Maths for (Games) Programmers Section 5 - Multivector Calculus

Multivector Derivatives
Introduction
Notations
Multivector Functions as Tensors
Fields     Tensors     Forms     Dyads     Multitensors     Extended Fields     Outtermorphisms   and Determinants     Eigenblades     Coordinate-based Tensor representations
Differentials
Directed Scalar Derivatives     Linearity of the Differential     Differentiating Exponentials     The Directed Chain Rule     Primary Differential     Second Primary Differential     Third Primary Differential     Secondary Differential     Lie Product
Undirected Derivatives
1-derivative     Ñ     Useful Ñ results     Ñ in Alternate Coordinate Systems     Monogenic Functions     Laplacian     Ñ2     Useful Ñ2 results     Multiderivative operator       Ñ     Difference between Derivatives and Differentials     Curl     ÑÙ     Partial Undirected Derivate     Secondary Undirected Derivative        ÑÞ     Simplicial Derivative     Ñ(r)     Conveyed Derivative     Ñ     Operating on and with Ñ
The Undirected Chain Rule     The Kinematic Rules     Taylor's Formula     Contraction and Trace     Covariance     Symmetry and Skewsymmetry
Characterising General Functions
Connections     Directed Multivector Derivatives
Multivector Fractals

Multivector Manifolds
Curves and Manifolds
Extended Mapspace     Submanifolds     Embedded Frame     Inverse Embedded Frame     Local Orientation     The Metric     M-Curve as an M-blade-valued field     Projector 1-multitensor     ¯
Integration over an M-curve
Fourier Transform
Differentiation within an M-curve
Directed Tangential Derivative     Ð¯     Undirected Tangential 1-Derivative     Ñ
Fundamental Theorem of Calculus
Basic Form     Greens Functions     General Form
Poles and Residues
Cauchy's Theorem

Manifold Restricted Tensors
Further differentials and derivatives
Operator Notations        Tangential 1-differential  Ñ     Directed Coderivative      Ð()     Coderivative     Ñ     Projection Differential (1.2)-tensor     ¯Ñ     Projection Second Differential (1;3)-tensor     ¯Ñ2     Squared Projection Differential (1;3)-tensor     (¯Ñ)2     Shape (<0.2>;1)-multitensor     [Ñ¯]     Squape 1-multitensor     [Ñ¯]2     Ricci 1-tensor       ([Ñ¯]2¯)
Curvature 2-tensor     [Ñ¯]×
Scalar Curvature (0;1)-tensor         R = ÑÞ [Ñ¯]2¯     Einstein 1-tensor         (1-½ÑÞ)[Ñ¯]2¯

The Coordinate based approach
Streamline Coordinates
Introduction
Lie Derivative
Lie Bracket     Lie Drag     Lie Derivative     Covariant Frame     Covectors     Parallel transport     Linear Connection     Directed Coderivative     Geodesics
Curvature
1-Curvaturem     2-Curvature   as ortho bi-circle     Curvature Measures     Curvature Mesaures of Polyhedral 2-curves     Curvature as loop integative limit     Curvature as Coderivative Operator     Symmetries and Bianci Relations
Tortion

References/Source Material

Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998, 2014
Web Source: www.iancgbell.clara.net/maths
Latest Edit: 04 Oct 2014.