Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.
A) 2y + 2z + 2x
B) 3y + 2z + x
C) y + z + x
D) 2y -2 z + 2x
E) y + 2z + 3x

Question

Quizplus · Accepted Answer

The divergence of a vector field $ \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} $ is given by $ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} $. For $ \mathbf{F} = (xy + xz)\mathbf{i} + (yz + yx)\mathbf{j} + (zx + zy)\mathbf{k} $, we compute the partial derivatives: $ \frac{\partial P}{\partial x} = y + z $, $ \frac{\partial Q}{\partial y} = z + x $, and $ \frac{\partial R}{\partial z} = x + y $. Adding these gives $ 2y + 2z + 2x $.

Compute the Divergence for the Vector Field F = (Xy