Problem
In vector calculus, the gradient of a scalar function $T$, is the vector function $\mathbfit{\nabla}T$, which points in the direction of the maximum increase of $T$, and whose magnitude is the rate of increase of $T$ along this direction. If $T$ is a scalar function of 3 variables $T = T(x, y, z)$, and $\mathbfit{\hat{x},\hat{y},\hat{z}} $ are the unit vectors parallel to $x$, $y$, and $z$ axes, respectively, derive the mathematical definition of $\mathbfit{\nabla}T$ using a theorem on partial derivatives: $dT=\left(\frac{\partial T}{\partial x}\right) dx + \left(\frac{\partial T}{\partial y}\right) dy + \left(\frac{\partial T}{\partial z}\right) dz$.