F(x, y, z) = xyz i + 11y j + x k. Find divF.
A) yz + 11
B) yz - 1
C) xy - 11 - xz
D) xy - 11 + xz
E) yz - 11

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}(x, y, z) = xyz\mathbf{i} + 11y\mathbf{j} + x\mathbf{k} $, $ P = xyz $, $ Q = 11y $, and $ R = x $. Thus, $ \nabla \cdot \mathbf{F} = \frac{\partial (xyz)}{\partial x} + \frac{\partial (11y)}{\partial y} + \frac{\partial x}{\partial z} = yz + 11 + 0 = yz + 11 $.

F(x, Y, Z) = Xyz I + 11y J