The maximum space rate of change of the function which is increasing direction of line function is known as

Explanation:

The gradient of the scalar function:

The magnitude of the gradient is equal to the maximum rate of change of the scalar field and its direction is along the direction of greatest change in the scalar function.

Let ϕ be a function of (x, y, z)

Then grad \(\left( ϕ \right)=\hat{i}\frac{\partial ϕ }{\partial x}+\hat{j}\frac{\partial ϕ }{\partial y}+\hat{k}\frac{\partial ϕ }{\partial z}\)

Divergence of the vector function:

The net outward flux from a volume element around a point is a measure of the divergence of the vector field at that point.

Divergence of a function \(\vec F = {F_1}\hat i + {F_2}\hat j + {F_3}\hat k\)

\( \nabla \cdot \vec F = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial y}} + \frac{{\partial {F_3}}}{{\partial z}}\)

Gauss divergence theorem:

It states that the surface integral of the normal component of a vector function \(\vec F\) taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function \(\vec F\) taken over a volume enclosed by the closed surface ‘S’.

\(\underset{S}{\mathop \iint }\,\vec{F}.\hat{n}ds=\iiint\limits_{V}{\nabla .\vec{F}dv}\)

Stokes theorem:

It states that the line integral of a vector filed \(\vec F\) around any closed surface C is equal to the surface integral of the normal component of the curl of vector \(\vec F\) over an unclosed surface ‘S’.

\(\underset{C}{\mathop \oint }\,\vec{F}.\overrightarrow{dr}=\underset{S}{\mathop \iint }\,curl~\vec{F}.\hat{n}ds\)

Important Points

Curl of the scalar function and divergence of the scalar function are not defined.