A vector-valued function \(\bm{r}\colon \mathbf{R}^2 \to \mathbf{R}^3\) can be visualized as a parametrically-defined surface \(S\) in three-dimensional space; the domain \(\mathbf{R}^2\) is literally a plane being embedded in \(\mathbf{R}^3.\) For a surface \(S\) with parameterization \(\bm{r}(s,t) = \bigl\langle x(s,t), y(s,t), z(s,t)\bigr\rangle\) we refer to \(s\) and \(t\) as the parameters and to \(x(s,t)\) and \(y(s,t)\) and \(z(s,t)\) as the component functions or coordinate functions. Note that many writers conventionally use \(u\) and \(v\) as the names of the parameters. A surface \(S\) is closed if for some parameterization the entire surface is traced out for some bounded interval of the parameters \(s\) and \(t,\) bounding some expanse on its interior. A vector-valued function is smooth if the derivatives of its component functions exist for all orders. A surface itself is smooth if it has a smooth parameterization.
Define \(\bm{r}_s\) and \(\bm{r}_t\) as the first-order partial derivatives \(\bigl\langle \frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s} \bigl\rangle\) and \(\bigl\langle \frac{\partial x}{\partial t}, \frac{\partial y}{\partial t}, \frac{\partial z}{\partial t} \bigl\rangle\) respectively. The vectors \(\bm{r}_s\) and \(\bm{r}_t\) will be tangent to the surface and \(\mathbf{N} = \bm{r}_s \times \bm{r}_t\) will be normal to the tangent plane. A parameterization \(\bm{r}\) of a surface is regular if \({\bm{r}_s \times \bm{r}_t \neq \bm{0}}\) at any point. A surface has two orientations, one for each consistent choice of direction of normal vectors across the surface. For a closed surface we conventionally refer to the “outward” orientation as the positive orientation. The surface area of \(S\) over some region \(R\) in the \(st\)-plane is given by \({\iint_R \bigl|\bm{r}_s \times \bm{r}_t\bigr| \,\mathrm{d}A}\,.\) For a continuous scalar field \(f\) defined on \(S,\) the surface integral of \(f\) over \(S\) is \[ \iint_S f(x,y,z) \,\mathrm{d}S = \iint_R f(\bm{r})\,\bigl| \bm{r}_s \times \bm{r}_t\bigr| \,\mathrm{d}A \,. \]
For an oriented surface \(S\) with normal vector \(\mathbf{N}\) and a continuous vector field \(\bm{F}\) defined on \(S,\) the surface integral of \(\bm{F}\) across \(S\), also called the flux of \(\bm{F}\) across \(S,\) is \[ \iint_S \bm{F}\cdot\mathrm{d}\mathbf{S} = \iint_S \bm{F}\cdot\mathbf{N} \,\mathrm{d}S = \iint_R \bm{F}\cdot (\bm{r}_s \times \bm{r}_t) \,\mathrm{d}A \,. \]