6.9. THE DERIVATIVE AND THE CARTESIAN PRODUCT 153

and to say that x→ Df (x) is continuous is the same as saying that each of these partialderivatives is continuous. Written out in more detail,

f (x+v)−f (x) = Df (x)v+o(v) =n

∑k=1

∂f

∂xk(x)vk +o(v)

Thus Df (x)v = ∑nk=1

∂f∂xk

(x)vk. Now consider the second derivative.

D2f (x)(w)(v) =

Df (x+w)v−Df (x)v+o(w)(v)=n

∑k=1

(∂f

∂xk(x+w)− ∂f

∂xk(x)

)vk +o(w)(v)

=n

∑k=1

(n

∑j=1

∂ 2f (x)

∂x j∂xkw j +o(w)

)vk +o(w)(v) = ∑

j,k

∂ 2f (x)

∂x j∂xkw jvk +o(w)(v)

and so D2f (x)(w)(v) = ∑ j,k∂ 2f(x)∂x j∂xk

w jvk. Hence D2f is continuous if and only if each of

these coefficients x→ ∂ 2f(x)∂x j∂xk

is continuous. Obviously you can continue doing this and

conclude that Dkf is continuous if and only if all of the partial derivatives of order up to kare continuous. ■

In practice, this is usually what people are thinking when they say that f is Ck. But asjust argued, this is the same as saying that the r linear form x→Drf (x) is continuous intothe appropriate space of linear transformations for each r ≤ k.

Of course the above is based on the assumption that the first k derivatives exist and givestwo equivalent formulations which state that these derivatives are continuous. Can anythingbe said about the existence of the derivatives based on the existence and continuity of thepartial derivatives? As pointed out, if the partial derivatives exist and are continuous, thenthe function is differentiable and has continuous derivative. However, I want to emphasizethe idea of the Cartesian product.

6.9 The Derivative and the Cartesian ProductThere are theorems which can be used to get differentiability of a function based on exis-tence and continuity of the partial derivatives. A generalization of this was given above.Here a function defined on a product space is considered. It is very much like what waspresented above and could be obtained as a special case but to reinforce the ideas, I will doit from scratch because certain aspects of it are important in the statement of the implicitfunction theorem.

The following is an important abstract generalization of the concept of partial derivativepresented above. Insead of taking the derivative with respect to one variable, it is taken withrespect to several but not with respect to others. This vague notion is made precise in thefollowing definition. First here is a lemma.

Lemma 6.9.1 Suppose U is an open set in X×Y. Then the set, Uy defined by

Uy ≡ {x ∈ X : (x,y) ∈U}