## PH 422 (Vector Fields) Math Bits

#### Divergence (40 min)

• Definition of divergence (Lecture) 20 min
• Visualizing Divergence (Maple Visualization) 20 min Students practice estimating divergence from graphs of various vector fields.

#### Divergence Theorem (20 min)

• Reading: GVC § Divergence Theorem
• Derivation of the Divergence Theorem (lecture). We follow “div, grad, curl and all that”, by Schey. The Divergence theorem is almost a lemma based on the definition of divergence. Draw a diagram of an arbitrary volume divided into lots of little cubes. Calculate the sum of all the fluxes out of all the little cubes (isn't this a strange sum to consider!!) and argue that the flux out of one cube is the flux into the adjacent cube unless the cube is on the boundary.

#### Stokes' Theorem

• Reading: GVC § Stokes' Theorem
• Derivation of Stokes' Theorem (lecture). We follow “div, grad, curl and all that”, by Schey

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