James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A48
appendix F Proofs of Theorems
Section 14.3
Clairaut’s Theorem Suppose f is defined on a disk D that contains the point
sa, bd. If the functions f xy and f yx are both continuous on D, then f xy sa, bd − f yx sa, bd.
ProoF For small values of h, h ± 0, consider the difference
Dshd − f f sa 1 h, b 1 hd 2 f sa 1 h, bdg 2 f f sa, b 1 hd 2 f sa, bdg
Notice that if we let tsxd − f sx, b 1 hd 2 f sx, bd, then
Dshd − tsa 1 hd 2 tsad
By the Mean Value Theorem, there is a number c between a and a 1 h such that
tsa 1 hd 2 tsad − t9scdh − hf f x sc, b 1 hd 2 f x sc, bdg
Applying the Mean Value Theorem again, this time to f x , we get a number d between b
and b 1 h such that
Combining these equations, we obtain
f x sc, b 1 hd 2 f x sc, bd − f xy sc, ddh
Dshd − h 2 f xy sc, dd
If h l 0, then sc, dd l sa, bd, so the continuity of f xy at sa, bd gives
Similarly, by writing
lim
h l 0
Dshd
h 2 − lim
sc, dd l sa, bd f xysc, dd − f xy sa, bd
Dshd − f f sa 1 h, b 1 hd 2 f sa, b 1 hdg 2 f f sa 1 h, bd 2 f sa, bdg
and using the Mean Value Theorem twice and the continuity of f yx at sa, bd, we obtain
lim
h l 0
Dshd
h 2
− f yx sa, bd
It follows that f xy sa, bd − f yx sa, bd.
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Section 14.4
8 Theorem If the partial derivatives f x and f y exist near sa, bd and are continuous
at sa, bd, then f is differentiable at sa, bd.
ProoF Let
Dz − f sa 1 Dx, b 1 Dyd 2 f sa, bd
According to (14.4.7), to prove that f is differentiable at sa, bd we have to show that we
can write Dz in the form
Dz − f x sa, bd Dx 1 f y sa, bd Dy 1 « 1 Dx 1 « 2 Dy
where « 1 and « 2 l 0 as sDx, Dyd l s0, 0d.
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