2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 167 • The dot product (resp. cross product) of two diFerentiable vector-valued functions on I is diFerentiable on I : d dt b −→ f ( t ) · −→ g ( t ) B = v f ′ ( t ) · −→ g ( t ) + −→ f ( t ) · vg ′ ( t ) d dt b −→ f ( t ) × −→ g ( t ) B = v f ′ ( t ) × −→ g ( t ) + −→ f ( t ) × vg ′ ( t ) . Compositions: 12 If t ( s ) is diFerentiable function on J and takes values in I , then the composition ( −→ f ◦ t )( s ) is diFerentiable on J : d dt b −→ f ( t ( s )) B = d −→ f dt dt ds = v f ′ ( t ( s )) d ds [ t ( s )] . Proof. The proofs of diFerentiability 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. ±or example, to prove diFerentiability of α ( t ) −→ f ( t ), write −→ f ( t ) = ( x ( t ) ,y ( t ) ,z ( t )); then α ( 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 last expression, combined with Remark 2.3.14 , immediately gives us the second statement. The proof of the ²rst statement, which is similar, is left to you
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.