# surface integrals

(1.7 hours to learn)

## Summary

A surface integral is the integral of a function over a surface. Important cases include surface area and flux, where the function is the dot product of the surface normal with a vector field.

## Context

This concept has the prerequisites:

- vector fields (Flux is defined in terms of a vector field.)
- multiple integrals (Surface integrals are defined in terms of multiple integrals.)
- cross product (The definition of the surface integral includes a cross product.)
- dot product (The definition of flux includes a dot product.)
- partial derivatives (Partial derivatives are used in computing surface integrals.)

## Goals

- Know the definition of a surface integral in three dimensions.

- Be able to integrate a function over a surface in three dimensions.

- As a special case, be able to compute surface area.

- Be able to compute the flux across a surface in three dimensions.

- Derive the formula for a surface integral for a surface given as z = f(x, y).

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Multivariable Caclulus (2010)

Video lectures for MIT's introductory multivariable calculus class.

## -Paid-

→ Multivariable Calculus

An introductory multivariable calculus textbook.

Location:
Section 15.5, "Surface integrals," pages 1047-1055

## Supplemental resources (the following are optional, but you may find them useful)

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 8.4, "Surface integrals and flux," pages 367-375

Additional dependencies:

- differential forms
- pullback
- determinant

Other notes:

- This presents a more rigorous treatment than the other resources.

## See also

-No Additional Notes-