Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 8

Table of contents
- a 8
  - Solution a
- b 8
  - Solution b

a

Solution

First take

F_{y} = x e^{s i n (x y)} cos (x y)

, then differentiate with respect to

x

. The product rule and chain rule give

F_{x y} = e^{s i n (x y)} [cos (x y) + x y (cos^{2} (x y) - sin (x y))] .

The function is smooth, so the two mixed partial derivatives agree.

b

Solution

For the divergence in Cartesian coordinates, differentiate each component with respect to its own coordinate:

\nabla \cdot G = 2 x z + 3 y^{2} z + 0 = z (2 x + 3 y^{2}) .

The third component contributes zero because it is independent of

z

.

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