Module 1 : A Crash Course in Vectors
Lecture 4 : Gradient of a Scalar Function
Gradient of a Scalar Function :
Consider a scalar field such as temperature $T(x,y,z)$ in some region of space. The distribution of temperature may be represented by drawing isothermal surfaces or contours connecting points of identical temperatures,
 
  One can draw such contours for different temperatures. If we are located at a point $\vec r$ on one of these contours and move away along any direction other than along the contour, the temperature would change.
  The change $\Delta T$ in temperature as we move away from a point $P(x,y,z)$ to a point $Q(x+\Delta x, y+\Delta y, z+ \Delta z)$ is given by
 
\begin{displaymath}\Delta T = \frac{\partial T}{\partial x}\Delta x + \frac{\par... ...T} {\partial y}\Delta y + \frac{\partial T}{\partial z}\Delta z\end{displaymath}
  where the derivatives in the above expression are partial derivatives.
   
1
2