Surface Integrals of Vector Fields

Calculus 3 · Axiom Academy

LESSON Surface Integrals: Flux Through Surfaces How much of a vector field pierces an oriented surface S — and why flipping the surface's orientation flips the flux. 1. What Flux Measures: the Piercing Component The flux of a field through a surface S adds up how much of the field pierces it. At each point we take — the field dotted with the unit normal — which keeps only the component perpendicular to S . A field running along the surface contributes nothing. flux = the field dotted with the unit normal, summed over S only the perpendicular part gets through ( = angle between and ) 2. Parametric Surfaces and the Vector Area Element Most surfaces come parametrized as . The two tangent vectors and span a tiny parallelogram — the area element — and their cross product is perpendicular to it, pointing along the normal. and the vector area element is . The length is exactly the area the parallelogram covers — so it cancels the denominator in , leaving just the raw cross product in the integral over the parameter domain D : 3. Orientation and Sign: Flip the Normal, Flip the Flux A surface has two sides, so two possible unit normals. Choosing an orientation means picking one. Swap to the other normal — — and every changes sign, so the whole flux negates . Positive: the field crosses the surface the same way points (outward flow for a closed surface). Negative: the field crosses against (inward flow for a closed surface).

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