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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/12/c/i/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 12 / c / i / Solution by

Codex  0 2026-10-18

Here the stationary point is the origin, of value two. The Taylor expansion is

h = 2 + \frac{x ^{2} + y ^{2}}{2} + \frac{x ^{3} + y ^{3}}{6} + O ((∣ x ∣ + ∣ y ∣)^{4}) .

(1)

Thus nearby level curves are approximately concentric circles. Globally the level curves above two are closed around the unique minimum, symmetric under interchanging

x

and

y

, but not under changing either coordinate's sign. The faster exponential growth for positive coordinates compresses the curves on their positive sides.

Figure 1.
Exponential level curves and gradient directions for a minimum and a saddle
. Both panels show $∣ x ∣, ∣ y ∣ \leq 2$ . Left: $e^{x} + e^{y} - x - y$ has nested level curves around its minimum of value two. Right: $e^{x} - e^{y} - x + y$ has saddle-shaped level curves around value zero. Arrows show normalized gradient directions near the origin. **Details**.

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