By K. Donner
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Additional info for Extension of Positive Operators and Korovkin Theorems
Example text
A notable exception occurs in the theory of elasticity, where the biharmonic equation contains fourth-order derivatives, but that is perhaps stretching things a bit. 2 1. Find all first- and second-order partial derivatives, checking the equality of the mixed derivatives, for the following functions: (i) X 3y2 + 4 xy4 (ii) In(x 2 + y2) (iii) eXY cos(x + y) 2 2. Show thatf(x, y) = In(x + y2) satisfies the partial differential equation ',> alj alj -+ 0 ax2 -ay= 2 This is called the Laplace equation in two-dimensional rectangular coordinates.
Each of them may be functions of any number of variables - these do not necessarily have to appear as derivatives or coefficients in the PDEs. The question obviously arises about the solution of PDEs. Even the simplest require care: Solve the following PDEs: (ii) az (x, ~ = x ay (i) az (x, y) = 0 ax 2z (iii) a (x, y) axay = 0 2z (iv) a (x, y) ax2 = 0 Differentiation of Functions of More Than One Variable 47 (i) When Zx is evaluated it is assumed that y remains constant. So, for any function ofy, say g(y), we have ag(y) = 0 ax So one solution of Zx = 0 is certainly z = g(y), an arbitrary function of y alone.
For example, a::{x == :y (:):~) ==fxyy a~rl== a~(:A:y(;~))==hyxx and so on. ; In practice, there is no great call for derivatives higher than second order - as you might suspect from the number of times you have had to deal with ordinary derivatives of greater than second order. A notable exception occurs in the theory of elasticity, where the biharmonic equation contains fourth-order derivatives, but that is perhaps stretching things a bit. 2 1. Find all first- and second-order partial derivatives, checking the equality of the mixed derivatives, for the following functions: (i) X 3y2 + 4 xy4 (ii) In(x 2 + y2) (iii) eXY cos(x + y) 2 2.