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The oriented area vector transforms by the parameter Jacobian determinant

Statement

If $ψ = φ \circ h$ , then $ψ_{s} \times ψ_{t} = (\det D h) (φ_{u} \times φ_{v}) \circ h$ .

If $ψ = φ \circ h$ , then $J_{ψ} = (J_{φ} \circ h) ∣ \det D h ∣$ .

Facts & Assumptions

Given: A regular reparametrization $ψ = φ \circ h$ , with $h = (h_{1}, h_{2})$ .

[L1]

The chain rule and the coordinate interpretation of total derivatives give $ψ_{s} = (h_{1})_{s} (φ_{u} \circ h) + (h_{2})_{s} (φ_{v} \circ h)$ and the analogous formula for $ψ_{t}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , A total derivative computes every directional derivative, and its matrix is the Jacobian, Surface reparametrizations and their orientation sign).

[L2]

The cross product is bilinear and alternating (The cross product is bilinear, alternating, and orthogonal to both factors), and area density is the cross-product norm (The surface area density is the norm of the cross product of the parameter tangents).

Proof

technique · direct

1.1

Substitute the two formulas from [L1] into $ψ_{s} \times ψ_{t}$ . By [L2], the equal-vector terms vanish and the remaining terms combine to $((h_{1})_{s} (h_{2})_{t} - (h_{2})_{s} (h_{1})_{t}) (φ_{u} \times φ_{v}) \circ h$ .

givenL1L2algebra

2.1

The scalar coefficient in step 1.1 is $\det D h$ , proving the signed area-vector formula.

step 1.1algebra

3.1

Taking Euclidean norms, using $∥ c w ∥_{2} = ∣ c ∣ ∥ w ∥_{2}$ , and applying [L2] gives $J_{ψ} = (J_{φ} \circ h) ∣ \det D h ∣$ .

step 2.1L2algebra

4.1

The first identity retains the determinant sign, while only the norm identity replaces it by an absolute value, as asserted.

step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

University of Toronto MAT237 notes, Section 5.3, An Invariance Property (standard reference, not scraped)
M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)