Question 2. Vector calculus
(a) Consider a vector field \(\vec{A} = xyz\hat{x} + y^2\hat{y} + (x + y + z)\hat{z}\).
(i) Calculate its divergence \(\nabla \cdot \vec{A}\) at point \((x, y, z) = (1,2,3)\).
[5 marks]
(ii) Calculate its curl \(\nabla \times \vec{A}\) at the same point.
[5 marks]
(iii) Calculate its line integral \(\int \vec{A} \cdot d\vec{l}\) along a straight line from
\((1,2,3)\) to \((1,3,3)\).
[5 marks]
(b) Calculate the value of \(\nabla \cdot \nabla \times \vec{A}\), where \(\vec{A}\) is an arbitrary vector field.
[5 marks]
(c) Given a vector field \(\vec{A} = x\hat{x} + y\hat{y} + z\hat{z}\), calculate its surface integral
\(\oint_S \vec{A} \cdot d\vec{a}\) over the enclosed surface of a cylinder. The cylinder has
a radius of 2 and a height of 3. The centres of its two ends are at
\((x, y, z) = (0,0,0)\) and \((0,0,3)\), respectively.
