台湾交通大学高等微积分classbfs1210092454161514

Λ (γ ) = ∫ | γ ′(t ) | dt.
a
b
Theorem Let γ :[a, b] → R k be a curve on [a, b] . If c ∈ (a, b) , let γ 1 and γ 2 be
the restriction of γ on [a, c] and [c, b] , respectively, then Λ (γ ) = Λ (γ 1 ) + Λ (γ 2 ) . Proof : Exercise.
c b b
∫
and
a
f d α + ∫ f dα = ∫ f d α .
c a
(c) If f ∈ R (α1 ) and f ∈ R (α 2 ) on [a, b] , then f ∈ R (α1 + α 2 ) on [a, b]
∫
[a, b] and
b
a
f d (α1 + α 2 ) = ∫ f dα1 + ∫ f dα 2 ;
∫
b
a
(f1 + f 2 )dα = ∫ f1dα + ∫ f 2 dα ,
a a
b
b
∫
[c, b] , and
b
a
cf dα = c ∫ f dα .
a
b
(b) If f ∈ R (α ) on [a, b] and if a < c < b , then f ∈ R (α ) on [a, c] and on
corresponding mapping of [a, b] into R k .
[a, b] , to say that f ∈ R (α ) means that f j ∈ R (α ) for j = 1,… , k .
If this is the
case, we define
∫
In other words,
2 月 26 日課堂摘要(一) INTEGRATION OF VECTOR-VALUED FUNCTIONS Definition Let f1 ,… , f k be real functions on [a, b] , and let f = ( f1 ,… , f k ) be the
If α increases monotonically on
For a ≤ x ≤ b , define
x a
F( x) = ∫ f (t )dt.
Then F is continuous on [a, b] ; furthermore, If f is continuous at a point x0 of [a, b] , then F is differentiable at x0 , and F′( x0 ) = f ( x0 ).
∫
Outline of the proof :
b
a
fd α ≤ ∫ | f | d α .
a
b
∫
b
a
f dα = ∑
i =1
2
k
(∫
b
a
fi dα
)
2
= ∑ ∫ fi
b i =1 a
k
(∫
b
a
f i d α dα = ∫
)
b k
a
∑( f ∫
i i =1
b
a
f i d α dα
)
By Schwartz inequality, we have
Definition
Let γ 1 :[a, b] → R k be a curve on [a, b] and let φ :[c, d ] → [a, b] be a
continuous and strictly monotonic mapping from [c, d ] onto [a, b] . Then the composition function γ 2 = γ 1 φ is a curve having the same graph as γ 1 . Two curve
Λ (γ ) = sup Λ ( P, γ ),
where the supremum is taken over all partitions of [a, b] .
Examples
π x sin , if x ≠ 0. 1. f :[0,1] → R defined by f ( x) = is not rectifiable. 2x 0, if x = 0.
If γ is one-to-one, γ is called an arc. If γ (a ) = γ (b), γ is said to be a closed curve.
Note : 1. Here a curve is a mapping, not the range of γ (called the graph of γ ). 2. Different curves may have the same graph.
γ 1 and γ 2 so related are called equivalent.
Let C be the common graph of two equivalent curves γ 1 and γ 2 . If φ is strictly increasing, we say φ is said to be orientation preserving --- γ 1 and γ 2 trace out C “ in the same direction”. If φ is strictly decreasing, we say φ is said to be orientation reserving --- γ 1 and γ 2 trace out C “ in opposite direction”.
Theorem Suppose that γ 1 and γ 2 be two equivalent curves, then Λ (γ 1 ) = Λ (γ 2 ) .
Proof : Rudin p.142, #19.
a a
b
b
If f ∈ R (α ) on [a, b] and c is a positive constant, then f ∈ R (cα ) on
b b
∫
a
f d (cα ) = c ∫ f dα .
a
Theorem If f maps [a, b] into R k , if f ∈ R (α ) for some monotoni increasing function α on [a, b] , then | f |∈ R (α ) , and
b
a
fd α =
(∫
b
a
f1dα ,… , ∫ f k dα .
a
b
)
∫ fd α
is the point in R k whose jth coordinate is
∫ f dα .
j
Theorem (a) If f ∈ R (α ) , f1 ∈ R (α ) and f 2 ∈ R (α ) on [a, b] , then f1 + f 2 ∈ R (α ) on [a, b] , cf ∈ R (α ) on [a, b] for every constant c, and
Definition
Let γ :[a, b] → R k be a curve on [a, b] and let P = {t0 ,… , tn } be a
partition of [a, b] , define
Λ ( P, γ ) = ∑ | γ (ti ) − γ (ti −1 ) |.
i =1 n
Theorem If f and F map [a, b] into R k , if f ∈ R on [a, b] , and if F′ = f , (t )dt = F(b) − F(a).
RECTIFIABLE CURVES
Definition [ a, b] .
A continuous mapping γ :[a, b] → R k is called a curve in R k on
2 π x sin , if x ≠ 0. is rectifiable. 2. f :[0,1] → R defined by f ( x) = 2x 0, if x = 0.
Theorem If γ ′ is continuous on [a, b] , then γ is rectifiable, and
If the set of numbers
{Λ( P, γ ) | P is a partition of [a, b]}
is bounded then we say γ
is rectifiable and its length ( or arc length) , denoted by, Λ (γ ) , is defined by
∑( f ∫
i i =1
k
b
a
f i dα ≤ f ⋅
)
∫
b
a
f dα .
Corollary
If f ∈ R (α ) on [a, b] and if | f ( x) |≤ M on [a, b] , then
∫
b
a
f dα ≤ M [α (b) − α (a )].
Theorem Let f ∈ R (α ) on [a, b] .