Find the divergence of the vector yi + zj + xk.

The divergence of a vector field (mathbf{F} = Pmathbf{i} + Qmathbf{j} + Rmathbf{k}) is given by the formula:

[

nabla cdot mathbf{F} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z}

]

For the vector field (mathbf{F} = ymathbf{i} + zmathbf{j} + xmathbf{k}), we identify:

– (P = y)

– (Q = z)

– (R = x)

Now we compute the partial derivatives:

1. (frac{partial P}{partial x} = frac{partial y}{partial x} = 0)
2. (frac{partial Q}{partial y} = frac{partial z}{partial y} = 0)
3. (frac{partial R}{partial z} = frac{partial x}{partial z} = 0)

Adding these together, we find:

[

nabla cdot mathbf{F} = 0 + 0 + 0 = 0

]

Therefore, the divergence of the vector field (ymathbf{i} + zmathbf{j} + xmathbf{k}) is:

[