Curve

2024. 12. 23. 17:51·Mathematics/Calculus
목차
  1. Curve
  2. Velocity, Speed, Unit Tangent Vector
  3. Simple Curve
  4. Limit and Continuity of Curves
  5. Theorem 1
  6. Differentiability of Curves
  7. Smooth
  8. Length
  9. Arc Length Parameter
  10. Integrals

Curve

Definition 1. We call the vector function r:(a,b)→R3r:(a,b)→R3 a curve. We can parametrize curves by r(t)=⟨f(t),g(t),h(t)⟩r(t)=⟨f(t),g(t),h(t)⟩ where t∈(a,b)t∈(a,b). 

Velocity, Speed, Unit Tangent Vector

Definition 2. Let rr be a curve. Then
(1) v(t)=drdtv(t)=drdt is the velocity vector or the tangent vector of rr,
(2) |v||v| is the speed of rr,
(3) T=v|v|T=v|v| is the unit tangent vector.
(4) the tangent line is the line passing through r(t)r(t) in the direction r′(t)r′(t). 

Simple Curve

Definition 3. The curve rr is simple if it has no further self-intersections; that is, if t1,t2∈(a,b),t1≠t2t1,t2∈(a,b),t1≠t2, then r(t1)≠r(t2).r(t1)≠r(t2). [각주:1]

Limit and Continuity of Curves

Definition 4. Let r(t)=⟨f(t),g(t),h(t)⟩r(t)=⟨f(t),g(t),h(t)⟩ be a curve with domain DD, and let L∈R3L∈R3. We say that rr has limit LL as tt approaches t0t0 and write limt→t0r(t)=Llimt→t0r(t)=L if, for every number ϵ>0ϵ>0, there exists a corresponding number δ>0δ>0 such that for all t∈Dt∈D |r(t)−L|<ϵ whenever 0<|t−t0|<δ.|r(t)−L|<ϵ whenever 0<|t−t0|<δ.
Definition 5. A curve r(t)r(t) is continuous at a point t=t0t=t0 in its domain if limt→t0r(t)=r(t0).limt→t0r(t)=r(t0). The curve is continuous if it is continuous at every point in its domain.

Theorem 1

Theorem 1. Let r(t)=⟨f1(t),f2(t),f3(t)⟩r(t)=⟨f1(t),f2(t),f3(t)⟩ be a curve with differentiable functions f1,f2,f3f1,f2,f3. Then limt→t0r(t) exists ⟺∀j∈{1,2,3},limt→t0fi(t) exists. limt→t0r(t) exists ⟺∀j∈{1,2,3},limt→t0fi(t) exists. Moreover, limt→t0r(t)=⟨limt→t0f1(t),limt→t0f2(t),limt→t0f3(t)⟩.limt→t0r(t)=⟨limt→t0f1(t),limt→t0f2(t),limt→t0f3(t)⟩.
Proof. (⟹)(⟹) Let limt→t0r(t)=L=⟨L1,L2,L3⟩limt→t0r(t)=L=⟨L1,L2,L3⟩. Then ∀ϵ,∃δ>0∀ϵ,∃δ>0 such that |r(t)−L|<ϵ|r(t)−L|<ϵ whenever 0<|t−t0|<δ0<|t−t0|<δ. This means that |fi(t)−Li|<ϵ,∀i|fi(t)−Li|<ϵ,∀i whenever 0<|t−t0|<δ0<|t−t0|<δ. Thus limits of each component exist and limt→t0fi(t)=Li,∀ilimt→t0fi(t)=Li,∀i.

