Kenneth Kuttler

198 CHAPTER 7. THE DERIVATIVE

The way this is usually used is in the following corollary which has already been ob-tained. Remember the matrix of Df (x). Recall that if a function is C1 in the sense thatx→Df (x) is continuous then all the partial derivatives exist and are continuous. The nextcorollary says that if the partial derivatives do exist and are continuous, then the function isdifferentiable and has continuous derivative.

Corollary 7.9.6 Let U be an open subset of Fn and let f :U → Fm be C1 in the sensethat all the partial derivatives of f exist and are continuous. Then f is differentiable and

f (x+v) = f (x)+n

∑k=1

∂f

∂xk(x)vk +o(v) .

Similarly, if the partial derivatives up to order k exist and are continuous, then the functionis Ck in the sense that the first k derivatives exist and are continuous.

7.10 Mixed Partial DerivativesContinuing with the special case where f is defined on an open set in Fn, I will next con-sider an interesting result which was known to Euler in around 1734 about mixed partialderivatives. It was proved by Clairaut some time later. It turns out that the mixed partialderivatives, if continuous will end up being equal. Recall the notation fx =

∂ f∂x = De1 f and

fxy =∂ 2 f∂y∂x = De1e2 f .

Theorem 7.10.1 Suppose f : U ⊆ F2→ R where U is an open set on which fx, fy,fxy and fyx exist. Then if fxy and fyx are continuous at the point (x,y) ∈U, it follows

fxy (x,y) = fyx (x,y) .

Proof: Since U is open, there exists r > 0 such that B((x,y) ,r)⊆U. Now let |t| , |s|<r/2, t,s real numbers and consider

∆(s, t)≡ 1st{

h(t)︷︸︸︷f (x+ t,y+ s)− f (x+ t,y)−

h(0)︷︸︸︷( f (x,y+ s)− f (x,y))}. (7.16)

Note that (x+ t,y+ s) ∈U because

|(x+ t,y+ s)− (x,y)| = |(t,s)|=(t2 + s2)1/2

≤(

)1/2

=r√2< r.

As implied above, h(t) ≡ f (x+ t,y+ s)− f (x+ t,y). Therefore, by the mean value theo-rem from one variable calculus and the (one variable) chain rule,

∆(s, t) =1st(h(t)−h(0)) =

1st

h′ (αt) t

=1s( fx (x+αt,y+ s)− fx (x+αt,y))

for some α ∈ (0,1) . Applying the mean value theorem again,

∆(s, t) = fxy (x+αt,y+β s)

198 CHAPTER 7. THE DERIVATIVEThe way this is usually used is in the following corollary which has already been ob-tained. Remember the matrix of Df (x). Recall that if a function is C! in the sense thatx — Df (x) is continuous then all the partial derivatives exist and are continuous. The nextcorollary says that if the partial derivatives do exist and are continuous, then the function isdifferentiable and has continuous derivative.Corollary 7.9.6 Let U be an open subset of F" and let f :U — F" be C' in the sensethat all the partial derivatives of f exist and are continuous. Then f is differentiable andfete) =F@)+¥ SF aw) +0(v).k=l O%kSimilarly, if the partial derivatives up to order k exist and are continuous, then the functionis C* in the sense that the first k derivatives exist and are continuous.7.10 Mixed Partial DerivativesContinuing with the special case where f is defined on an open set in F”, I will next con-sider an interesting result which was known to Euler in around 1734 about mixed partialderivatives. It was proved by Clairaut some time later. It turns out that the mixed partialderivatives, if continuous will end up being equal. Recall the notation f. = of = De, f and_ Pf _Sry ~ Oydx Dejerf-Theorem 7.10.1 Suppose f :U CF? +R where U is an open set on which Soffry and fy, exist. Then if fry and fy are continuous at the point (x,y) € U, it followstry (x,y) = fyx (x,y) .Proof: Since U is open, there exists r > 0 such that B((x,y),r) C U. Now let |t|,|s| <r/2,t,s real numbers and considerh(t) (0)A(sst) =F O+hy+9) Fete —Flerts)—-Fa) (716)Note that (x+t,y-+s) € U because2l(e+ny+8)—(%y)] = [les] =(P+5°)"rye Wee Z—+— =——z<r4.4 V2As implied above, h(t) = f (x+t,y+s)—f(x+t,y). Therefore, by the mean value theo-rem from one variable calculus and the (one variable) chain rule,IAAst) = + (h()—n(0)) = 4H! (ae)t= “(fe e+ ata +8) — fle tarry)for some a € (0,1). Applying the mean value theorem again,A(s,t) = fry (x+ at,y + Bs)