apostol.pdf

# Entonces 11 vii r puesto que t es lineal es tav tahy

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Entonces 11 vII < r. Puesto que T¿ es lineal es Ta(v) = Ta(hy) = hTa(y). Por 10 tanto (8.10) nos da (8.11) fea + hy) - fea) = T( ) + Ihlllyll E( ) h a y h a, v . Ya que [u] ~ O cuando h ~ O Y ya que Ihl/h = -1-1, el segundo miembro de (8.11) tiende al límite L(y) cuando h ~ O. Por consiguiente el primer miembro tiene el mismo límite. Esto demuestra (8.8). Para deducir (8.9) utilizamos la linealidad de Ts. Si y = (y" ... ,Yn) tene- mos y = 2~=1 Ykek , luego 8.12 Gradiente de un campo escalar La fórmula del teorema 8.5, que expresa i'(«; y) como una combinación lineal de los componentes de y, puede escribirse como un producto escalar, n ['(a; y) = 2 Dd(a)Yk = Vf(a) . y, k=1

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