Vector calculus

Working with vectors is a very fundamental part of any scientific theory.

Related to the gradient, divergence and curl of scalar and vector fields, we can write the following short-hand notations. If we introduce a short-hand notation for the partial derivative as \[\begin{equation} \require{physics} \partial_i = \pdv{r_i} % \thinspace , \end{equation}\] the \(i\)-th component of the gradient of a scalar field can be written as \[\begin{equation} \qty( \grad{f} )_i = \partial_i f % \thinspace . \end{equation}\]

Any dot product between two vectors can be written as (Einstein summation convention implied) \[\begin{equation} \vb{A} \vdot \vb{B} = A_i B_i % \thinspace , \end{equation}\] so the divergence of a vector field becomes \[\begin{equation} \boldsymbol{\nabla} \vdot{\vb{A}} = \partial_i A_i % \thinspace . \end{equation}\]

An element of the outer product of two vectors can be written as \[\begin{equation} \qty( \vb{A} \cross \vb{B} )_i % = \epsilon_{ijk} A_j B_k % \thinspace , \end{equation}\] by introducing the three-dimensional Levi-Civita tensor \[\begin{equation} \epsilon_{ijk} = % \begin{cases} +1 &\text{if } ijk \text{ is an even permutation of } 123 \\ -1 &\text{if } ijk \text{ is an odd permutation of } 123 \\ 0 &\text{otherwise} \thinspace . \end{cases} \end{equation}\]

The curl of a vector field then becomes \[\begin{align} \qty( \curl{\vb{A}} )_i = \epsilon_{ijk} \partial_j A_k % \thinspace . \end{align}\] Useful properties of the 3D Levi-Civita tensor are related to Kronecker deltas: \[\begin{equation} \epsilon_{ijk} \epsilon_{ilm} % = \delta_{jl} \delta_{km} % - \delta_{jm} \delta_{lk} % \end{equation}\] and: \[\begin{equation} \epsilon_{ikl} \epsilon_{jkl} = 2 \delta_{ij} % \thinspace . \end{equation}\]