Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 1 / b

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- Solution b

Solution

The divergence theorem states that, for a continuously differentiable vector field

H

on a bounded region

V

with piecewise smooth boundary

S

and outward unit normal

\hat{n}

,

\int_{V} \nabla \cdot H d V = \int_{S} H \cdot \hat{n} d S .

To apply it, first derive the divergence of a cross product using the Levi-Civita symbol and the product rule:

\nabla \cdot (F \times G) = ϵ_{ijk} \partial_{i} (F_{j} G_{k}) = ϵ_{ijk} (\partial_{i} F_{j}) G_{k} + ϵ_{ijk} F_{j} \partial_{i} G_{k} = G \cdot (\nabla \times F) - F \cdot (\nabla \times G) .

In the second term the minus sign comes from interchanging the first two indices of the Levi-Civita symbol. The divergence theorem with

H = F \times G

now yields

\int_{V} [G \cdot (\nabla \times F) - F \cdot (\nabla \times G)] d V = \int_{S} (F \times G) \cdot \hat{n} d S .

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