Paper 3, Section I, C

Vector Calculus | Part IA, 2012

Define what it means for a differential Pdx+QdyP d x+Q d y to be exact, and derive a necessary condition on P(x,y)P(x, y) and Q(x,y)Q(x, y) for this to hold. Show that one of the following two differentials is exact and the other is not:

y2dx+2xydyy2dx+xy2dy\begin{aligned} &y^{2} d x+2 x y d y \\ &y^{2} d x+x y^{2} d y \end{aligned}

Show that the differential which is not exact can be written in the form gdfg d f for functions f(x,y)f(x, y) and g(y)g(y), to be determined.

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