Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 12 / d

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- Solution d

Solution

The zero level set is especially simple:

f = 0 ⟺ x = 0 or y = 0 or x^{2} + y^{2} = 1 .

The two axes and the unit circle divide the plane into regions where the signs alternate. Inside the circle,

f

has the opposite sign to

x y

; outside it has the same sign. Closed level curves surround the two interior local minima and two interior local maxima; the zero curves cross at the five saddle points of a scalar function. Farther out, the nonzero level curves are open within the displayed square.

Figure 1.
Contours, stationary points and gradient directions of the quartic surface
. Left: signed level curves and the exact zero set consisting of both axes and the unit circle. Right: normalized gradient directions, including arrows on the zero curves. The nine stationary points are marked in both panels. Details.

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