Talk:Function of several real variables/to do

Summary of the material to be added to this article:

List created by D.Lazard (talk) 09:49, 27 June 2013 (UTC)
 * Relation univariate multivariate: By fixing all variables but one one gets a univariate real function. If the function is continuous (resp. differentiable) the same is true for these univariate functions, but the converse is false: if all these univariate functions are continuous (resp. differentiable) this is not always true for the multivariate function.
 * Partial derivatives, differential and gradient. Partial derivatives of higher order.
 * Classes of functions: continuous, differentiable, $C^{k}$, $C^{∞}$, analytic
 * Taylor expansion
 * Analytic prolongation in the analytic case, which makes that the domain of the function is almost always left implicit. Difficulty to make it explicit
 * Tangent hyperplane to the graph of the function, expressed in term of the gradient
 * Stationary or critical points, that are those where the gradient is zero (differentiable case)
 * Maxima and minima: At a local minimum, the gradient is zero and the Hessian matrix is positive semidefinite. A point at which the gradient is zero and the Hessian matrix is positive definite is a local minimum
 * Stationary points at which the Hessian matrix is not semidefinite, saddle points
 * Convexity: If the Hessian matrix is everywhere positive definite, the function is convex and the function has a unique minimum, and there are efficient algorithms to find it (fundamental in optimization)