Exploring Vector Fields: Divergence, Curl, and Their Relationship

In this article, we will dive into the world of vector calculus and explore the concepts of divergence and curl for a specific vector field. Let's consider the vector field:

F = (7yz)i + (6xz)j + (6xy)k

where i, j, and k are the standard unit vectors in the x, y, and z directions, respectively.

A) Computing the Divergence of F

The divergence of a vector field F = (P, Q, R) is defined as:

div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Applying this formula to our vector field F, we get:

div F = ∂(7yz)/∂x + ∂(6xz)/∂y + ∂(6xy)/∂z
     = 7z + 6z + 6x
     = 7z + 6z + 6x
     = 13z + 6x

Therefore, the divergence of the vector field F is 13z + 6x.

B) Computing the Curl of F

The curl of a vector field F = (P, Q, R) is defined as:

curl F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k

Applying this formula to our vector field F, we get:

curl F = (∂(6xy)/∂y - ∂(6xz)/∂z)i + (∂(7yz)/∂z - ∂(6xy)/∂x)j + (∂(6xz)/∂x - ∂(7yz)/∂y)k
     = (6x - 6x)i + (7y - 6y)j + (6z - 7z)k
     = 0i + (y)j + (-z)k

Therefore, the curl of the vector field F is (0)i + (y)j + (-z)k.

C) Computing the Divergence of the Curl of F

To compute the divergence of the curl of F, we simply apply the divergence formula to the result of the curl computation:

div curl F = ∂(0)/∂x + ∂(y)/∂y + ∂(-z)/∂z
           = 0 + 1 + (-1)
           = 0

Therefore, the divergence of the curl of the vector field F is 0.

In conclusion, we have successfully computed the divergence, curl, and the divergence of the curl for the given vector field F = (7yz)i + (6xz)j + (6xy)k. These vector calculus operations are essential in understanding and analyzing the properties of vector fields, with applications in various fields, such as physics, engineering, and mathematics.