Question

Evaluate ∫C∇f.dr using fundamental theorem of line integral, where f(x,y,z)=cos (πx)+sin(πy)-xyz and C is any path that starts at (1,1/2,2) and ends at (2,1,-1)

Answer 1

$f(x,y,z)$ is differentiable (because each component of $\nabla f$ is continuous), so the gradient theorem holds. This simply means

$\displaystyle\int_C\nabla f\cdot\mathrm dr=f(2,1,-1)-f\left(1,\frac12,2\right)=3-(-1)=4$