### MA 5221 Tensor Analysis and Applications

#### Course Details

Curvilinear coordinates:

Transformation of coordinates, Orthogonal curvilinear coordinates, unit vectors in curvilinear systems, arc length and volume elements, Gradient, divergence and curl, special orthogonal coordinate systems, general curvilinear coordinates. ( 8 L)

Tensor Algebra:

The concept of tensor and law of transformation of it�s components, the summation convention, invariants, covariant and contravariant vectors, covariant, contravariant and mixed tensors, symmetric and skew- symmetric tensors, outer and inner multiplications of tensors, quotient law of tensors, metric tensors, associated tensors, reciprocal tensor of a tensor. (16 L)

Tensor Calculus:

Christoffel Symbols of first and second kind, properties of Christoffel symbols, Covariant differentiation of tensors, the intrinsic or absolute derivative, gradient, divergence and curl in tensor form, Ricci�s theorem, Riemann-Christoffel curvature tensor, Ricci�s tensor, Einstein tensor, Geodesics, Riemannian and Geodesic coordinates, Geodesic curves in space, Serret-Frenet formulae, Geodesic curvature, First and second fundamental forms of a surface.

#### Course References:

Text Books:

1. Introduction to Tensor Analysis and the calculus of moving surfaces by Pavel Grinfeld, Springer, New York, 2010

2. Tensor Analysis by I. S. Sokolnikoff, John-Wiley, 2000.
References:

1. A brief on Tensor Analysis by James G. Simmonds, Springer-Verlag, New York, 1994

2. Tensor Analysis by Leonid P. Lebedev and Michael J. Cloud, World Scientific Publishing Co., London, 2003

3. Tensor Algebra and Tensor Analysis for Engineers with Applications to Continuum Mechanics by Mikhail Itskov, Springer, 2013.