Integral of a Vector Field over a Surface

Topics: Calculus - Vector Field - Surface

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Let $F:{\mathbb{R}}^3\to V_3$ be a vector field, and let $\Phi :R\to {\mathbb{R}}^3$ be the parametrisation of a surface.

The integral of $F$ over such a surface is defined as:

$${\iint }_{\Phi }F\cdot dS:={\iint }_{R}F(\Phi (u,v))\cdot (T_u\times T_v)dA$$