Vector Calculus | Part IA, 2001

Suppose that

u=y2sin(xz)+xy2zcos(xz),v=2xysin(xz),w=x2y2cos(xz)u=y^{2} \sin (x z)+x y^{2} z \cos (x z), \quad v=2 x y \sin (x z), \quad w=x^{2} y^{2} \cos (x z)

Show that udx+vdy+wdzu d x+v d y+w d z is an exact differential.

Show that

(0,0,0)(π/2,1,1)udx+vdy+wdz=π2\int_{(0,0,0)}^{(\pi / 2,1,1)} u d x+v d y+w d z=\frac{\pi}{2}

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