Differentiation and Integral
of vector function
3104 김동은
Definition of vector function
2
F(x) = 2x + 10
14
3
(t) =< 2t, 3t − 2, t >
< 6,7,3 >
Differential of vector function
Let r t = (f(t), g(t), h(t))
r(t + ∆t) − r(t) (f(t + ∆t), g(t + ∆t), h(t + ∆t))
r′(t) = lim
=
∆→
∆t
∆t
f(t + ∆t) g(t + ∆t) h(t + ∆t)
= lim
,
,
= f′(t), g′(t), h′(t)
∆→
∆t
∆t
∆t
Differential of vector function
Example 1
Unit tangent vector
r′(t)
T(t) =
r′(t)
Q t = r t + k × r′(t)
Differential rules
Example 2
Integral of vector function


 r t dt =  {f t i + g t j + h t k} dt





=  f t dt × i +  g(t)dt × j +  h(t)dt × k



Example 3
(0,a)
Calculate the electric field of point P

cos
sin
E=

i+
j d
4
x
x
P(x,0)