2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS167•The dot product (resp. cross product) of two differentiablevector-valued functions onIis differentiable onI:ddtbracketleftBig−→f(t)·−→g(t)bracketrightBig=vectorf′(t)·−→g(t) +−→f(t)·vectorg′(t)ddtbracketleftBig−→f(t)×−→g(t)bracketrightBig=vectorf′(t)×−→g(t) +−→f(t)×vectorg′(t).Compositions:12Ift(s)is differentiable function onJand takes valuesinI, then the composition(−→f◦t)(s)is differentiable onJ:ddtbracketleftBig−→f(t(s))bracketrightBig=d−→fdtdtds=vectorf′(t(s))dds[t(s)].Proof.The proofs of differentiability of linear combinations, as well asproducts or compositions with real-valued functions, are most easily doneby looking directly at the corresponding coordinate expressions. Forexample, to prove differentiability ofα(t)−→f(t), write−→f(t) = (x(t),y(t),z(t));thenα(t)−→f(t) = (α(t)x(t),α(t)y(t),α(t)z(t)).The ordinary product rule applied to each coordinate in the lastexpression, combined with Remark2.3.14, immediately gives us the secondstatement. The proof of the first statement, which is similar, is left to you
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