Understanding Vector Divergence Calculation for xi + yj + zk

This explanation details the calculation of the divergence for a given vector field.

What is Vector Divergence?

The divergence is a vector calculus operation that measures the rate at which a vector field flows outwards from a given point. It essentially quantifies the "source" or "sink" strength at that point. If the divergence is positive, it indicates an outward flow (a source); if negative, an inward flow (a sink); and if zero, the flow is incompressible (no net flow in or out).

The Vector Field

The vector field provided is:

F = xi + yj + zk

In this vector field:

• The component in the x-direction is $P(x, y, z) = x$.
• The component in the y-direction is $Q(x, y, z) = y$.
• The component in the z-direction is $R(x, y, z) = z$.

Calculating the Divergence

The divergence of a vector field $F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k$ is calculated using the Del operator ($\nabla$) dotted with the vector field:

$\nabla \cdot F = (\frac{\partial}{\partial x}i + \frac{\partial}{\partial y}j + \frac{\partial}{\partial z}k) \cdot (Pi + Qj + Rk) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$

Step-by-Step Calculation:

1. Find the partial derivative of P with respect to x:

   Here, $P = x$. So, $\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(x) = 1$.

2. Find the partial derivative of Q with respect to y:

   Here, $Q = y$. So, $\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(y) = 1$.

3. Find the partial derivative of R with respect to z:

   Here, $R = z$. So, $\frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(z) = 1$.

4. Sum the partial derivatives:

   The divergence is the sum of these results:

   $\nabla \cdot F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} = 1 + 1 + 1 = 3$

Conclusion

The divergence of the vector field $F = xi + yj + zk$ is 3.