 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/6/a/solution

For a differentiable scalar function of

n

real variables, its derivative is the linear map

D f (x) [h] = \sum_{i = 1}^{n} \frac{\partial f}{\partial x _{i}} (x) h_{i} = \nabla f (x)^{T} h,

where the column gradient is

\nabla f = (\partial f / \partial x_{1}, \dots, \partial f / \partial x_{n})^{T}

. Its Hessian is the

n \times n

H_{f} = (\partial^{2} f / \partial x_{i} \partial x_{j})_{i, j}

. For a

C^{2}

function, equality of mixed partial derivatives makes this matrix symmetric.

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