5.1 ベクトル関数

1.

(a)

$\displaystyle {\bf F}'(t)$	$\displaystyle =$	$\displaystyle ((\cos{t})',(\sin{t})')$
	$\displaystyle =$	$\displaystyle (-\sin{t}, \cos{t}\cos{t})$
$\displaystyle \Vert{\bf F}\Vert$	$\displaystyle =$	$\displaystyle \sqrt{\cos^2{t} + \sin^2{t}} = 1$

(b)

$\displaystyle {\bf F}'(t)$	$\displaystyle =$	$\displaystyle ((1+2t)',(3-t)',(2+3t)')$
	$\displaystyle =$	$\displaystyle (2,-1,3)$
$\displaystyle \Vert{\bf F}\Vert$	$\displaystyle =$	$\displaystyle \sqrt{4 + 1 + 9} = \sqrt{14}$

(c)

$\displaystyle {\bf F}(t)$	$\displaystyle =$	$\displaystyle (\int 1\; dt, \int 2t\; dt)$
	$\displaystyle =$	$\displaystyle (t, t^2)$

(d)

$\displaystyle {\bf F}(t)$	$\displaystyle =$	$\displaystyle (\int e^{t}\; dt, \int \sqrt{2}\; dt, \int e^{-t}\; dt)$
	$\displaystyle =$	$\displaystyle (e^{t}, \sqrt{2}t, -e^{-t})$