152 CHAPTER 6. THE DERIVATIVE

The same pattern applies to taking higher order derivatives. For example, D3f (x) ≡D(D2f (x)

)and D3f (x) may be considered as a trilinear map having values in Y . In

general Dkf (x) may be considered a k linear map. This means

(u1, · · · ,uk)→ Dkf (x)(u1) · · ·(uk)

has the property u j→ Dkf (x)(u1) · · ·(u j) · · ·(uk) is linear.Also, instead of writing D2f (x)(u)(v) , or D3f (x)(u)(v)(w) the following notation

is often used.D2f (x)(u,v) or D3f (x)(u,v,w)

with similar conventions for higher derivatives than 3. Another convention which is oftenused is the notation Dkf (x)vk instead of Dkf (x)(v, · · · ,v) .

Note that for every k, Dkf maps U to a normed vector space. As mentioned above,Df (x) has values in L (X ,Y ) ,D2f (x) has values in L (X ,L (X ,Y )) , etc. Thus it makessense to consider whether Dkf is continuous. This is described in the following definition.

Definition 6.7.2 Let U be an open subset of X , a normed vector space, and letf : U → Y. Then f is Ck (U) if f and its first k derivatives are all continuous. Also,Dkf (x) when it exists can be considered a Y valued multi-linear function. Sometimesthese are called tensors in case f has scalar values.

6.8 Some Standard NotationIn the case where X = Rn there is a special notation which is often used to describe higherorder mixed partial derivatives. It is called multi-index notation.

Definition 6.8.1 α = (α1, · · · ,αn) for α1 · · ·αn positive integers is called a multi-index, as before with polynomials. For α a multi-index, |α| ≡ α1 + · · ·+αn, and if x ∈ X,

x= (x1, · · · ,xn),

and f a function, define

xα ≡ xα11 xα2

2 · · ·xαnn , Dαf(x)≡ ∂ |α|f(x)

∂xα11 ∂xα2

2 · · ·∂xαnn

.

Then in this special case, the following is another description of what is meant by a Ck

function.

Definition 6.8.2 Let U be an open subset of Rn and let f : U → Y. Then for k anonnegative integer, a differentiable function f is Ck if for every |α| ≤ k, Dαf exists andis continuous.

Theorem 6.8.3 Let U be an open subset of Rn and let f : U → Y. Then if Drf (x)exists for r ≤ k, then Drf is continuous at x for r ≤ k if and only if Dαf is continuous at xfor each |α| ≤ k.

Proof: First consider the case of a single derivative. Then as shown above, the matrixof Df (x) is just

J (x)≡(

∂f∂x1

(x) · · · ∂f∂xn

(x))