²,V²,W²)$

... in orthogonal coordinate systems:

$(∇Φ)_i=÷{(e⃗̄_1)_i}{U}⋅÷{∂}{∂x̄_1}(Φ)+÷{(e⃗̄_2)_i}{V}⋅÷{∂}{∂x̄_2}(Φ)+÷{(e⃗̄_3)_i}{W}⋅÷{∂}{∂x̄_3}(Φ)$

$(∇⃗∙a⃗)=÷{1}{U⋅V⋅W}⋅[÷{∂}{∂x̄_1}(V⋅W⋅a_1)+÷{∂}{∂x̄_2}(U⋅W⋅a_2)+÷{∂}{∂x̄_3}(U⋅V⋅a_3)]$

$∆(Φ)=÷{1}{U⋅V⋅W}⋅[÷{∂}{∂x̄_1}(÷{V⋅W}{U}⋅÷{∂}{∂x̄_1}(Φ_1))+÷{∂}{∂x̄_2}(÷{U⋅W}{V}⋅÷{∂}{∂x̄_2}(Φ_2))+÷{∂}{∂x̄_3}(÷{U⋅V}{W}⋅÷{∂}{∂x̄_3}(Φ_3))]$

$(rot v⃗)_1=÷{1}{V⋅W}⋅[÷{∂}{∂x̄_2}(W⋅v_3)-÷{∂}{∂x̄_3}(V⋅v_2)]$

$(rot v⃗)_2=÷{1}{U⋅W}⋅[÷{∂}{∂x̄_3}(W⋅v_1)-÷{∂}{∂x̄_1}(U⋅v_3)]$

$(rot v⃗)_3=÷{1}{U⋅V}⋅[÷{∂}{∂x̄_1}(V⋅v_2)-÷{∂}{∂x̄_2}(U⋅v_1)]$

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .