(a) Let F = 2xz²i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf

Find curl(F x G) = V × (F × G). F(x, y, z) = i + 8xj + 4yk G(x, y, z) = xi - yj + zk

Apply the translation theorem to find the inverse Laplace transform of the following function. 9s+12 F(s) = S²-85+41 L₁' {F(s)} = Table of Laplace transforms f(t)=L¹{F(s)} L{f(t)} = F(s) 1 1 (s > 0) S 1 5² n! t ta eat n+1 s [(a+1) n+1 s' 1 s-a (s > 0) (s> 0) (s>0) (s>a) f(t)=L¹{F(s)} cos kt sin kt cosh kt sinh kt (please enter an expression using t as the variable) eatin L{f(t)} = F(s) S s²+k² k s²+k² S s²-K² k 2 s-k n! (s-a)+1 (s > 0) (s > 0) (s> |k|) (s> |kl) (s>a)

Suppose F(x, y) = (x² + 3y, 7x − 7y²). Use Green's Theorem to calculate the circulation of F around the perimeter of the triangle C oriented counter-clockwise with vertices (6, 0), (0, 3), and (-6, 0). Jo F. dr =