Table of Contents

### PH 422 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)

### PH 425 Math Bits

##### Matrix Manipulations (45 minutes)

- Review of matrix manipulations (Lecture, 45 minutes)

##### Introduction to Bra-ket notation (15 minutes)

- Introduction to bra-ket notation (Lecture, ?? minutes) - Emphasize normalization and finding components. Add a good SWBQ or small group activity.

#### Unit: Operators and Transformations in Linear Systems

##### Linear Transformations (1 1/2 hr)

- Linear Transformations Activity (Small Group Activity)

##### Tangible Metaphor for Complex Vectors (10 minutes)

- Visualizing Complex Two Component Vectors (Kinesthetic Activity)

##### Properties of Linear Vector Spaces (30 min)

- Properties of linear vector spaces (Optional Lecture, ?? minutes)

##### Inner Products and Norms

- Calculating Inner Products and Normalization of Vectors (Lecture, ?? minutes)

Write these lecture notes. Emphasize complex vectors.

##### Matrix Components (40 min)

- Finding Matrix Elements (Small Group Activity) Work in comments about Fourier series into wrap-up.

##### Rotation Matrices in 2 and 3 Dimensions (10 min)

- Rotation Matrices in 2 and 3 Dimensions (Lecture, ?? minutes).

#### Unit: Eigenvalues and Eigenvectors

##### Eigenvalues & Eigenvectors (2 hr)

- Eigenvectors and Eigenvalues Activity (Small Group Activity)

##### Special Properties of Hermitian Matrices (40 min)

- Properties of Hermitian matrices (Lecture, ?? minutes)
- Diagonalization of matrices (Lecture, ?? minutes)
- Diagonalizing Matrices (Small Group Activity).

##### Commuting Matrices (10 min)

- Properties of commuting matrices (Lecture, ?? minutes)

### PH 423 Math Bits

##### Partial Derivatives (40 minutes)

- Quantifying Change (Small Group PDM Activity)
- Isowidth and Isoforce Stretchability (Small Group PDM Activity)

##### Potential Energy and the Partial Derivative Machine (1 hour 40 minutes)

- Math pre-test (Class actvity, 15 minutes)
- Easy and Hard Derivatives (Small Group PDM Activity)
- Potential Energy of a Spring (Lecture 5 minutes)
- Potential Energy of an Elastic System (Integrated PDM Lab)

##### Total Differentials and Partial Derivatives (1 hour 30 minutes)

- Total Differentials (Lecture 10 minutes)
- Evaluating Total Differentials: "Zapping with d" (Small Group Activity)
- What You Can Do with Total Differentials (Lecture 5 minutes)
- Upside Down Derivatives (Small Group PDM Activity)
- Cyclic Chain Rule (Small Group PDM Activity)
- Deriving Change of Variables (Small Group PDM Activity)

##### Maxwell Relations and Legendre Transforms

- Mixed Partials and Maxwell Relations (Lecture 30 minutes)
- Legendre Transforms (Lecture 5 minutes)