Pre-Quantum Electrodynamics

Consider a function \(T\) of the 3 variables \(x, y, z\) noted as \(T({\bf r})\). This is commonly known as a scalar field.

Its change under a displacement by \(d{\bf l}\) is given by

\[ \Delta_{d{\bf l}} T = T({\bf r} + d{\bf l}) - T({\bf r}) = {\boldsymbol \nabla} T \cdot d{\bf l} + \mbox{O}(dl^2) \label{Gr(1.35)} \]

in which

\[ {\boldsymbol \nabla} T \equiv \frac{\partial T}{\partial x} ~\hat{\bf x} + \frac{\partial T}{\partial y} ~\hat{\bf y} + \frac{\partial T}{\partial z} ~\hat{\bf z} \label{Gr(1.36)} \]

is a vector called the gradient of \(T\).