Chapter 4: Partial Differentiation
Section 4.11: Differentiability
Show that the first partial derivatives of the function hx,y in Table 4.11.1 are continuous.
To show that the first partial derivatives
are continuous functions, first obtain the estimates
≤x4+2 x2y2+y4+x4+2 x2y2+y4
To show that hx is a continuous function, show that the limit of y⁢x4+4⁢x2⁢y2−y4x2+y22 is zero as x,y approaches the origin. To do this, show that the difference between this rational function and the purported limit of zero indeed goes to zero as x,y approaches 0,0. Using the first estimate from above, obtain
which immediately implies that the limit is zero.
Similarly for hy, the second estimate from above leads to
which again implies that the limit is zero.
Since, for x,y≠0,0 both hx and hy are rational functions, Maple's bivariate limit command applies, and gives the following results immediately.
limity⁢x4+4⁢x2⁢y2−y4x2+y22,x=0,y=0 = 0
limitx⁢x4−4⁢x2⁢y2−y4x2+y22,x=0,y=0 = 0
Since the limiting value and the declared value agree for each partial derivative, these first partials are continuous functions.
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