Exercise 9.11

Question

\def\where{\,|\,}
Exercise 11: 
    If $f$ and $g$ are differentiable real functions in $\R n$, prove that
    $$
        \del(fg)=f\del g+g\del f
    $$
    and that $\del(1/f)=-f^{-2}\del f$ wherever $f\ne 0$.

analambanomenos · Accepted Answer

(\textit{analambanomenos})
    \begin{align*}
        \del(fg) &= \sum_{i=1}^n\big(D_i(fg)\big)\v e_i \
        &= \sum_{i=1}^n\big(f(D_ig)+g(D_if)\big)\v e_i \
        &= f\sum_{i=1}^n(D_ig)\v e_i + g\sum_{i=1}^n(D_if)\v e_i \
        &= f\del g+g\del f \[3ex]
        0 &= \del(1) \
        &= \del(f\cdot f^{-1}) \
        &= f\del(f^{-1}) + f^{-1}\del f \
        f\del(f^{-1}) &= -f^{-1}\del f \
        \del(f^{-1}) &= -f^{-2}\del f
    \end{align*}

Exercise 9.11

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Exercise 9.11

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