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.

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

- Applications:
- 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

- Applications:
- 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

- Applications:
- Fluid Dynamics: Conservation of Mass, Euler Equations, Conservation of Vorticity