What is the gradient?

Question

Consider the function S(H, T, V)=(V+10)e^(-1/H)+(T-80)^2+V^3. What is the gradient of the function S(H, T, V) at the point (1, 81, 3)?

Answer 1

S(H,T,V) = (V + 10)e^-1/H + (T - 80)² + V³

∇S(H,T,V) = <∂S/∂H, ∂S/∂T, ∂S/∂V>

= <(V+10)e^-1/H/H²), 2(T-80), e^-1/H+3V²>

∇S(1,81,3) = <(3+10)e^-1/1, 2(81-80), e^-1/1+3*3²>

= <13/e, 2, 27+1/e>