Let S Rn be open and let f : Rn R. We recall that, for xo = (xo , xo , , xo ) S 1 2 n the partial derivative of f at the point xo with respect to the component xj is defined as f (xo , xo , , , xo , xo + h, xo , , xo ) f (xo ) 1 2
In this section, we introduce one of the most important ideas in the theory of optimization, that of a convex set. We discuss other ideas which stem from the basic definition, and in particular, the notion of a convex function which will be im
Our final topic in this first part of the course is that of convex functions. Again, we will concentrate on the situation of a map f : Rn R although the situation can be generalized immediately by replacing Rn with any real vector space V
MATH 535: Introduction to Partial Dierential Equations Spring11 (F.J. Sayas) Some dierential equations (Refresher for Section 1.2)
Linear dierential equations of order one
The general form of a linear dierential equation of order one is y + a(x)y = f (x).