9c6c8fc5bd-p-75.pdf

# 9c6c8fc5bd-p-75.pdf - 7.3 CAMPOS CONSERVATIVOS E...

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7.3. CAMPOS CONSERVATIVOS E IRROTACIONALES 137 M´as a´un, cuando g es riemanniana y γ es una curva arbitraria en Q (no necesariamente contenida en S ) tal que γ (0) = p S , | γ (0) = 1, entonces: d dt | t =0 f γ ( t ) = g p (grad f p , γ (0)) = grad f p · cos β, siendo β el ´angulo que forman grad f p y γ (0) seg´un g p . En particular, si γ (0) apunta en la direcci´on de grad f p entonces cos β = 1 y la derivada anterior es m´axima. En resumen, grad f p es perpendicular a S en p y apunta en la direcci´ on de m´axima variaci´on de f , en sentido creciente , con m´odulo igual a la m´axima derivada. 7.3. Campos conservativos e irrotacionales Definiciones 7.3.1 Sea ( Q, g ) una variedad semi-riemanniana. (i) Diremos que un campo vectorial X X ( Q ) es conservativo si X = grad f para alguna funci´on f C ( Q ) , esto es, si X es exacta ( X = df ). (ii) Diremos que un campo vectorial X X ( Q ) es irrotacional si X es cerrada, esto es, si la 2-forma rotacional dX es nula.
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