Introduction In this post I go over the basics of index notation for calculus. This is the notation that was invented by Einstein and also known in machine learning community as einsum. It serves as a convenient way to supress summations in formulas, by viewing repeated indices as being summed over. In the field of tensor calculus and in particular fluid dynamics, this notation can come in handy when deriving complex formulas involving $\nabla, \nabla \cdot, \nabla^2$....

Green’s Three Identities are the fundamental results for vector calculus, which is widely used in basic PDE theories and fluid mechanics. Here I present the results. Assume that $U \subset \mathbb{R}^n$ be open and has $C^1$ boundary, then: Integration by Part Result 1 (Integration by Part): for $u,v \in C^1(\bar{U})$, we have: $$\int_U u_{x_i} v dx + \int_U v_{x_i} u dx = \int_{\partial U} uv \hat{n}^i dS, \quad (i = 1,\cdots, n)$$...