Surface Integral

Surface Integrals of Scalar Fields

Given a scalar-valued function 𝑓 and surface 𝑆. The surface integral of 𝑓 over 𝑆 is defined as:

• ${\iint }_{S}fdS={\iint }_{T}f(\mathbf{r}(s,t))\Vert \frac{\partial \mathbf{r}}{\partial s}⨯\frac{\partial \mathbf{r}}{\partial t}\Vert dsdt$

where:

• 𝐫(𝑠,𝑡) is the parametrization of 𝑆
• (𝑠,𝑡) varies in some region 𝑇 in the plane
• ||·|| is the magnitude of the cross-product of the partial derivatives of 𝐫(𝑠,𝑡)

Surface Integrals of Vector Fields

TODO