Pre-Quantum Electrodynamics

\[ {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} \label{Gr(1.48)} \]

where \({\cal P}\) is a path from point \({\bf a}\) to point \({\bf b}\).

Example: work done by a force, \(W = \int {\bf F} \cdot d{\bf l}\).

Integral over a closed loop:

\[ \oint {\bf v} \cdot d{\bf l} \label{Gr(1.49)} \]

\[ \int_{\cal S} {\bf v} \cdot d{\bf a} \label{Gr(1.50)} \]

Over a closed surface:

\[ \oint {\bf v} \cdot d{\bf a} \]

\[ \int_{\cal V} T d\tau \label{Gr(1.51)} \]

In Cartesian coordinates:

\[ d\tau = dx dy dz \label{Gr(1.52)} \]