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Engineering Calculus Notes 179

# Engineering Calculus Notes 179 - 167 2.3 CALCULUS OF...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 167 The dot product (resp. cross product) of two differentiable vector-valued functions on I is differentiable on I : d dt bracketleftBig −→ f ( t ) · −→ g ( t ) bracketrightBig = vector f ( t ) · −→ g ( t ) + −→ f ( t ) · vectorg ( t ) d dt bracketleftBig −→ f ( t ) × −→ g ( t ) bracketrightBig = vector f ( t ) × −→ g ( t ) + −→ f ( t ) × vectorg ( t ) . Compositions: 12 If t ( s ) is differentiable function on J and takes values in I , then the composition ( −→ f t )( s ) is differentiable on J : d dt bracketleftBig −→ f ( t ( s )) bracketrightBig = d −→ f dt dt ds = vector f ( t ( s )) d ds [ t ( s )] . Proof. The proofs of differentiability of linear combinations, as well as products or compositions with real-valued functions, are most easily done by looking directly at the corresponding coordinate expressions. For example, to prove differentiability of α ( t ) −→ f ( t ), write −→ f ( t ) = ( x ( t ) ,y ( t ) ,z ( t )); then α ( t ) −→ f ( t ) = ( α ( t ) x ( t ) ( t ) y ( t ) ( t ) z ( t )) . The ordinary product rule applied to each coordinate in the last expression, combined with Remark 2.3.14 , immediately gives us the second statement. The proof of the first statement, which is similar, is left to you
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