(⟸)(⟸) Suppose that limt→t0fi(t)=Li,∀ilimt→t0fi(t)=Li,∀i. Then ∀ϵ>0,∃δi∀ϵ>0,∃δi such that |fi(t)−Li|<ϵ√3|fi(t)−Li|<ϵ√3 whenever 0<|t−t0|<δi0<|t−t0|<δi. Take δ=min{δ1,δ2,δ3}δ=min{δ1,δ2,δ3}. If L=⟨L1,L2,L3⟩L=⟨L1,L2,L3⟩, then we have |r(t)−L|≤√3∑i=1(fi(t)−Li)2<√ϵ33⋅3=ϵ whenever 0<|t−t0|<δ.◼|r(t)−L|≤ ⎷3∑i=1(fi(t)−Li)2<√ϵ33⋅3=ϵ whenever 0<|t−t0|<δ.■

Differentiability of Curves

Definition 6. The curve r(t)=⟨f(t),g(t),h(t)⟩r(t)=⟨f(t),g(t),h(t)⟩ has a derivative (is differentiable) at tt if f,gf,g and hh have derivatives at tt. The derivative is the curve r′(t)=drdt=limΔt→0r(t+Δt)−r(t)Δt=⟨f′(t),g′(t),h′(t)⟩.r′(t)=drdt=limΔt→0r(t+Δt)−r(t)Δt=⟨f′(t),g′(t),h′(t)⟩. The curve is differentiable if it is differentiable at every point of its domain.

Theorem 1의 존재로 인해 우리는 곡선의 극한, 연속, 미분가능성은 component-wise하게 정의할 수 있다. 때문에 각 component function인 scalar function들이 극한을 가지는지, 연속인지, 미분가능한지 체크해서 취합해주면 된다.

Smooth

Definition 7. The curve rr is smooth if the velocity vector, i.e., the velocity of rr, that is, v=drdtv=drdt is continuous and never 00. 

속도가 영벡터가 된다면 곡선을 따라 움직이는 입자가 멈췄다가 다시 움직인다고 물리적으로 해석할 수 있다. 그러면 말그대로 부드럽지 않고 방향을 전환하여 곡선이 꺾이는 등의 불상사가 발생할 수도 있다. 때문에 이러한 상황이 발생하지 않는 곡선을 smooth하다고 말한다.

Length

Definition 8. The length of a smooth curve r(t)=⟨f(t),g(t),h(t)⟩,a≤t≤br(t)=⟨f(t),g(t),h(t)⟩,a≤t≤b, that is traced exactly once as tt increases from t=at=a to t=bt=b, is L=∫ba|v|dt=∫ba√(dxdt)2+(dydt)2+(dzdt)2dt.L=∫ba|v|dt=∫ba√(dxdt)2+(dydt)2+(dzdt)2dt.

곡선의 '길이'로 생각할 수도 있고, 물리적으로는 곡선을 따라 움직이는 입자의 '이동거리'라고 생각할 수 있다. 이 정의에 근거해서 arc length parameter ss를 정의하자.

Arc Length Parameter

Definition 9. If we choose a base point P(t0)P(t0) on a smooth curve CC parametrized by tt, each value of tt determines a point P(t)=(f(t),g(t),h(t))P(t)=(f(t),g(t),h(t)) on CC and a directed distance s(t)=∫tt0|v(τ)|dτ,s(t)=∫tt0|v(τ)|dτ, measured along C from the base point. We call s an arc length parameter for the curve. 

Integrals

Definition 10. The definite integral of a continuous curve r(t)=⟨f(t),g(t),h(t)⟩ is defined by ∫bar(t)dt=⟨∫baf(t)dt,∫bag(t)dt,∫bah(t)dt⟩. R(t) is antiderivative of r(t) if R′(t)=r(t). We use the notation ∫r(t)dt=R(t)+C for some constant vector C. 
  1. https://product.kyobobook.co.kr/detail/S000002271869 [본문으로]
저작자표시 (새창열림)
  1. Curve
  2. Velocity, Speed, Unit Tangent Vector
  3. Simple Curve
  4. Limit and Continuity of Curves
  5. Theorem 1
  6. Differentiability of Curves
  7. Smooth
  8. Length
  9. Arc Length Parameter
  10. Integrals
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