vector calculus identities with einstein notation

The resulting vector is the vector with its tail at the tail of and its head at the head of (adding “tip-to-tail”): Vectors can be added-to add two vectors and, just align the tip of to the tail of. Here, is the length of the vector and is a vector in the direction of with a length of (note that the “hat” notation is substandard in mathematics, although it is quite standard in physics). A vector can be expressed in terms of these two properties: Vectors have a “magnitude,” or length, and a direction. There are precise definitions for vectors and vector fields provided by linear algebra, and vectors/vector fields have been subjected to many degrees of abstraction and generalization in mathematics, but let’s just take for granted that vectors represent these sorts of “arrows” in space. Specifically, let’s handle them in an intuitive sense. Part 1: Vectors and Functions with Vectorsįirst, let’s talk about vectors. Obviously, this is not meant to be a substitute for a more rigorous foundation with more computational practice. Note: I presume basic knowledge of calculus in this post.

So what does this all mean? Let’s break it down, arduously and tediously, from the nitty-gritty of the vector calculus requisite for understanding the elegance of these equations. One popular formulation of Maxwell’s equations is the following: It is fascinating to me that Maxwell’s equations can so succinctly and elegantly express so many phenomena, from electric and magnetic interactions to light (electromagnetic waves).

Specifically, vector calculus is the language in which (classical) electromagnetism is written. Vector calculus is an extremely interesting and important branch of math with very relevant applications in physics.

Vector calculus identities with einstein notation