Tensor Analysis : Lecture 01

This is the first lecture on a hopefully short series of lectures on Tensor analysis. Here I will discuss the old fashioned "index gymnastics" approach to the topic. While I am sure most will agree that the more abstract approach that follows from a thorough understanding of differential geometry is aesthetically more pleasing, there is no substitute to the coordinate based approach as far as practical applications are concerned. In this lecture I have introduced the "physicist's version" of the definition of a vector. I begin with describing 3-vectors (vectors under rotation), and then go on to 4-vectors (vectors under Lorentz transformation). After describing properties of the transformation matrix for these two cases, I go on to describe the definition of a vector under a general coordinate transformation. I then introduce the Einstein summation convention - a tool that will make calculations substantially simpler in the lectures to come. I wrap up this version with a slightly philosophical digression on how a differential geometric approach (which we will not discuss in this course) helps in making some of the results that we talk about more natural.