Surface Integral

Let $\Phi(s,t): D \mapsto\mathbb{R}^3$ be a continuous function where $D\subset\mathbb{R}^2$ is open and connected (or with some/all of its boundary points).

If $\Phi$ is injective, then we say that $\Phi$ is a parameterized surface, and $\Phi(D)$ is the surface parameterized by $\Phi$.

Moreover, $D$ is called a connected set if every 2 points in $D$ can be joined by a path in $D$.

Let $\Phi:D\mapsto\mathbb{R}^3$ be a parameterized surface of class $C^1$ where $D\subset \mathbb{R}^2$. Let $(a,b)\in D$, then the standard normal vector is defined by

$$\mathbf{N}(a,b)=\mathbf{T}_s(a,b)\times \mathbf{T}_t(a,b)$$

The surface $S=\Phi(D)$ is smooth at $(a,b)$ if $\mathbf{N}(a,b)\ne 0$. If $S$ is smooth at every point, then it’s said to be smooth.

The surface area of a smooth surface $S=\Phi(D)$ is

$$\iint_D \|\mathbf{N}(s,t)\|\, dA$$

The formula applies when the surface is smooth except at finitely many points.

Let $\Phi:D\mapsto\mathbb{R}^3, D\subset\mathbb{R}^2$, $f:S\mapsto\mathbb{R}, S=\Phi(D)$

The scalar surface integral of $f$ along $\Phi$ is

$$\iint_D f(\Phi(s,t))\|\mathbf{N}(s,t)\|\, dsdt$$

The vector surface integral of $\mathbf{F}$ along $\Phi$ is

$$\iint_D \mathbf{F}(\Phi(s,t))\cdot \mathbf{N}(s,t)\, dsdt$$

A smooth and connected surface $S$ is orientable if we can define a single unit normal vector at each point of $S$ so that the normal vectors vary continuously. Otherwise, $S$ is non-orientable.

Let $S$ be a bounded, smooth, orientable surface in $\mathbb{R}^3$ with a given orientation.

We say the boundary $\partial S$ is oriented consistently if the orientation of each closed curve is $\partial S$ is chosen such that the right-hand rule is satisfied, i.e., if we use the fingers of the right hand to curl in the direction of the curve, then the thumb will point in the direction of the normal to $S$.

Let $F:X\mapsto\mathbb{R}^3$ be a vector field of class $C^1$, where $X\subset\mathbb{R}^3$ contains $S$. Then

$$\iint_S \nabla\times\mathbf{F}\cdot d\mathbf{S}=\oint_{\partial S}\mathbf{F}\cdot d\mathbf{s}$$

Let $D$ be a bounded solid region in $\mathbb{R}^3$. Suppose $\partial D$ is (union of finitely many) smooth, closed, orientable surfaces which are oriented by normals pointing aways from $D$.

Let $\mathbf{F}:X\mapsto\mathbb{R}^3$ be a vector field of class $C^1$, where $X\subset\mathbb{R}^3$ contains $D$, then

$$\oiint_{\partial D}\mathbf{F}\cdot d\mathbf{S}=\iiint_D \nabla\cdot\mathbf{F}\, dV$$