1. §1.2 Differential Calculus
Christopher Crawford
PHY 311

2. Key Points up to Now Linear spaces Linear operators
Linear combinations / projections -> basis / components
Dot product reduces; Cross product builds up dimension (area, vol.)
Orthogonal projections (Dot = parallel, Cross = perpendicular)
Linear operators
Most general transformation is a rotation * stretch
Rotations (orthogonal) appear in coordinate transformations
Stretches (symmetric) occur in orthogonal directions (eigenspaces)

3. Where are we heading? Differential spaces
Everything follows from the differential (d) and chain rule (partials)
Differential (line, area, vol.) elements are ordered by dimension
The derivative increases to one higher dimension
There is only ONE 1st derivative: d or in different dimensions
There is only ONE 2nd derivative: the Laplacian
Affine space/curvilinear coordinates
Operations on points and vectors: affine combination
Position vector: connection between point and vector
Coordinates: used to parameterize a volume / surface / curve
Differential d is more natural than for curvilinear coordinates
2 classes: Integration / Stokes’ theorems / Poincaré lemma
2 classes: Delta fn / Green’s fn / Helmholtz theorem / fn spaces
These 4 types of fundamental theorems map directly onto electrodynamic principles (and all classical fields)

4. Outline Differential operator – `d’ Calculus of a single variable
Partial differentials – chain rule Gradient, vector differential (del operator) Differential line, area, volume elements (dl, da, d¿) Relation between d, , dr
Curl and Divergence – differential in higher dimensions Geometric interpretation (boundary)
Laplacian – unique 2nd derivative: curvature Projection into longitudinal / transverse components
Unified approach to all derivatives using differential `d’

5. Differential operator
Definition
Infinitesimal
Relation between differentials
Becoming finite: ratio / infinite sum
Chain rule

6. Partial differentials
Chain rule
Partial derivative
List of differentials

7. Gradient – del operator
Separate out vectors
Differential operator
Del operator
Line element
Relation between them
Differential basis: dx, dy, dz
Example: d (x2y)

8. Example 2d vs. 3d gradients

9. Higher dimensional derivatives
Curl – circular flow
Divergence – outward flux
Derivative lies on the boundary
It is a higher dimensional density
More detail in Integral / Stokes / Gauss section

10. Product Rules Combine vector and derivative rules
How many distinct products? (combinations of dot,cross)

11. 2nd derivative: the Laplacian
Net curvature of a scalar function; Net ??? of a vector function?
How many 2nd derivatives? (combinations of dot, cross)

12. Projections of the Laplacian

13. Unification of vector derivatives
Three rules: a) d2=0, b) dx2 =0, c) dx dy = - dy dx
Differential (line, area, volume) elements as transformations