Conservative fields and potential functions

Problem 12.253 · medium

1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x \sin{\left(y \right)}\right)\\\frac{d}{d y} \cos{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(y \right)}\\- \sin{\left(y \right)}\end{matrix}\right] \]
   ∂Q/∂x = ∂P/∂y: F is conservative.✓ Proved
2. Integrate the first component in x, then fix the 'constant' (a function of the other variables) by matching the other components.
   Reviewed
3. \[ \left[\begin{matrix}\frac{\partial}{\partial x} x \cos{\left(y \right)}\\\frac{\partial}{\partial y} x \cos{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}\cos{\left(y \right)}\\- x \sin{\left(y \right)}\end{matrix}\right] \]
   f = x*cos(y) has gradient F.✓ Proved
4. \[ - 2 \cos{\left(2 \right)} + 2 \cos{\left(1 \right)} \]
   ∫_C F·dr = f(end) − f(start), for any path.✓ Proved