##### Line, Surface and Volume Integrals
c.m.ic.lsv

###### Line Integrals

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

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:

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

###### Surface Integrals

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

Over a closed surface:

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

###### Volume Integrals

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

In Cartesian coordinates:

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

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42