Gradient of Scalar field
We all know that a scalar field can be solved more easily as compared to vector field. Therefore, it is better to convert a vector field to a scalar field. Not all vector fields can be changed to a scalar field; however, many of them can be changed.
The relation between the two types of fields is accomplished by the term gradient. Hence, gradient of a vector field has a great importance for solving them.
The gradient ‘grad f’ of a given scalar function f(x, y, z) is the vector function expressed as
Grad f = (df/dx) i + (df/dy) j + (df/dz) k
We can also show the above formula in terms of ‘nabla’ and the new form of this formula is expressed by the image below.
Let us discuss an example.
If a function is a f(x, y, z) = 2x+ yz – 3y2, then grad f= f= 2i+ (z-6y)j+ yk. Actually, the term df/dx is the partial differentiation with respect to variable x.
Properties of gradient
- We can change the vector field into a scalar field only if the given vector is differential. The given vector must be differential to apply the gradient phenomenon.
- The gradient of any scalar field shows its rate and direction of change in space.
Having discussed the gradient, we turn next to the divergence.