Page 55 - Vector Analysis

§2.5 Properties of Differentiable Functions 51

Proof. To simplify the notation, let y0 = f (x0), A = (Df )(x0) P B(Rn, Rm), and B =
(Dg)(y0) P B(Rm, Rℓ). Let ε ą 0 be given. By the differentiability of f and g at x0 and y0,
there exists δ1, δ2 ą 0 such that if }x ´ x0}Rn ă δ1 and }y ´ y0}Rm ă δ2, we have

}f (x) ´ f (x0) ´ A(x ´ x0)}Rm ď min ␣1, ε 1) (}x ´ x0}Rn ,
2(}B} +

}g(y) ´ g(y0) ´ B(y ´ y0)}Rℓ ď ε 1) }y ´ y0}Rm .
2(}A} +

Define

u(h) = f (x0 + h) ´ f (x0) ´ Ah @ }h}Rn ă δ1 ,
v(k) = g(y0 + k) ´ g(y0) ´ Bk @ }k}Rm ă δ2 .

Then if }h}Rn ă δ1 and }k}Rm ă δ2,

}u(h)}Rm ď }h}Rn , ε and ε

}u(h)}Rm ď 2(}B} + 1) }h}Rn }v(k)}Rℓ ď 2(}A} + 1) }k}Rm .

Let k = f (x0 + h) ´ f (x0) = Ah + u(h). Then lim k = 0; thus there exists δ3 ą 0 such that
hÑ0
}k}Rm ă δ2 whenever }h}Rn ă δ3 .

Since
()

F (x0 + h) ´ F (x0) = g(y0 + k) ´ g(y0) = Bk + v(k) = B Ah + u(h) + v(k)
= BAh + Bu(h) + v(k) ,

we conclude that if }h}Rn ă δ = mintδ1, δ3u,

}F (x0 + h) ´ F (x0) ´ BAh}Rℓ ď }Bu(h)}Rℓ + }v(k)}Rℓ ď }B}}u(h)}Rm + ε 1) }k}Rm
2(}A} +
ε ε( )ε
ε
ď 2 }h}Rn + 2(}A} + 1) }A}}h}Rn + }u(h)}Rm ď 2 }h}Rn + 2 }h}Rn = ε}h}Rn

[] ˝
which implies that F is differentiable at x0 and (DF )(x0) = BA.

Example 2.50. Consider the polar coordinate x = r cos θ, y = r sin θ. Then every function
f : R2 Ñ R is associated with a function F : [0, 8) ˆ [0, 2π) Ñ R satisfying

F (r, θ) = f (r cos θ, r sin θ) .