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Answers to Selected Exercises 779 12.3.24 (p. 663) u(x,y) = ∞ X n=1 αn cosh nπx/b cosh nπa/b sin nπy b , αn = 2 b Z b 0 g(y) sin nπy b dy u(x, y) = 96 π5 ∞ X n=1 cosh(2n −1)πx (2n −1)5 cosh(2n −1)π sin(2n −1)πy. 12.3.25 (p. 664) u(x,y) = ∞ X n=1 αn cosh(2n −1)πx/2b cosh(2n −1)πa/2b cos (2n −1)πy 2b , αn = 2 b Z b 0 g(y... | Elementary Differential Equations with Boundary Value Problems_Page_789_Chunk3201 |
780 Answers to Selected Exercises 12.3.32 (p. 664) u(x,y) = −a π ∞ X n=1 αn n e−nπy/a sin nπx a , αn = 2 a Z a 0 f(x) sin nπx a dx u(x) = 4 ∞ X n=1 (1 + (−1)n2) n4 e−ny sin nx 12.3.33 (p. 664) u(x, y) = −2a π ∞ X n=1 αn 2n −1e−(2n−1)πy/2a cos (2n −1)πx 2a , αn = 2 a Z a 0 f(x)cos (2n −1)πx 2a dx u(x, y) = 5488 π3 ∞ X n... | Elementary Differential Equations with Boundary Value Problems_Page_790_Chunk3202 |
Answers to Selected Exercises 781 αn = 2 γ Z γ 0 g(θ) sin (2n −1)πθ 2γ dθ, n = 1, 2, 3,. . . 12.4.6 (p. 673) u(r, θ) = α0 + ∞ X n=1 αn rnπ/γ ρnπ/γ cos nπθ γ α0 = 1 γ Z γ 0 f(θ) dθ, αn = 2 γ Z γ 0 f(θ) cos nπθ γ dθ, n = 1, 2, 3,. . . 12.4.7 (p. 673) vn(r, θ) = rn nρn−1 (αn cos nθ + sin nθ) u(r, θ) = c + ∞ X n=1 rn nρn−1... | Elementary Differential Equations with Boundary Value Problems_Page_791_Chunk3203 |
782 Answers to Selected Exercises 13.1.12 (p. 685) y = sinh(x −a) sinh(b −a) Z b x F(t) sinh(t −b) dt + sinh(x −b) sinh(b −a) Z x a F(t) sinh(t −a) dt 13.1.13 (p. 685) y = −sinh(x −a) cosh(b −a) Z b x F(t) cosh(t −b) dt −cosh(x −b) cosh(b −a) Z x a F(t) sinh(t −a) dt 13.1.14 (p. 685) y = −cosh(x −a) sinh(b −a) Z b x F(... | Elementary Differential Equations with Boundary Value Problems_Page_792_Chunk3204 |
Answers to Selected Exercises 783 If ω = n (positive integer), then Z π 0 F(t) cos nt dt = 0 is necessary for existence of a solution. In this case, y = −1 n cos nx Z π x F(t) sin nt dt + sin nx Z x 0 F(t) cos nt dt + c1 cos nx with c1 arbitrary. 13.1.20 (p. 685) y1 = B1(z2)z1 −B1(z1)z2 13.1.21 (p. 685) (a) G(x,t) ... | Elementary Differential Equations with Boundary Value Problems_Page_793_Chunk3205 |
784 Answers to Selected Exercises 13.1.26 (p. 686) α(ρ + δ) −βρ ̸= 0 G(x,t) = (β −αt)(ρ + δ −ρx) α(ρ + δ) −βρ , 0 ≤t ≤x, (β −αx)(ρ + δ −ρt) α(ρ + δ) −βρ , x ≤t ≤1 13.1.27 (p. 686) αδ −βρ ̸= 0 G(x, t) = (β cos t −α sin t)(δ cos x −ρ sin x) αδ −βρ , 0 ≤t ≤x, (β cos x −α sin x)(δ cos t −ρ sin t) αδ −βρ... | Elementary Differential Equations with Boundary Value Problems_Page_794_Chunk3206 |
Answers to Selected Exercises 785 13.2.15 (p. 697) (a) λ = 0 isn’t an eigenvalue (b) −1.0664054 y = cosh √ −λx (c) 1.5113188, 8.8785880, 21.2104662, 38.4805610 y = cos √ λ x 13.2.16 (p. 697) (a) λ = 0 isn’t an eigenvalue (b) −1.0239346 y = √ −λ cosh √ −λx −sinh √ −λx (c) 2.0565705, 9.3927144, 21.7169130, 38.9842177 y =... | Elementary Differential Equations with Boundary Value Problems_Page_795_Chunk3207 |
Index A Abel’s formula, 199–202, 468 Accelerated payment, 139 Acceleration due to gravity, 151 Airy’s equation, 319 Amplitude, of oscillation, 271 time-varying, 279 Amplitude–phase form, 272 Aphelion distance, 300 Apogee, 300 Applications, of first order equations, 130–192 autonomoussecondorder equations, 162–179 coolin... | Elementary Differential Equations with Boundary Value Problems_Page_797_Chunk3208 |
788 Index Circuit, RLC. See RLC circuit Closed Circuit, 289 Coefficient(s) See also Constant coefficient equations computing recursively, 322 Fourier, 587 in Frobenius solutions, 352–358 undetermined, method of, 229–248, 475–496 principle of superposition and, 235 Coefficient matrix, 516, 516 Competition, species, 6, 541 ... | Elementary Differential Equations with Boundary Value Problems_Page_798_Chunk3209 |
