 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/f2/a/i/solution

F

x

D F_{x} : R^{n} \to R^{m}

such that

F (x + h) = F (x) + D F_{x} (h) + o (∥ h ∥) .

m \times n

J_{F} (x) = (\partial F_{i} / \partial x_{j}) (x)

. When the total derivative exists, this is its matrix in standard bases:

D F_{x} (h) = J_{F} (x) h

. Existence of partial derivatives alone need not imply existence of the total derivative.

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