# Tag Archives: Surface integrals

## Math2111: Chapter 4: Surface integrals. Section 3: Surface integrals of vector fields

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In Section 2 we have seen how to calculate surface area and integrate scalar fields over surfaces. We now establish an extension similar to the step from scalar line integrals to vector line integrals.

We assume again that the surfaces we consider are smooth, that is, they are assumed to be images of parameterised surfaces $\boldsymbol{\Phi}:D\to\mathbb{R}^3$ for which:

• $D$ is a non-empty, compact and Jordan-measurable subset of $\mathbb{R}^2$;
• the mapping $\boldsymbol{\Phi}$ is one-to-one;
• $\boldsymbol{\Phi}$ is continuously differentiable;
• the normal vector $\boldsymbol{n}=\frac{\partial \boldsymbol{\Phi}}{\partial u} \times \frac{\partial \boldsymbol{\Phi}}{\partial v} \neq \boldsymbol{0}$ except possibly at a finite number of points;

(Notice, the condition that ${}D$ is compact can also be replaced by the condition that the surface $S=\{\boldsymbol{\Phi}(u,v): (u,v)\in D\}$ is compact.)

Let $D\subseteq \mathbb{R}^2$ be Jordan measurable. By a parameterised surface we mean a continuously differentiable function $\boldsymbol{\Phi}:D\to\mathbb{R}^3$. The image $\boldsymbol{\Phi}(D)$ is called surface.