Index 789 Dirac delta function, 452 Direction fields for first order equations, 16–27 Dirichlet, Peter G. L., 662 Dirichlet condition, 662 Dirichlet problem, 662 Discontinuity, jump, 398 removable, 408 Distributions, theory of, 453 Divergence of improper integral, 393 Divergent power series, 306 E Eccentricity of orbit, ... | Elementary Differential Equations with Boundary Value Problems_Page_799_Chunk3210 |
790 Index piecewise continuous constant equations with, 430–439 Fourier coefficients, 587 Fourier series, 586–616 convergence of, 589 cosine series, 603–604 convergence of, 607 mixed, 606 defined 588 even and odd functions, 592–589 sine series, 605 convergence of, 605 mixed, 609 Fourier solutions of partial differential ... | Elementary Differential Equations with Boundary Value Problems_Page_800_Chunk3211 |
Index 791 separation of variables to solve, 618 Heat flow lines, 185 Heaviside’s method, 407, 412 Hermite’s equation, 322 Heun’s method, 116 Higher order constant coefficient homogeneousequa- tions, 475–487 characteristic polynomial of, 482–478 fundamental sets of solutions of, 480 general solution of, 476–482 Homogeneou... | Elementary Differential Equations with Boundary Value Problems_Page_801_Chunk3212 |
792 Index boundary conditions, 649–651 defined, 649 formal solutions of, 651–662 in polar coordinates, 666–673 for semi-infinte strip, 660 Laplace transforms, 393–461 computation of simple, 393–396 of constant coefficient equations with impulses 452–461 with piecewise continuous forcing functions, 430–439 convolution, 440... | Elementary Differential Equations with Boundary Value Problems_Page_802_Chunk3213 |
Index 793 linear indeopendence of, 521, 524 trivial and nontrivial solution of, 521 Wronskian of solution set of, 523 nonhomogeneous, 516 variation of parameters for, 568–576 solutions to initial value problem, 515–517 Lines of force, 185 Liouville, Joseph, 689 local truncation error, 100–102 numerical methods with O(h... | Elementary Differential Equations with Boundary Value Problems_Page_803_Chunk3214 |
794 Index existence and uniqueness of solutions of, 56–72 transformation into separable equations, 62–72 Nonoscillatory solution, 358 Nontrivial solutions of homogeneous linear first order equations, 30 of homogeneous linear higher order equations, 465 of homogeneous linear second order equations, 194 of homogeneous lin... | Elementary Differential Equations with Boundary Value Problems_Page_804_Chunk3215 |
Index 795 of higher order constant coefficient homoge- neous equations, 475–478 Chebyshev, 322 indicial, 343, 351 Taylor, 309 trigonometric, 602 Polynomial operator, 475 Population growth and decay, 2 Positive half-plane, 552 Potential equation, 649 Power series, 306–319 convergent, 306–307 defined, 306 differentiation o... | Elementary Differential Equations with Boundary Value Problems_Page_805_Chunk3216 |
796 Index second, 425–427 Simple harmonic motion, 269–273 amplitude of oscillation, 271 natural frequency of, 272 phase angle of, 272 Simpson’s rule, 127 Singular point, 319 irregular, 342 regular, 342–347 Solution(s), 9–10 See also Frobenius solutions Non- trivial solutions Series solutions of linear second order equa... | Elementary Differential Equations with Boundary Value Problems_Page_806_Chunk3217 |
Index 797 Undamped autonomoussecond order equations, 164– 171 pendulum, 173–169 spring-mass system, 164–166 stability and instabilty conditions for, 170–171 Undamped motion, 268 Underdamped motion, 279 Underdamped oscillation, 291 Undetermined coefficients for linear higher order equations, 487–496 forcing functions, 48... | Elementary Differential Equations with Boundary Value Problems_Page_807_Chunk3218 |
