This book presents the fundamentals of modern tensor calculus for
students in engineering and applied physics, emphasizing those
aspects that are crucial for applying tensor calculus safely in
Euclidian space and for grasping the very essence of the smooth
manifold concept. After introducing the subject, it provides a
brief exposition on point set topology to familiarize readers with
the subject, especially with those topics required in later
chapters. It then describes the finite dimensional real vector
space and its dual, focusing on the usefulness of the latter for
encoding duality concepts in physics. Moreover, it introduces
tensors as objects that encode linear mappings and discusses affine
and Euclidean spaces. Tensor analysis is explored first in
Euclidean space, starting from a generalization of the concept of
differentiability and proceeding towards concepts such as
directional derivative, covariant derivative and integration based
on differential forms. The final chapter addresses the role of
smooth manifolds in modeling spaces other than Euclidean space,
particularly the concepts of smooth atlas and tangent space, which
are crucial to understanding the topic. Two of the most important
concepts, namely the tangent bundle and the Lie derivative, are
subsequently worked out.
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