Multivariate Calculus

• a function is continuous at x in S if for every e>0 there exists a d>0 such that:


  ||x-y||<d implies ||f(x)-f(y)|| <e
• Weierstrass' Theorem: every continuous real-valued function defined on a non-empty compact set of Rn (closed and bounded set) has global maximum and minimum
• the kth partial derivative is a measure of how fast the function changes when its kth argument, xk, changes, but all other arguments remain constant
• the gradient or jacobian is the n-vector of partial derivatives (also represented with upside down triangle)
• if f is differentiable, then it is continuous and the partial derivatives exist
• A function is of class C1 if the partial derivatives of the component functions of f exist and are continuous. If f is C1, then f is differentiable
• the Hessian is the symmetrics n-square matrix of second-order partial derivatives of a function
• If the Hessian is negative definite, then x* is a strict local maximum
• The eigenvalue test: a matrix is negative definite if and only if all its eigenvalues are strictly negative
• The determinant test: A matrix is negative definite if and only if the sign of the determinant of the first k rows and columns is alternating in k (with first being negative)