Tensor Analysis

Department: 
MATH
Course Number: 
7586
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
no regular schedule

Review of linear algebra, multilinear algebra, algebra of tensors, co- and cotravariant tensors, tensors in Riemann spaces, geometrical interpretation of skew tensors.

Prerequisites: 

MATH 2403 and either MATH 2406 or MATH 4305

Course Text: 

No text

Topic Outline: 
  • Algebraic Theory of Tensors with Application to the Understanding of Crystals: - Review of linear algebra, multilinear algebra, algebra of tensors, co- and contravariant tensors, tensors in Riemann spaces, geometrical interpretation of skew tensors
    • Applications:
      • Geometry of crystals, invariant tensors,
      • Dual basis, reciprocal lattice, x-ray crystallography
      • Applications to lattice geometry
  • General Coordinates and Tensor Fields: - Vector-fields, tensor-fields, transformation of tensors, transformation of differential equations, gradient and Laplace operator in general coordinates
    • Applications:
      • Mechanics: D'Alembert principle and Lagrangian mechanics
      • Emphasis on co-variance of the Euler-Lagrange equations
      • Motion of a particle on surfaces and in the Schwarzschild metric
  • Elasticity: Strain Tensor, Tensor of Elasticity, Motions in an Elastic Body, Elastic Moduli of Crystals
  • Electromagnetism: Solution of Boundary Value Problems in Suitable Coordinates
  • Differentiation and Integration of Tensors: - Transformation properties of the gradient, differentiation of skew tensors, covariant differentiation, divergence, curl and Stokes' theorem - Torsion tensor and curvature tensor as examples from geometry
    • Applications:
      • Electromagnetism: Field tensor, field energy tensor
  • Fluid Dynamics: Conservation of Mass, Euler Equations, Conservation of Vorticity