Introduction:

In mathematics the, Del operator comes under the topic vector calculus. Del operator is nothing but the vector differential operator which is represented as nabla symbol. Del operator is used in the different dimensional domain such as vector field in divergence and curl, scalar field in gradient∇. Del operator is a mathematical notation which is used only for these three operations to make the equation easier to remember and also for writing. There are three meaning for Del operator in derivative operation in vector calculus which is divergence, gradient, and curl. These operators are viewed as dot, scalar, and cross products.

 

Formulas for Del Operators:

The general form Del operator is given as,

∇ = i (∂/∂x) + j(∂/∂y)+ k(∂/∂z).

The coordinates x, y, z are the scalar field which are single-valued function

 When we multiply the Del operator with a scalar field A(x, y, z) we get the vector field which is named as gradient. The gradient of A is given as, 

∇A = i (∂A /∂x) + j(∂A /∂y) + k(∂A /∂z).

 

When the Del operator product Ñ with a vector field v(x, y, z) we get a scalar value which is named as divergence. Then divergence of V is given as,

 ∇.V= (∂Vx /∂x) + (∂Vy /∂y)+ (∂Vz/∂z).                      

When the Del operator cross product with vector v(x, y, z) it gives the vector value which is named as curl. Then curl of v is given as,

∇xV= i((∂Vz/∂z) - (∂Vy /∂y))+ j ((∂Vx /∂x) - (∂Vz/∂z))+ k((∂Vy /∂y) - (∂Vx /∂x)).

Examples:

Example1: Find the divergence of V = (3x, 2xy, 4z)

Solution:

Then divergence of V is given as,

 ∇.V= (∂Vx /∂x) + (∂Vy /∂y)+ (∂Vz/∂z).

                              Where,

.                       ∂Vx /∂x = 3,

                        ∂Vy /∂y = 2y and

                        ∂Vz /∂z = 4

                        ∇.V = 3 + 2x + 4.

                                = 7 + 2x.

The divergence of V = 7 + 2x.

Example 2: find the gradient of V (4y, 3xz, and 2 x)?

Solution:

The gradient of A is given as, 

                                     Where,

                                    ∇A = i (∂A /∂x) + j(∂A /∂y) + k(∂A /∂z).

                                     (∂A /∂x) = 0

                                     (∂A /∂y) = 3z

(∂A /∂z) = 0

∇A = 3zj.

The gradient of A = 3zj

 Practice problem:

Problem 1: Find the divergence for the given data V= (4xz, 3x, 5yz)?

Answer is 4z + 5y

Problem 2: Find the Gradient of the given data A = (4yz, 3y2x, 4y)?

                            Answer is  6xyj.