Vector Integrals

1 Definite Integral

Consider a vector function v : [ a , b ] → V₃ , ...i.e. v ( t ) ∊ V₃ , t ∊ [ a , b ]. ...Let

v_n ...= ...

n – 1
∑		v ( tr ) δnt
r = 0

where ...δ_nt = ( b – a ) ⁄ n , ...and ...t_r = a + r δ_nt. ...Now ...δ_nt → 0 as n → ∞. define the definite integral of v over [ a , b ] as

⌠	b
│		v (t) dt
⌡	a

...:= ...

limit		vn
n → ∞

2 Integral Of Coordinates

if v is expressed in terms of its coordinate functions:

v (t) ...= ...i v_x (t) + j v_y (t) + k v_z (t)

then

⌠	b
│		v (t) dt
⌡	a

...:= ...i

⌠	b
│		vx (t) dt
⌡	a

+ ...j

⌠	b
│		vy (t) dt
⌡	a

+ ...k

⌠	b
│		vz (t) dt
⌡	a

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