University of Pennsylvania
School of Engineering and Applied Science
Department of Mechanical Engineering

# Spring 2009

Instructor:     Dr. P.S. Ayyaswamy
Office:           229B Towne
Phone:           215 898-8362
Email:             ayya@seas.upenn.edu

Office Hour:  Wednesdays, 2-3pm

TA:                 Evan Galipeau
Phone:            215-898-5346

Email:             galipeau@seas.upenn.edu

Office Hour:  Tuesdays, 2:10-4:00pm in 337 Towne

Topics to be covered:

1. Vector Analysis:  scalar and vector products, rigid body rotation, moment of a force, triple product, derivatives of  a vector, dynamics of a system of particles, acceleration of a particle along a space curve, fixed and moving frames of reference, Coriolis acceleration;  binormal, trihedral coordinate systems, Frenet - Serret formulae; partial derivatives of a vector, gradient, divergence and curl, surface integral, Gauss divergence theorem, Green's identities, vector function, Stokes and Green's Theorems; general coordinates, reciprocal system of vectors, base vectors, covariant and contra variant components, arc length, orthogonal coordinate system, gradient, divergence and curl in general coordinates, orthogonal curvilinear coordinates, cylindrical and spherical coordinates.

2. Theory of Matrices:  transposition, multiplication, partitioning, linear transformations, inverse of a matrix, involutary matrix, conjugate matrix, Hermitian, skew Hermitian, Unitary, and orthogonal matrices; direct sum, determinant of a matrix, rank, adjoint, linear system of equations, Gauss elimination, echelon form, Gauss-Jordan elimination, Cramer's rule, eigen values and eigen vectors of algebraic systems,  forms in matrix theory, inner product and norm; Vector spaces and span:  linear independence of eigen vector, basis of eigen vectors, important vector spaces of the Rn  family, basis of eigen vectors, diagonalization of a matrix, quadratic forms, canonical forms; vector spaces:  positiveness, Triangle inequality, Schwarz inequality, Basis for an n-tuple space.

3. Variational calculus: minimization of functions, shortest distance between two points, Euler-Lagrange equation; one-dimensional steady state heat conduction in a rod: variational statement, subdomain creation, integration and differentiation, minimum of the integral; Finite element formulation through matrix representation, computational aspects, Least-squares approach; Hilbert Space, Weak formulation , Galerkin technique, Method of weighted residuals, Introduction to MATLAB and FEMLAB and the solution of a problem using FEMLAB.

4. Tensors: Einstein summation, principle of equivalence and inverse of the transformation, contra and co-variant components of a tensor, tensors of arbitrary order, Kronecker delta, Permutation symbol, Properties of matrices using tensor notation, generalized Kronecker delta; transformation equation, concept of distance, weighted tensors, examples from dynamics, affine tensors, cartesian tensors, transitive property; dyads and polyads, outer product, symmetric and skew-symmetric systems, idemfactor, double dot products;  Tensor operations: addition, subtraction, contraction, outer and inner products, quotient law; Special tensors: metric tensor, arc length, metric tensor in various coordinate systems, conjugate metric tensor, associated tensors, Euclidean and Riemannian spaces, invariants, detailed study of cartesian tensors;  Physical components of a tensor;  Tensor calculus: derivative of a tensor, covariant differentiation, Christoffel symbols of the first and second kind, covariant derivative of a covariant and a contravariant tensor, evaluation of Christoffel symbols, covariant derivatives of higher rank tensors, Ricci's theorem;  Intrinsic differentiation and generalized acceleration; Tensor notation for gradient, divergence, curl and Laplacian;  Geodesics, equations of geodesics, geodesic coordinates.

List of books for ENM511:  (These are on Reserve in the library)

1. G. Arfken: Mathematical methods for physicists

2. R. Aris:  Vectors, tensors and the basic quations of fluid mechanics

3. H. Jeffreys:  Cartesian tensors

4. M. R. Spiegel:  Theory and problems of vector analysis and an introduction to Tensor Analysis

5. B. Noble and J. W. Daniel:  Applied Linear Algebra

6. R. S. Schechter:  Variational method in Engineering

7. J. N. Reddy:  An Introduction to the Finite Element Method

Required Text

1. Notes for the Foundations of Engineering Mathematics Course ENM511, by M. A. Carchidi

2. J. H. Heinbockel:  Introduction to Tensor calculus and Continuum Mechanics

Distributed Notes

Notes on Tensor Analysis developed by me will be given during class meetings.

Home work  30%,
Mid Term 30%
2nd Final 40%

Home Work Policy

Each assigned home work must be worked and submitted exactly a week from the day assigned.  Late home work submissions will be given no credit.