School of Engineering and Applied Science

Department of Mechanical Engineering

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.

**Grading Policy**

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.

SEAS

University of Pennsylvania Library System