Tensors under general coordinate transformations 2

In this lecture I begin (after the customary recap) with an example of transformation of vector components under the well known Cartesian to polar coordinate transformation. I then address the apparent conflict between the resulting polar components and the ones we had learnt about in high school. With this example out of the way, I then go on to discuss some essential features of tensors like contractions and the quotient rule. In the final part of this lecture I explain why partial derivatives of (co)vector fields are not tensors. I then introduce an additional structure - the connection and show how they should transform to ensure that we can get a version of the derivative that is tensorial - the covariant derivative.