Section 17.4

Flux Integrals

Quantifying how much "stuff" (water, electricity, heat) passes through a surface.

Introduction

If represents the velocity of a flowing fluid, then the Flux is the volume of fluid passing through the surface per unit time.

Only the component of that is normal/perpendicular to the surface actually goes through it.

The Formula

Vector Surface Integral

We integrate . When parameterized, the calculation simplifies beautifully:

Note: The definition of depends on orientation. For closed surfaces, "positive" is outwards.

Visualizing Flux

Interactive: Flow Through Cylinder

Worked Examples

Example 1: Flux through a Disk

Find flux of through disk in plane (normal up).

1. Logic Check:

Field is constant UP. Normal is UP. They are parallel.

Flux = (Field Strength) * (Area) = .

2. Calculation Check:

Integral of 3 dA = 3 * Area = 3π.

Example 2: Radial Flux through Sphere

Find flux of through sphere .

1. Parameterize:

on sphere.

Field on surface .

2. Normal Vector:

For sphere, (points outward).

3. Dot Product:

Since , .

Product = .

4. Integrate:

Example 3: Paraboloid Cap

Find flux of through (oriented up).

1. Parameterize (Graph):

Normal .

2. Dot Product:

3. Integrate over Unit Disk:

Switch to Polar. .

The term vanishes (symmetry). Left with .