Divergence of (r²r)

To find the divergence of (r²r) vector, we need to take the dot product of the gradient operator (∇) with the vector (r²r).

Step 1: Expressing r² in terms of its components
Given that the position vector r = xi + yj + zk, we can find r² by taking the dot product of r with itself:
r² = (xi + yj + zk) · (xi + yj + zk)
= x²i·i + y²j·j + z²k·k + 2xyi·j + 2xzi·k + 2yzj·k
= x² + y² + z² + 2xyi·j + 2xzi·k + 2yzj·k

Step 2: Finding the gradient of r²
The gradient of r² is given by:
∇(r²) = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k (x² + y² + z² + 2xyi·j + 2xzi·k + 2yzj·k)

Step 3: Taking the dot product with r
Now, we need to take the dot product of ∇(r²) with r to find the divergence of (r²r):
∇(r²) · r = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k · (xi + yj + zk)
= (∂/∂x)(xi) + (∂/∂y)(yj) + (∂/∂z)(zk)
= (∂/∂x)(xi) + (∂/∂y)(yj) + (∂/∂z)(zk)
= (∂/∂x)(xi) + (∂/∂y)(yj) + (∂/∂z)(zk)
= (∂/∂x)(x) + (∂/∂y)(y) + (∂/∂z)(z)
= x + y + z

Therefore, the divergence of (r²r) is x + y + z.

Conclusion
The divergence of (r²r) is x + y + z. This means that the vector field (r²r) is diverging radially outward from the origin. The value of the divergence depends on the coordinates x, y, and z.