§1.2 Differential Calculus Christopher Crawford PHY 416G 2014-09-08.

§1.2 Differential Calculus

Christopher CrawfordPHY 416G

2014-09-08

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!

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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

• 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 ontoelectrodynamic principles (and all classical fields)

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Outline• Differential operator – `d’

Calculus of a single variable: chain rule, FTVC•

• Partial differentials – partial chain ruleGradient, vector differential (del operator)Differential line, area, volume elements (dl, da, d¿)Relation between d, , dr

• Curl and Divergence – differential `d’ in higher dimensionsGeometric interpretation (boundary)

• Laplacian – unique 2nd derivative: curvatureProjection into longitudinal / transverse components

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Differential operator• Definition

– Infinitesimal – Relation between differentials– Becoming finite: ratio / infinite sum

– Chain rule

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Partial differentials• Partial differential

• Chain rule

• Partial derivative

• List of differentials

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Gradient – del operator• Separate out vectors

– Differential operator– Del operator– Line element

• Relation between them– Differential basis: dx, dy, dz

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Example: d (x2y)

Example 2d vs. 3d gradients

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Higher dimensional derivatives• Curl – circular flow • Divergence – outward flux

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– 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) d2=0 , b) dx dy = - dy dx , c) dx2=0

• Differential (line, area, volume) elements as transformations

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Summary of 3 derivatives• Three rules: a) d2=0, b) dx2 =0, c) dx dy = - dy dx• Differential (line, area, volume) elements as transformations

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

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2nd derivative: the Laplacian• Net curvature of a scalar function;

Net ??? of a vector function?• How many 2nd derivatives? (combinations of dot, cross)

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Projections of the Laplacian

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