§1.2 Differential Calculus Christopher Crawford PHY 416G

Key Points up to Now Linear spaces – Linear combinations / projections -> basis / components – Dot product reduces; Cross product builds up dimension (area, vol.) – Orthogonal projection (Dot = parallel, Cross = perpendicular) products – Affine space of points, position vector Linear operators – Most general transformation is a rotation * stretch – Rotations (orthogonal) appear in coordinate transformations – Stretches (symmetric) occur in orthogonal directions (eigenspaces) – APPLICATION to functions and differential operators! 2

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 1 st derivative: d or in different dimensions – There is only ONE 2 nd derivative: the Laplacian 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) 3

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

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

Partial differentials Partial differential Chain rule Partial derivative List of differentials 6

Gradient – del operator Separate out vectors – Differential operator – Del operator – Line element Relation between them – Differential basis: dx, dy, dz 7 Example: d (x 2 y)

Example 2d vs. 3d gradients 8

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

Unification of vector derivatives Three rules: a) d 2 =0, b) dx dy = - dy dx, c) dx 2 =0 Differential (line, area, volume) elements as transformations 10

Summary of 3 derivatives Three rules: a) d 2 =0, b) dx 2 =0, c) dx dy = - dy dx Differential (line, area, volume) elements as transformations 11

Product Rules Combine vector and derivative rules How many distinct products? (combinations of dot,cross) 12

2 nd derivative: the Laplacian Net curvature of a scalar function; Net ??? of a vector function? How many 2 nd derivatives? (combinations of dot, cross) 13

Projections of the Laplacian 14