Exercise 3. Let a: I → R", and 3: J → R" be a pair of differentiable curves. Show that ((at), 8(1)))' = (a't), B() + (at), 8'(t)) and (a(t). a'(t)) ||a(t)|| %3D

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Exercise 3
Exercise 3. Let a: I → R", and B: J → R" be a pair of differentiable
curves. Show that
(lat), B())' = (a'(t), 8() + (a(t), "(t)
and
(la(t)|) = (a).a'(t))
|a(t)||
(Hìnt: The first identity follows immediately from the definition of the inner-
product, together with the ordinary product rule for derivatives. The second
identity follows from the first once we recall that || || := (, )/2).
Transcribed Image Text:Exercise 3. Let a: I → R", and B: J → R" be a pair of differentiable curves. Show that (lat), B())' = (a'(t), 8() + (a(t), "(t) and (la(t)|) = (a).a'(t)) |a(t)|| (Hìnt: The first identity follows immediately from the definition of the inner- product, together with the ordinary product rule for derivatives. The second identity follows from the first once we recall that || || := (, )/2).
Expert Solution
Step 1

We have to prove that ddtαt, βt=α't, βt+αt, β't

where αt and βt are differentiable curves.

We have to use the definition of derivatives to prove that the above

property holds.

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