Module 1 : A Crash Course in Vectors
Lecture 4 : Gradient of a Scalar Function
  Note that the change $dT$ involves a change in temperature with respect to each of the three directions. We define a vector called the gradient of $T$, denoted by $\nabla T$ or grad $T$ as
\begin{displaymath}\nabla T = \hat\imath\frac{\partial T}{\partial x} + \hat\jm... ...c{\partial T}{\partial y} + \hat k\frac{\partial T}{\partial z}\end{displaymath}
  using which, we get
 
\begin{displaymath}dT = \nabla T\cdot(\hat\imath dx + \hat\jmath dy + \hat k dz)= \nabla T\cdot d\vec r\end{displaymath}
  Note that $\nabla T$, the gradient of a scalar $T$ is itself a vector. If $\theta$ is the angle between the direction of $\nabla T$ and $d\vec r$,
 
\begin{displaymath}dT = \mid\nabla T\mid\mid d\vec r\mid \cos\theta = (\nabla T)_r\mid d\vec r\mid\end{displaymath}
  where $(\nabla T)_r$ is the component of the gradient in the direction of $d\vec r$. If $d\vec r$ lies on an isothermal surface then $dT=0$. Thus, $\nabla T$ is perpendicular to the surfaces of constant $T$. When $d\vec r$ and $\nabla T$ are parallel, $\cos\theta=1$$dT$ has maximum value. Thus the magnitude of the gradient is equal to the maximum rate of change of $T$ and its direction is along the direction of greatest change.
  The above discussion is true for any scalar field $V$. If a vector field can be written as a gradient of some some scalar function, the latter is called the potential of the vector field. This fact is of importance in defining a conservative field of force in mechanics. Suppose we have a force field $\vec F$ which is expressible as a gradient
 
\begin{displaymath}\vec F = \nabla V\end{displaymath}
   
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