For the vector field
F(x,y,z) = e^y i + xe^y j + (z+1)e^z k, where
C: r(t) = t i + t^2 j + t^3 k, 0 ≤ t ≤1
Find a function ƒ such that F = gradƒ
and use the above to evaluate ∫C F•dr along C
Can you prove that F is a conservative field?
Vector fields that are conservative have integrals that are path independent.
Can you show that $\displaystyle \phi = xe^y + ze^z \quad \Rightarrow \quad \nabla \phi = F$?
What is $\displaystyle \phi (1) - \phi (0)$?