DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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110 Appendix. Review of Linear Algebra and CalculusIn particular, if α(0) = P and α ′ (0) = V ∈ R n , then (f ◦α) ′ (0) = ∇f(P ) · V. This is somewhatsurprising, as the rate of change of f along α at P depends only on the tangent vector and onnothing more subtle about the curve.Proposition 2.3. D V f(P )=∇f(P ) · V. Thus, the directional derivative is a linear functionof V.Proof. If we take α(t) =P + tV, then by definition of the directional derivative, D V f(P )=(f ◦α) ′ (0) = ∇f(P ) · V. □Another important consequence of the chain rule, essential throughout differential geometry, is thefollowingProposition 2.4. Suppose S ⊂ R n is a subset with the property that any pair of points of Scan be joined by a C 1 curve. Then a C 1 function f : S → R with ∇f = 0 everywhere is a constantfunction.Proof. Fix P ∈ S and let Q ∈ S be arbitrary. Choose a C 1 curve α with α(0) = P andα(1) = Q. Then (f ◦α) ′ (t) =∇f(α(t)) · α ′ (t) =0for all t. Itisaconsequence of the Mean ValueTheorem in introductory calculus that a function g :[0, 1] → R that is continuous on [0, 1] andhas zero derivative throughout the interval must be a constant. Therefore, f(Q) =(f ◦α)(1) =(f ◦α)(0) = f(P ). It follows that f must be constant on S. □We will also have plenty of occasion to use the vector version of the product rule:Proposition 2.5. Suppose f, g : R → R 3 are differentiable. Then we have(f · g) ′ (t) =f ′ (t) · g(t)+f(t) · g ′ (t) and(f × g) ′ (t) =f ′ (t) × g(t)+f(t) × g ′ (t).Last, from vector integral calculus, we recall the analogue of the Fundamental Theorem ofCalculus in R 2 :Theorem 2.6 (Green’s Theorem). Let R ⊂ R 2 be a region, and let ∂R denote its boundarycurve, oriented counterclockwise (i.e., so that the region is to its “left”). Then∫∫∫ ( ∂QP (u, v)du + Q(u, v)dv =∂u − ∂P )dudv.∂v∂RRdcR∂RabFigure 2.1
§2. Calculus Review 111Proof. We give the proof here just for the case where R is a rectangle. Take R =[a, b] × [c, d],as shown in Figure 2.1. Now we merely calculate, using the Fundamental Theorem of Calculusappropriately:∫∫R( ∂Q∂u − ∂P )dudv =∂v=∫ dc∫ dc∫ b=∫=a∂R(∫ ba)∂Q∂u du dv −∫ ba(∫ d(Q(b, v) − Q(a, v))dv −∫ bP (u, c)du +∫ dccaQ(b, v)dv −P (u, v)du + Q(u, v)dv,)∂P∂v dv du( )P (u, d) − P (u, c) du∫ baP (u, d)du −∫ dcQ(a, v)dvas required.□EXERCISES A.2♯ 1.Suppose f :(a, b) → R n is C 1 and nowhere zero. Prove that f/‖f‖ is constant if and only iff ′ (t) =λ(t)f(t) for some continuous scalar function λ. (Hint: Set g = f/‖f‖ and differentiate.Why must g ′ · g = 0?)2. Suppose α: (a, b) → R 3 is twice-differentiable and λ is a nowhere-zero twice differentiablescalar function. Prove that α, α ′ , and α ′′ are everywhere linearly independent if and only ifλα, (λα) ′ , and (λα) ′′ are everywhere linearly independent.3. Let f, g : R → R 3 be C 1 vector functions. Supposef ′ (t) =a(t)f(t)+b(t)g(t)g ′ (t) =c(t)f(t) − a(t)g(t)for some continuous functions a, b, and c. Prove that the parallelogram spanned by f(t) andg(t) lies in a fixed plane and has constant area.♯ *4.Prove that for any continuous vector-valued function f :[a, b] → R 3 ,wehave∥∫ ba∫ bf(t)dt∥ ≤ ‖f(t)‖dt.a♯ 5.Let R ⊂ R 2 be a region. Prove that∫area(R) =∂R∫udv = − vdu = 1 ∫−vdu + udv.∂R 2 ∂R
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DIFFERENTIALGEOMETRY:A First Course
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4 Chapter 1. Curves2πb2πaFigure 1
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6 Chapter 1. CurvesAlternatively, s
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8 Chapter 1. CurvesAn important obs
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10 Chapter 1. Curvesof the road, st
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12 Chapter 1. CurvesSummarizing,κ(
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14 Chapter 1. Curvescurvature κ. I
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16 Chapter 1. CurvesFigure 2.2Conve
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18 Chapter 1. Curvesalso Exercise 2
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20 Chapter 1. Curvesto the curve β
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22 Chapter 1. Curvesthe osculating
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24 Chapter 1. CurvesWe turn now to
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26 Chapter 1. Curvesintersect C eit
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28 Chapter 1. Curvesh(s,t)α(t)α(s
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30 Chapter 1. Curvesy( x(s) ,y(s))
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32 Chapter 1. Curvesɛ2ɛ 1ɛ 3CFig
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CHAPTER 2Surfaces: Local Theory1. P
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36 Chapter 2. Surfaces: Local Theor
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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