Surface Integrals
Table of Contents
1. Surfaces
- Connected
- A subset \( D \) is connected if every 2 points in \( D \) can be joined by a path in \( D \).
- Parameterized Surface
- Let \( \Phi:D\mapsto\mathbb{R}^{3} \) be a continuous function where \( D\subset\mathbb{R}^{2} \) is open and connected. If \( \Phi \) is injective, except possibly along boundary \( \partial D \), then we say \( \Phi \) is a parameterized surface, and \( \Phi(D) \) is the surface parameterized by \( \Phi \).
Let \( \Phi:D\mapsto\mathbb{R}^{3} \) be a parameterized surface of class \( C^{1} \) where \( D\subset\mathbb{R}^{2} \). Let \( (a,b)\in D \). The standard normal vector wrt. the parameterization \( \Phi \) is defined by
\[ \mathbf{N}(a,b) = \mathbf{T}_{s}(a,b) \times \mathbf{T}_{t}(a,b) \]
Here, \( \mathbf{T}_{s} = \frac{\partial \Phi}{\partial s}(s, b) \) and \( \mathbf{T}_{t} = \frac{\partial \Phi}{\partial t}(a,t) \) denotes the tangent vectors to the coordinate curves.
We say that the surface \( S=\Phi(D) \) is smooth at the point \( \Phi(a,b) \) if \( \mathbf{N}(a,b)\ne 0 \). If \( S \) is smooth at every point, then it’s said to be smooth.
1.1. Area of Surface
Let \( \Phi:D\mapsto\mathbb{R}^{3} \) be a smooth parameterized surface of class \( C^{1} \) where \( D\subset\mathbb{R}^{2} \). Then, the surface area of \( S=\Phi(D) \) is
\[ \iint_{D} \left\| \mathbf{N}(s,t) \right\|\, dA \]
2. Surface Integrals
2.1. Scalar Surface Integral
Let \( \Phi:D\mapsto\mathbb{R}^{3} \) be a smooth parameterized surface where \( D\subset\mathbb{R}^{2} \). Let \( f:S\mapsto\mathbb{R} \) be a continuous function where \( S = \Phi(D) \). The scalar surface integral of \( f \) along \( \Phi \) is defined by
\[ \iint_{\Phi} f\, dS = \iint_{D} f(\Phi(s,t)) \| \mathbf{N}(s,t) \|\, ds\, dt \]
2.2. Vector Surface Integral
Let \( \Phi:D\mapsto\mathbb{R}^{3} \) be a smooth parameterized surface where \( D\subset\mathbb{R}^{2} \). Let \( \mathbf{F}:S\mapsto\mathbb{R}^{3} \) be a continuous function where \( S = \Phi(D) \). The vector surface integral of \( \mathbf{F} \) along \( \Phi \) is defined by
\[ \iint_{\Phi} \mathbf{F}\cdot d\mathbf{S} = \iint_{D} \mathbf{F}(\Phi(s,t))\cdot \mathbf{N}(s,t)\, ds\, dt \]