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Jeramiah Campos

Jeramiah Campos

Answered question

2022-06-22

Let γ : [ 0 , 2 π ] R 3 be
γ ( t ) = ( 4 t , cos ( 3 t ) , sin ( 3 t ) ) .
Justify that γ is differentiable.

Answer & Explanation

drumette824ed

drumette824ed

Beginner2022-06-23Added 19 answers

Step 1
A vector valued function f : R n R m is differentiable if its component functions f i : R n R are differentiable. Here, n = 1 and m = 3 The component functions are,
f 1 ( t ) = 4 t f 2 ( t ) = cos ( 3 t ) f 3 ( t ) = sin ( 3 t )
Hopefully it is clear that each of these component functions are differentiable. Then, we have that f is differentiable and its derivative is the Jacobian (matrix of partial derivatives),
f = ( t f 1 t f 2 t f 3 ) = ( 4 3 sin ( 3 t ) 3 cos ( 3 t ) )

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