MULTIVARIABLE CALCULUS Don Shimamoto | Multivariable_Calculus_Shimamoto_Page_1_Chunk3219 |
Multivariable Calculus Don Shimamoto Swarthmore College | Multivariable_Calculus_Shimamoto_Page_5_Chunk3220 |
ii ©2019 Don Shimamoto ISBN: 978-1-7082-4699-0 This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Unless noted otherwise, the graphics in this work were created using the software packages: Cin- derel... | Multivariable_Calculus_Shimamoto_Page_6_Chunk3221 |
Contents Preface vii I Preliminaries 1 1 Rn 3 1.1 Vector arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The matrix of a linear transformation . . . . . . . . . . . . . . . . .... | Multivariable_Calculus_Shimamoto_Page_7_Chunk3222 |
. . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 More surfaces in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 The equation of a plane in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 Open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | Multivariable_Calculus_Shimamoto_Page_7_Chunk3223 |
iv CONTENTS 3.6 Some properties of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.7 The Cauchy-Schwarz and triangle inequalities . . . . . . . . . . . . . . . . . . . . . . 71 3.8 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9 Exercises... | Multivariable_Calculus_Shimamoto_Page_8_Chunk3224 |
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Real-valued functions: integration 121 5.1 Volume and iterated integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2 The double integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | Multivariable_Calculus_Shimamoto_Page_8_Chunk3225 |
. . . . . . . . . . . . . . . . . . . . 178 | Multivariable_Calculus_Shimamoto_Page_8_Chunk3226 |
CONTENTS v 7.4 Change of variables for n-fold integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.5 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 V Integrals of vector fields 195 8 Vector fields 197 8.1 Examples of vector fields . . . . . . . . . . . . .... | Multivariable_Calculus_Shimamoto_Page_9_Chunk3227 |
. . 257 10.7 A more substitution-friendly notation for surface integrals . . . . . . . . . . . . . . . 259 10.8 Independence of parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.9 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 11 Working w... | Multivariable_Calculus_Shimamoto_Page_9_Chunk3228 |
vi CONTENTS | Multivariable_Calculus_Shimamoto_Page_10_Chunk3229 |
Preface This book is based on a course that I taught several times at Swarthmore College. The material is standard, though this particular course is geared towards students who enjoy learning mathematics for its own sake. As a result, there is a priority placed on understanding why things are true and a recognition tha... | Multivariable_Calculus_Shimamoto_Page_11_Chunk3230 |
viii PREFACE chapter is a success if the students come away believing that something interesting is going on and curious enough to want to learn more. As is always the case, the most useful way for students to learn the material is by doing problems, and this book is written to get to the exercises as quickly as possib... | Multivariable_Calculus_Shimamoto_Page_12_Chunk3231 |
Part I Preliminaries 1 | Multivariable_Calculus_Shimamoto_Page_13_Chunk3232 |
