Question

Divergence and Curl of a vector field are ___________

(a) Scalar & Scalar

(b) Scalar & Vector

(c) Vector & Vector

(d) Vector & Scalar

The question was asked in a national level competition.

The doubt is from Divergence and Curl of a Vector Field topic in portion Vector Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (b) Scalar & Vector

To explain I would say: Let \(\vec{f} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \)

div\(\vec{f} = (\frac{∂}{∂x} \hat{i} + \frac{∂}{∂y} \hat{j}+ \frac{∂}{∂z} \hat{k}).(a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \)

\( = \frac{∂a_1}{∂x}  + \frac{∂a_2}{∂y} + \frac{∂a_3}{∂z} \)      which is a scalar quantity.

Also curl \(\vec{f} =  \begin{vmatrix}

i & j & k\\

\frac{∂}{∂x} & \frac{∂}{∂y} & \frac{∂}{∂z}\\

a1 & a2 & a3\\

\end{vmatrix} \) = \(b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \)

Which is going to be a vector quantity as cross product of two vectors is again a vector, where dot product gives a scalar outcome.