Differentiable Functions
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
Convex Sets
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
Convex Functions
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).