Paper 3, Section I, B

Vector Calculus | Part IA, 2019


u=(2x+x2z+z3)exp((x+y)z)v=(x2z+z3)exp((x+y)z)w=(2z+x3+x2y+xz2+yz2)exp((x+y)z)\begin{aligned} u &=\left(2 x+x^{2} z+z^{3}\right) \exp ((x+y) z) \\ v &=\left(x^{2} z+z^{3}\right) \exp ((x+y) z) \\ w &=\left(2 z+x^{3}+x^{2} y+x z^{2}+y z^{2}\right) \exp ((x+y) z) \end{aligned}

Show that udx+vdy+wdzu d x+v d y+w d z is an exact differential, clearly stating any criteria that you use.

Show that for any path between (1,0,1)(-1,0,1) and (1,0,1)(1,0,1)

(1,0,1)(1,0,1)(udx+vdy+wdz)=4sinh1\int_{(-1,0,1)}^{(1,0,1)}(u d x+v d y+w d z)=4 \sinh 1

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