Prof. Hans Wilhelm Alt: Script Analysis III -- Tangent cone and directional derivative

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Tangent cone and directional derivative

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Since in this chapter we are interested into applications to function spaces, we shall denote here arguments by u etc. and not by x etc. (as in sect:tangentialraum).

Definition (Directional derivative) [definition:7-2]

Let X be a normed space, M⊂X , and F:M → Y with Y=R^l . (In general Y can be a BANACH space.)

[definition:7-2-(1)] For u∈M the tangent cone T_u(M) is defined as in sect:tangentialraum (here with respect to the norm in X ), hence

T_u(M) := {v ∈X ;

∃ (u_j)_{j ∈N} ∃ (r_j)_{j ∈N} : u_j∈M, r_j>0 ,

u_j → u in X,
u_j - u

r_j

→ v in X as j → ∞. }

If T_u(M) turns out to be a subspace, we call it tangent space (as in sect:tangentialraum).

[definition:7-2-(2)] Let u∈M . We say, F has a directional derivative ∂_v F(u)∈Y at point u in direction v∈T_u(M) , if for all (!) sequences (u_j)_{j ∈N} and (r_j)_{j ∈N} in the above definition the following holds:

∂_v F(u) =

lim
j → ∞

F(u_j) - F(u)

r_j

in Y .

With

v_j :=
1

r_j
(u_j-u) → v in X as j → ∞

this becomes

∂_v F(u) =

lim
j → ∞

F(u+r_jv_j) - F(u)

r_j

in Y .

Notice: This definition includes the fact, that the above limit is independent of the choice of the sequences.

We mention, that this is a generalization of the notion of an derivative known from Analysis II:

Remark

Let M be an open subset of Rⁿ , f:M → Y continuously differentiable. Then for x∈M and v∈Rⁿ the directional derivative ∂_v f(x) = Df(x)(v) exists.

Warning: The existence of a single partial derivative ∂_i f(x) at x∈M (as in Analysis II) does not (!) imply the existence of the directional derivative ∂_{e_i}f(x) .

Proof Remark. The term

(f(x+r_jv_j)-f(x))

r_j

can be formulated as

r_j

∫

( f(x+s v_j) ) ds

∫

Df(x+s r_j v_j)(v_j) ds

→

Df(x) as j → ∞.

Thus the notion "directional derivative" means, that one goes into a certain direction allowing fluctuations and computes the limit of difference quotients, whereas "partial derivative" means, that one restricts the mapping to a submanifold and computes derivatives there.

Version 1.7
H.W. Alt - 02.01.2007

Minima of functionals

Divergence type equations

First variation

Index