# 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.