Question

What is the divergence of the vector field  \( \vec{f} = 3x^2 \hat{i}+5xy^2\hat{j}+xyz^3\hat{k} \) at the point (1, 2, 3).

(a) 89

(b) 80

(c) 124

(d) 100

I got this question in semester exam.

My question is based upon Divergence and Curl of a Vector Field topic in section Vector Differential Calculus of Engineering Mathematics

Answer 1

Correct answer is (b) 80

Easy explanation: \( \vec{f} = 3x^2 \hat{i}+5xy^2\hat{j}+xyz^3\hat{k} \)

∴ div \(\vec{f}= ∇.\vec{f} = (\frac{∂}{∂x}\hat{i}+ \frac{∂}{∂y}\hat{j} +\frac{∂}{∂z}\hat{k}).(3x^2\hat{i} + 5xy^2\hat{j} + xyz^3\hat{k}) \)

\(= 6x +10xy+ 3xyz^2 \)

At the point (1, 2, 3)

div \(\vec{f} =  6(1)+10(1)(2)+3(1)(2)(3)^2 = 80. \)