25. Session 2008/09 Semester II
(a) Given the vector field $F = \cos x \mathbf{i} + \sin z \mathbf{j} + z \mathbf{k}$. Determine
(i) $\nabla \cdot F$.
(ii) $\nabla \times F$.
(iii) $\nabla (\nabla \cdot F)$.
(2 marks)
(3 marks)
(2 marks)
6
(b) An object moves in space with acceleration
SSCE 1993: Tutorial 3
$\mathbf{a}(t) = -t\mathbf{i} + 2\mathbf{j} + (2 - t)\mathbf{k}$.
When $t = 0$, the object is at the point $(1, 0, 0)$ with velocity $\mathbf{v}(0) = 2\mathbf{i} - 4\mathbf{j}$.
(i) Find the velocity $\mathbf{v}(t)$ and the position vector $\mathbf{r}(t)$.
(ii) What is the speed and the direction of motion when $t = 1$.
(5 marks)
(c) The greatest rate of change of a function $f(x, y, z)$ at a point $P$ in the direction of
$\mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k}$ is $2\sqrt{3}$. Find $\nabla f$ at $P$.
(5 marks)
