$($a$)($Optional$)$ Let $F$ be a gradient vector field with $F=$gradf for some $C^1$ function function $f$:$R^{n}R.$ Suppose that $c$:$[a,b]R^n$ is a $($piecewise$)C^1-$path. Prove that

${\displaystyle \underset{c}{\overset{\phantom{}}{}}}$gradf$*ds=f(c(b))-f(c(a))$

This is called the Fundamental Theorem of Line Integrals.

$($b$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(z^{3}2xy)dx{x}^{2}dy3x{z}^{2}dz$ where $C$ is the closed curve obtained by going from $,,)$ to $(1,-1,0)$ to $(-1,-1,0)$ to $(-1,1,0)$ in that order then back to $(1,1,0)($i$.$e$.$ along the circumference of the unit square on $xy-$plane$).$

$($c$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}sin(x)dy-cos(y)dz$ where $C$ is the closed curve obtained by going from $(1,0,0)$ to $(0,1,0)$ to $(0,0,1)$ then back to $(1,0,0).$ Does this result contradict part $($a$)?$