Just like we can define a curve in two-dimensional space parametrically where each coordinate is a function of a single variable, we can parametrically define a surface (manifold) in three-dimensional space where each of the three coordinates is a function of two variables. E.g. a surface \(\mathcal{S}\) can be defined as \[ \mathcal{S}(u,v) = x(u,v)\mathbf{i} +y(u,v)\mathbf{j} +z(u,v)\mathbf{k} \]
Show examples, with grid curves along which \(u\) and \(v\) are constant. Surfaces of revolution and tangent planes and graphs as examples.
Given a smooth surface \(S\) over a domain \(R\) define as \[ \mathcal{S}(u,v) = x(u,v)\mathbf{i} +y(u,v)\mathbf{j} +z(u,v)\mathbf{k} \] such that \(S\) is "covered just once" as \(u,v\) vary (rectifiable?) the surface area of \(S\) can be calculated as \[ \iint\limits_R \big| S_u \times S_v\big| \,\mathrm{d}A \] where \[ S_u = \frac{\partial x}{\partial u}\mathbf{i} + \frac{\partial y}{\partial u}\mathbf{j} + \frac{\partial z}{\partial u}\mathbf{k} \qquad S_v = \frac{\partial x}{\partial v}\mathbf{i} + \frac{\partial y}{\partial v}\mathbf{j} + \frac{\partial z}{\partial v}\mathbf{k} \]
Surface Integrals : Surface Area :: Line Integrals : Arclength.
Given a parametrically-defined surface \(S\) over domain \(R\) defined as \[ \mathcal{S}(u,v) = x(u,v)\mathbf{i} +y(u,v)\mathbf{j} +z(u,v)\mathbf{k} \] and a function \(f\) defined on \(S,\) the surface integral of \(f\) over \(S\) is \[ \iint\limits_S f(x,y,z) \,\mathrm{d}S = \iint\limits_R f\big(S(u,v)\big)\,\big| S_u \times S_v\big| \,\mathrm{d}A \]
Oriented surfaces need to be cleared up before talking about flux — surface integrals over vector fields.
If \(\bm{F}\) is a continuous vector field defined on an oriented surface \(S\) with normal vector \(\mathbf{n}_S,\) then the surface integral of \(\bm{F}\) over \(S\) can be calculated as \[ \iint\limits_S \bm{F}\cdot\mathrm{d}\mathbf{S} = \iint\limits_S \bm{F}\cdot \mathbf{n}_S \,\mathrm{d}S \] this is also called the flux of \(\bm{F}\) across \(S.\)
Electric flux example