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ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2012/9/b/iv/solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 9 / b / iv / Solution by

Codex  0 2026-10-31

Differentiating the projected parameterization gives

J = \frac{\partial ( x , y )}{\partial ( s , t )} = det (1 1 + 2 s 1 - 1) = - 2 (s + 1) .

(1)

It vanishes exactly when

s = - 1

. The image is the tangency half-line from part (iii),

x = t - 1

y = - t

t \geq 0

. This is a fold caustic: the parameterized surface itself remains regular, but its projection ceases to be locally invertible, so it cannot there be expressed as a differentiable single-valued graph over the plane.

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