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A partir de este, sabemos que tenemos \begin{equation*} @@ -1152,13 +1181,13 @@ urlcolor=black]{hyperref} \begin{equation*} \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n = \det\left( \delta_1, \delta_2, \dots, \delta_n \right) \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n = uf \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n. \end{equation*} - Al ser $u$ simplemente una unidad, hemos llegado a que $\bigwedge^n N = \bigwedge^n \derivaciones{f}$. Por último, lo que buscamos es quitar ese producto exterior. Ya sabemos que $ N \subseteq \derivaciones{f}$, por lo que solo nos hace falta ver la otra implicación. Sea $\delta \in \derivaciones{f}$. Por ser este un subconjunto de \derivaciones{\Kx}, lo podemos escribir como + Al ser $u$ simplemente una unidad, hemos llegado a que $\bigwedge^n N = \bigwedge^n \derivaciones{f}$. Lo que buscamos ahora es quitar ese producto exterior. Ya sabemos que $ N \subseteq \derivaciones{f}$, por lo que solo nos hace falta ver la otra implicación. Sea $\delta \in \derivaciones{f}$, que por ser \derivaciones{f} un subconjunto de \derivaciones{\Kx}, lo podemos escribir como \begin{equation*} \delta = p_1\partial_1 + p_2\partial_2 + \dots + p_n\partial_n. \end{equation*} Fijémonos ahora en la relación \ref{Relación Matricial Derivadas}. Recordando la regla de Cramer, sabemos que \begin{equation*} A^{-1} = \frac{1}{\det A} \mathrm{adj}\!\left( A \right) = \frac{1}{uf}\mathrm{adj}\!\left( A \right) = \frac{1}{f}B - \end{equation*} donde $\mathrm{adj}\!\left( A \right)$ es la matriz adjunta de $A$. $A^{-1}$ está en $\mathrm{Mat}\!\left( n \times n, \cuerpofracciones{\Kx} \right)$ pero los adjuntos de la matriz $A$ están en \Kx, por lo que podemos expresar $A^{-1}$ como vemos al final de la igualdad con $B \in \mathrm{Mat}\!\left( n \times n, \Kx\right)$. Por lo tanto, como + \end{equation*} donde $\mathrm{adj}\!\left( A \right)$ es la matriz adjunta de $A$. $A^{-1}$ está en $\mathrm{Mat}\!\left( n \times n, \cuerpofracciones{\Kx} \right)$ pero los adjuntos de la matriz $A$ están en \Kx, por lo que podemos expresar $A^{-1}$ como vemos al final de la igualdad con $B \in \mathrm{Mat}\!\left( n \times n, \Kx\right)$. Como \begin{equation*} \vectorhorizontalcuatro{\partial_1}{\partial_2}{\dots}{\partial_n} = \vectorhorizontalcuatro{\delta_1}{\delta_2}{\dots}{\delta_n}A^{-1} = \frac{1}{f}\vectorhorizontalcuatro{\delta_1}{\delta_2}{\dots}{\delta_n}B, \end{equation*} @@ -1167,11 +1196,13 @@ urlcolor=black]{hyperref} \delta &= \frac{b_1}{f}\delta_1 + \frac{b_2}{f}\delta_2 + \dots + \frac{b_n}{f}\delta_n, \quad \text{donde $b_i \in \Kx$, o también} \\ &= \overline{b}_1\delta_1 + \overline{b}_2\delta_2 + \dots + \overline{b}_n\delta_n, \quad \text{donde $\overline{b}_i \in \cuerpofracciones{\Kx}$.} \end{align*} - Para poder garantizar que estos $\overline{b}_i$ están en \Kx, acordémonos de que $\delta \in \derivaciones{f}$ y $\bigwedge^n \derivaciones{f} = \bigwedge^n N$, por lo que si en $\delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n$ sustituimos $\delta_i$ por $\delta$ nos queda + En este último paso vamos garantizar que estos $\overline{b}_i$ están\footnote{Más bien, se pueden ver como elementos de \Kx. Como por ejemplo $\frac{f^2}{f} \in \cuerpofracciones{\Kx}$, que realmente puede verse como $f$ en \Kx.} en \Kx. Para ello, acordémonos de que $\delta \in \derivaciones{f}$ y $\bigwedge^n \derivaciones{f} = \bigwedge^n N$, por lo que si en $\delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n$ sustituimos $\delta_i$ por $\delta$ nos queda \begin{equation*} \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_{i-1} \wedge \delta \wedge \delta_{i+1} \wedge \dots \wedge \delta_{n-1} \wedge \delta_n \in \bigwedge^n N. \end{equation*} - Percatémonos de que esta expresión, una vez eliminamos todos los productos exteriores nulos queda como $\overline{b}_i \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n \in \bigwedge^n N$, lo cual es solo posible si $\overline{b}_i \in \Kx$. Con esto concluimos que $\delta \in N$ y, por tanto, que $N = \derivaciones{f}$. + Percatémonos de que esta expresión, una vez eliminamos todos los productos exteriores nulos, queda como $\overline{b}_i \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n \in \bigwedge^n N$, lo cual es solo posible si $\overline{b}_i \in \Kx$. Con esto concluimos que $\delta \in N$ y, por tanto, que $N = \derivaciones{f}$. + + \todo[inline]{Para poner un ejemplo espero a la introducción, para que case con el ejemplo que allí pongamos.} \appendix \chapter{Un ejemplo completo de cálculo}