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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 3

Writing $R=x^2+y^2$, direct differentiation gives
$$
\frac{\partial u}{\partial x}
=\frac{-2x(x^2-3y^2)}{R^3},
\qquad
\frac{\partial v}{\partial y}
=\frac{2x(x^2-3y^2)}{R^3},
$$
so $\nabla\mathbin{\cdot}\mathbf u=0$. Likewise,
$$
\frac{\partial v}{\partial x}
=\frac{-2y(3x^2-y^2)}{R^3}
=\frac{\partial u}{\partial y},
$$
so the two-dimensional <vorticity> $\partial_xv-\partial_yu$ vanishes. The flow is therefore <incompressible flow>[incompressible] and <irrotational flow>[irrotational] away from the origin.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 7A / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 7A / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 7A / a / Solution

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