\[T=T(x,y,z)\]
.Suppose we have a curve
\[\mathbf{r}\]
in the material. The curve is parametrized in terms of its distance from sme point, so that \[\mathbf{r}= \mathbf{r}(s)\]
.The change in
\[T\]
as it goes from \[(x,y,z)\]
to \[(x+ \delta x , y+ \delta y , z+ \delta z )\]
along the curve is \[\ delta T = \frac{\partial T}{\partial x} \delta x + \frac{\partial T}{\partial y} \delta y + \frac{\partial T}{\partial z} \delta z = (\frac{\partial T}{\partial x} , \frac{\partial T}{\partial y} , \frac{\partial T}{\partial z}) \cdot ( \delta x, \delta y \ \delta z = \mathbf{\nabla} \cdot \delta \mathbf{r}\]
Hence
\[\ \frac{\delta T}{\delta s} = \mathbf{\nabla} \cdot \frac{\delta \mathbf{r}}{\delta s}\]
Let
\[\delta \mathbf{r} \rightarrow \mathbf{0}\]
then\[\ \frac{d T}{ds} = (\mathbf{\nabla} T) \cdot \frac{d \mathbf{r}}{ds}\]