Coordinate Systems in 3D

A small spherical increment: 1≤ρ≤1 + δρ, π/6≤ϕ≤π/6 + δϕ, π/4≤θ≤π/4 + δθ
            δρ = δϕ = δθ = 1/8  with a magnified view

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Derivation of  dV = |Det[(∂X[s, t]/∂r, ∂X[s, t]/∂s, ∂X[s, t]/∂t)]| dr ds dt

We partition r-s-t-space into smal rectangles and look at the image of one of them:

    δrδsδt-VolumeIncrement = { Ξ : Ξ = X[r + ρ, s + σ, t + τ], 0≤ρ≤δr, 0≤σ≤δs, 0≤τ≤δt}

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The edges of the volume increment are the following segments with their differential approximations:

    

X[r + δr, s, t] - X[r, s, t] = ∂X[r, s, t]/∂rδr + ε_r · δr
X[r, s + δs, t] - X[r, s, t] = ∂X[r, s, t]/∂sδs + ε_s · δs
X[r, s, t + δt] - X[r, s, t] = ∂X[r, s, t]/∂tδt + ε_t · δt

The volume of the box with these edges is given by the determinant:

Det[{X[r + δr, s, t] - X[r, s, t], X[r, s + δs, t] - X[r, s, t], X[r, s, t + δt] - X[r, s, t}]] =

    = Det[{∂X[r, s, t]/∂r, ∂X[r, s, t]/∂s, ∂X[r, s, t]/∂t}] · δr · δs · δt + ε · δr · δs · δt

with  ε≈0, when  δr≈0, δs≈0, δt≈0 (compute using properties of determinants.)  

The approximating parallelopiped with sides  ∂X/∂rδr, ∂X/∂sδs, ∂X/∂tδt,

δrδsδt-LinearIncrement = { Ζ + C : Ζ = (δs ∂X[s, t]/∂s) ρ + (δs ∂X[s, t]/& ... ;X[s, t]/∂t) τ, 0≤ρ≤1, 0≤σ≤1, 0≤τ≤1}

has volume  Det[{∂X[r, s, t]/∂r, ∂X[r, s, t]/∂s, ∂X[r, s, t]/∂t}] · δr · δs · δt.

We summarize this approximation by the volume differential formula:

    dV[r, s, t] = Det[{∂X[r, s, t]/∂r, ∂X[r, s, t]/∂s, ∂X[r, s, t]/∂t}] · dr · ds · dt

In the limit as the size of a  δrδsδt-partition of a  r-s-t-domain tends to zero, for any compact parameter domain  ,

    Underscript[∫, ] fV = Underscript[∑, Underscript[r, s, t∈ ... 399;, step δr, δs, δt]] f[r, s, t] · vol[δrδsδt - Increment] ≈

        ≈Underscript[∑, Underscript[r, s, t∈, step δr, δs, δt ... ]/∂r, ∂X[r, s, t]/∂s, ∂X[r, s, t]/∂t}] · δr · δs · δt

because

    | Underscript[∑, Underscript[r, θ, z∈, step δr, δθ, &# ... 948;r, δθ, δz]] δr · δs · δt = | ε_max | vol[] ≈0

This means that volume integrals are given in coordinates by the formula

    Underscript[∫, ] fV = Underscript[∫, ] f[X[r, s, t]] ɹ ... ∂r, ∂X[r, s, t]/∂s, ∂X[r, s, t]/∂t}] | r s t

Created by Mathematica  (July 2, 2004)