 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2004/f/f1/solution

D f_{a} : R^{m} \to R^{n}

satisfying

f (a + h) = f (a) + D f_{a} (h) + o (∥ h ∥) as h \to 0.

This describes the best first-order linear approximation without choosing coordinates. The chain rule is

D (g \circ f)_{a} = D g_{f (a)} \circ D f_{a} .

The evaluation point

f (a)

in the outer derivative matters, particularly when composing a quadratic matrix map with the exponential.

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