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Table of Contents
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)
