In each of the following, compute the gradient of the function and evaluate this gradient at the given point. Determine at this point the maximum and minimum rate of change of the function. (a) φ(x,y,z)=xyz;(1,1,1) (5marks) (b) φ(x,y,z)=x 2
y−sin(xz);(1,−1,π/4) (5marks) (c) φ(x,y,z)=2xy+xe z
;(−2,1,6) (5marks) (d) φ(x,y,z)=cos(xyz);(−1,1,π/2) (5marks) 2. Let φ be a continuous scalar field with continuous first and second partial derivatives. In each of the following, compute ∇φ and verify explicitly that ∇×(∇φ)=0 (a) φ(x,y,z)=x−y+2z 2
(8marks) (b) φ(x,y,z)=18xyz+e x
(8marks) (c) φ(x,y,z)=−2x 3
yz 2
(8marks) (d) φ(x,y,z)=sin(xz)