Surface integral

A surface integral uses the area element induced by a parametrisation to integrate over a surface.

z = g (x, y)

has upward-oriented vector area element

d S = (- g_{x}, - g_{y}, 1) d x d y

and scalar area element

d S = 1 + g_{x}^{2} + g_{y}^{2} d x d y

. These follow from the cross product of the parametrization derivatives

(1, 0, g_{x})

and

(0, 1, g_{y})

Oriented surface element ( $d S$ )

r (u, v)

, the vector surface element is

d S = (r_{u} \times r_{v}) d u d v

, with the sign chosen to match the orientation.

A parametrised surface

r (u, v)

has area element

∣ r_{u} \times r_{v} ∣ d u d v

z = f (x, y)

has area element

1 + f_{x}^{2} + f_{y}^{2} d x d y

\int_{S} F \cdot n d S

measures flow through an oriented surface.

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