Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 17 / a / ii

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- Solution ii

Solution

Differentiate the flux:

Φ^{'} (z_{0}) = 3 π (1 - z_{0}^{2}), Φ^{''} (z_{0}) = - 6 π z_{0} .

On

- 1 < z_{0} < 1

,

Φ

is strictly increasing. Its limiting endpoint values are

Φ (- 1) = 0

and

Φ (1) = 4 π

, and

Φ (0) = 2 π

. It has horizontal limiting tangents at the endpoints, curves upward for

z_{0} < 0

, curves downward for

z_{0} > 0

, and has an inflection point at zero.

The derivative graph is a downward-opening parabola, approaching zero at both endpoints and attaining its maximum

3 π

at

z_{0} = 0

.

Figure 1.
Flux through the truncated unit sphere and its derivative
. Both curves are scaled by $π$ . Endpoint circles show limiting values because the prescribed truncation parameter lies in $(- 1, 1)$ . The flux increases from zero to $4 π$ , while its derivative is largest at the equator. Details.

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