Divergence Calculator

Divergence Calculator

Divergence calculator is used to finding the divergence of the given vector field with steps in no time.

What is a divergence of a function?

The divergence of a vector is a method that turns a vector field into a scalar value. We obtain the scalar field by differentiating the given vector field. It is also known as the dot product of a vector field with the nabla operator.

Formula of divergence:

A vector field  F(x, y, z) and the nabla operator ∇=(∂/∂x, ∂/∂y, ∂/∂z). Then the divergence of the vector will be

div. F = ∇.F = ∂/∂x(F(x,y,z)) + ∂/∂y(F(x,y,z)) + ∂/∂z(F(x,y,z))

Explanation of the divergence:

The divergence of the function may be positive, negative, or zero. Let F is a vector

• If ∇. F < 0; it means the divergence is negative which identifies the fluid is denser at the given point.
• If ∇. F > 0; it means the divergence is positive which identifies the fluid is less dense at the given point.
• If ∇. F = 0; it means the divergence is zero which identifies the density of the fluid as constant at a given point.

How do we get the divergence?

To understand the step-by-step solution, we have discussed an example below comprehensively.

Example:

Find the divergence of the function (cos(xyz), x², z³) at the point (2, 3, 4).

Solution:

Step 1: It is the dot product of ∇=(∂/∂x, ∂/∂y, ∂/∂z) with the function.

∇.(cos(xyz), x², z³)

Step 2: Now calculate the partial derivative of the function.

∂/∂x(cos(xyz)) = -yz sin(xyz)

∂/∂y(x²) = 0

∂/∂z(z³) = 3z²

Step 3: Combine the terms.

∇.(cos(xyz), x², z³) = -yzsin(xyz)+ 3z²

Step 4: Put the value of the terms in the above function.

= -3(4)sin((2)(3)(4))+ 3(4)²

= 48-12sin(24)