Question

Divergence of  \(\vec{f} (x, y, z) = e^{xy} \hat{i} -cos⁡y \hat{j}+(sinz)^2 \hat{k}.\)

(a) ye^xy+ cos⁡y + 2 sinz.cosz

(b) ye^xy– sin⁡y + 2 sinz.cosz

(c) 0

(d) ye^xy+ sin⁡y + 2 sinz.cosz

This question was posed to me during an online exam.

My query is from Divergence and Curl of a Vector Field in chapter Vector Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (d) ye^xy+ sin⁡y + 2 sinz.cosz

Easy explanation: \(div \vec{f} = (\frac{∂}{∂x}\hat{i} + \frac{∂}{∂y}\hat{j}+ \frac{∂}{∂z}\hat{k}) .( e^{xy} \hat{i} – cos⁡y \hat{j} + (sinz)^2  \hat{k}) \)

\( = \frac{∂}{∂x}(e^{xy}) + \frac{∂}{∂y}(-cos⁡y) + \frac{∂}{∂z}((sinz)^2) \)

\( = ye^{xy} + sin⁡y + 2 sinz.cosz .\)