Question

Find the divergence of this given vector \(\vec{r}=12x^6 y^6 \vec{i}+3x^3 y^3 z\vec{j}+3x^2 yz^2 \vec{k}\).

(a) \(12x^5 y^6+2x^3 yz+6x^2 yz\)

(b) \(72x^5 y^6+2x^3 yz+3x^2 yz\)

(c) \(72x^5 y^6+2x^3 yz+6x^2 yz\)

(d) \(6x^5 y^6+2x^3 yz+6x^2 yz\)

The question was asked during an interview.

I'd like to ask this question from Using Properties of Divergence and Curl in section Vector Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (c) \(72x^5 y^6+2x^3 yz+6x^2 yz\)

To explain: We know that divergence of a vector is given by

\(\bigtriangledown.\vec{r}=\frac{\partial(12x^6 y^6)}{\partial x}+\frac{\partial(3x^3 y^3 z)}{\partial y}+\frac{\partial(3x^2 yz^2)}{\partial z}\)

\(\bigtriangledown.\vec{r}=12×6x^5 y^6+x^3×2y×z+3x^2×y×2z\)

\(\bigtriangledown.\vec{r}=72x^5 y^6+2x^3 yz+6x^2 yz\).