## Texts

Most of the readings and all practice problems are from the course text:

Knowles, J. K. Linear Vector Spaces and Cartesian Tensors. New York, NY: Oxford University Press, 1998. ISBN: 9780195112542.

Abeyaratne, Rohan. Lecture Notes on the Mechanics of Elastic Solids. Vol. 1, A Brief Review of Some Mathematical Preliminaries. Free e-book, 2006.

## Further References

Gel'fand, I. M. Lectures on Linear Algebra. New York, NY: Dover, 1989. ISBN: 9780486660820.

Halmos, P. R. Finite Dimensional Vector Spaces. Princeton, NJ: Van Nostrand-Reinhold, 1958. ISBN: 9780387900933.

Gel'fand, I. M., and S. V. Fomin. Calculus of Variations. Englewood Cliffs, NJ: Prentice Hall, 1963.

Giaquinta, M., and S. Hilderbrandt. Calculus of Variations I. New York, NY: Springer, 1996. ISBN: 9780387506258.

Troutman, J. L. Variational Calculus with Elementary Convexity. New York, NY: Springer-Verlag, 1983. ISBN: 9780387907710.

LEC # Topics READINGS PRACTICE PROBLEMS
1 Vector space, linear independence, dimension of a vector space, basis for vector space, components of a vector pp. 1-8 Problems 1.1-1.11
2 Scalar product, length of vector, distance between vectors, angle between vectors, orthonormal basis pp. 9-17 Problems 1.12-1.20
3 Linear transformations, invariant subspace, eigenvalue problem pp. 18-20 and 23-26 Problems 2.1, 2.3, 2.6, and 2.17 (except questions about singular/non-singular/inverse transforms)
4 Null space, singular/non-singular linear transformations, inverse, components of a linear transformation Chapter 2 Problems 2.1-2.5, 2.8, 2.9, 2.11, and 2.15-2.17
5 Components of a linear transformation, components in different bases, scalar invariants, cartesian tensors, symmetric tensors, skew-symmetric tensors pp. 27-32 and 42-46 Problems 3.1-3.12
6 Eigenvalues of a symmetric tensor, principal basis, positive-definite tensor, orthogonal tensor, proper/improper, orthogonal tensor pp. 42-52 (except tensor products) and 56-57 Problems 3.13-3.18, 3.20, and 3.24-3.26
7 Tensor product of 2 vectors, polar decomposition of a non-singular tensor

pp. 44 and 57-59

Also read chapters 2 and 3 in Abeyaratne, Rohan. Lecture Notes on the Mechanics of Elastic Solids.

Problems 3.3-3.7, 3.11-3.16, 3.22, and 3.23