§1.2 Differential Calculus Christopher Crawford PHY 311 2014-01-22

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)

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)

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’

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

Partial differentials Chain rule Partial derivative List of differentials

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

Example 2d vs. 3d gradients

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

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

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

Projections of the Laplacian

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