SSCE $1993$

QUESTION $2(20$ MARKS$)$

a$)$ A curve in space is given by the equation

$r(t)=costi+sintj+ln(cost)k$

Find the unit normal vector $N(t)$ at the point $(1,0,0).$

$(6$ marks$)$

b$)$ Given the vector field $G(x,y,z)=lnzi-3ysinyj+e^{x}k$ and scalar function $f(x,y,z)=x^{2}y+2xy+z^2.$ Find

i$.$ gradf

$(2$ marks$)$

ii$.$ curl G

$(3$ marks$)$

iii. gradf$*(gradG).$

$(2$ marks$)$

c$)$ Consider a scalar field $P(x,y,z)$ defined by the equation

$P(x,y,z)=e^{(x^2+z^2)}+4x^2{z}^3{e}^{y^2}$

i$.$ At the point $A(1,2,0),$ find the directional derivatives of $P(x,y,z)$ towards the point $B(0,1,2).$

$(4$ marks$)$

ii$.$ Determine the unit vector representing the direction in which $P(x,y,z)$ changes the least at the point $A,$ and find the corresponding change of $P(x,y,z)$ in that direction.

$(3$ marks$)$