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## PH 422 (Vector Fields) Math Bits

#### Power Series Basics

- Reading: GVC § Power Series–Properties of Power Series
- Power Series (Lecture: 15 min)
- Approximating Functions with a Power Series (Maple/Mathematica)
*30 min* - Properties of Power Series (Lecture: 15 min)

#### dr(vector)

- Reading: GEM § 1.4
- Curvilinear Coordinates (lecture)
- Scalar Line Integral (lecture)

#### Derivatives of Scalar Fields

- Reading: GEM § 1.2.2
- Partial Derivatives (lecture)
- Curvilinear Basis Vectors (kinesthetic)
- Introducing $d\Vec{r}$ (lecture)
- Gradient (lecture)
- Visualizing Gradient (Maple/Mathematica)
- directional derivatives (lecture) (Optional)

#### 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.

#### Curl

- Circulation (lecture)
- Visualizing Curl (Maple)
- Definition of Curl (lecture). We follow “div, grad, curl and all that”, by Schey

#### Stokes' Theorem

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

#### Product Rules

- Reading: GVC § Product Rules–Integration by Parts
- Product Rules (lecture)
- Integration by Parts (lecture)