# Tag Archives: scalar surface integral

## Math2111: Chapter 4: Surface integrals. Section 2: Surface area and surface integrals of scalar 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 the following we assume that the surfaces 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.)