GM Week 4

GM Week 4

1. Level Sets

   • The level sets of a function f : U ⊂ R^n → R are sets of points Lk = {x ∈ U| f(x) = k} for each constant k in the range of f.
   • Level sets, also called contours in R 2 or isosurfaces in R3 , are subsets of U which are all mapped to the same value by f.
   • Typically contours will be curves in R 2 . To represent a function using contours one typically plots several contours with the corresponding values of k labelled in some way. These are call contour maps or contour plots.

2. Partial Derivatives in R^n

   • ∂f/∂x=lim h→0 (f(a + h, b) − f(a, b) )/h
   • • In words, the derivative is computed are follows: starting at the left, compute the partial derivative of f with respect to its first argument and multiply by the ordinary derivative of that argument function. Now do the same for the next argument of f and add that on. Continue until you get to the last component of argument of f.
       • Most aspects of partial differentiation are straightforward, almost trivial extensions of what you know from functions of one variable. However, the Chain Rule has a tendency to cause trouble. The reason is compact notation that is used, as in Eq. (8). It is assumed you understand where the functions are being evaluated, so be sure you do understand.

3. The Chain Rule from Vectors

   • Let us now re-approach the Chain Rule using vector notation. Given a function of n variables f : R n → R and a vector function r : R → R n , into same n-dimensions, we can compose these to obtain g = f ◦ r : R → R, with g(t) = f(r(t))

4. Directional Derivatives

   • The directional derivative of f(x, y) in the direction u and is denoted by Duf.
   • • While one can compute the directional derivative from the definition, it is more common to use the gradient vector as follows. Let g(t) = f(r(t)), where r(t) = x + tu with x and u fixed with u a unit vector
       • These have the same meaning as above: x will be the point where we evaluate the directional derivative and u is the direction. Note r(0) = x r0 (t) = u

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