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
Ñ
2
   
Useful
Ñ
2
results
   
Multiderivative operator
Ñ
   
Difference between Derivatives and Differentials
   
Curl
Ñ
Ù
   
Partial Undirected Derivate
¶
   
Secondary Undirected Derivative
Ñ
Þ
   
Simplicial Derivative
Ñ
(
r
)
   
Conveyed Derivative
Ñ
→
   
Operating on and with
Ñ
   
Adjoints
         
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
¯
Ñ
2
   
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
=
Ñ
Þ
[
Ñ
¯]
2
¯
   
Einstein 1-tensor (1-½
Ñ
Þ
)
[
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
.