The Physics Post

In the previous tutorial, we explored the metric tensor and basis. In this tutorial, we will delve into the concept of tensors and their properties related to coordinate transformations.  A tensor is a multidimensional array that obeys certain transformation rules under coordinate transformations. A vector is an example of a tensor, specifically one with rank 1. One of the key properties of a vector is that it possesses both magnitude and direction. The significance of direction lies in the fact that a vector can be transformed under a coordinate transformation.  Let's consider the basis vectors denoted by $g_i$, which can be expressed as $g_i = \frac{\partial \vec{r}}{\partial x^i}$. We also have another basis, denoted by $g'_i=\frac{\partial \vec{r}}{\partial x'^i}$.  By applying the chain rule, we can establish the following relationship between the two bases:  $\frac{\partial \vec{r}}{\partial x^j} = \frac{\partial \vec{r}}{\partial x'^k} \frac{\partial x'k}{\pa...