Chapter 1 Rn Let R denote the set of real numbers. Its elements are also called scalars. If n is a positive integer, then Rn is defined to be the set of all sequences x of n real numbers: x = (x1, x2, . . . , xn). (1.1) The elements of Rn are called points, vectors, or n-tuples. We follow the convention of indicating v... | Multivariable_Calculus_Shimamoto_Page_15_Chunk3233 |
4 CHAPTER 1. Rn Figure 1.1: Vector addition determined by x and y, as on the left of Figure 1.1. If we think of x + y as an arrow as well, it is one of the diagonals of the parallelogram, as shown on the right. Another way to reach x + y is to move the arrow representing y so that it retains the same length and directi... | Multivariable_Calculus_Shimamoto_Page_16_Chunk3234 |
1.1. VECTOR ARITHMETIC 5 Figure 1.3: Scalar multiplication Figure 1.4: The difference of two vectors In the case of R2, the coordinates are usually denoted by x, y, rather than x1, x2. Then R2 is the usual xy-plane. The notation is potentially confusing since x is also often used to denote the generic vector in Rn, as ... | Multivariable_Calculus_Shimamoto_Page_17_Chunk3235 |
6 CHAPTER 1. Rn 1.2 Linear transformations Linear transformations are functions that respect vector addition and scalar multiplication. More precisely: Definition. A function T : Rn →Rm is called a linear transformation if: • T(x + y) = T(x) + T(y) and • T(cx) = c T(x). The conditions must be satisfied for all x, y in ... | Multivariable_Calculus_Shimamoto_Page_18_Chunk3236 |
1.3. THE MATRIX OF A LINEAR TRANSFORMATION 7 Figure 1.6: So are reflections. On the other hand: T(x) + T(y) = (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2). Both expressions equal the same thing, so T(x + y) = T(x) + T(y). Similarly, T(cx) = T(cx1, cx2, cx3) = (cx1, cx2) while c T(x) = c(x1, x2) = (cx1, cx2). Thus T(cx) = c... | Multivariable_Calculus_Shimamoto_Page_19_Chunk3237 |
8 CHAPTER 1. Rn For example, in R3, if x = (1, 2, 3) and y = (4, 5, 6), then x·y = 1·4+2·5+3·6 = 4+10+18 = 32. The dot product satisfies a variety of elementary properties, such as x · y = y · x. The ones we shall use are pretty obvious, so we won’t bother listing them out, though please see Exercises 5.5–5.10 if you’d... | Multivariable_Calculus_Shimamoto_Page_20_Chunk3238 |
1.3. THE MATRIX OF A LINEAR TRANSFORMATION 9 To use the matrix A to compute T(x) in a systematic way, we observe the convention that vectors are identified with matrices having a single column. Thus: x = x1 x2 ... xn and aj = a1j a2j ... amj represent vectors in Rn and Rm, respectively. So... | Multivariable_Calculus_Shimamoto_Page_21_Chunk3239 |
10 CHAPTER 1. Rn Once you get the hang of it, this may be the simplest way to find a formula for a rotation. For the reflection in the x1-axis, (1.6) gives: T(x) = 1 0 0 −1 x1 x2 = 1 · x1 + 0 · x2 0 · x1 −1 · x2 = x1 −x2 = (x1, −x2). We didn’t need matrix methods to come up with this formula, but at least ... | Multivariable_Calculus_Shimamoto_Page_22_Chunk3240 |
1.5. THE GEOMETRY OF THE DOT PRODUCT 11 Proposition 1.8. Composition of linear transformations corresponds to matrix multiplication. That is, let S : Rm →Rp and T : Rn →Rm be linear transformations with matrices A and B, respectively, with respect to the standard bases. Then S ◦T is a linear transformation, and its mat... | Multivariable_Calculus_Shimamoto_Page_23_Chunk3241 |
12 CHAPTER 1. Rn Definition. The norm, or magnitude, of a vector x = (x1, x2, . . . , xn) in Rn, denoted by ∥x∥, is defined to be: ∥x∥= q x2 1 + x2 2 + · · · + x2n. For instance, in R3, if x = (1, 2, 3), then ∥x∥= √1 + 4 + 9 = √ 14. The following simple property gets used a lot. Proposition 1.10. If x ∈Rn, then x · x =... | Multivariable_Calculus_Shimamoto_Page_24_Chunk3242 |
1.5. THE GEOMETRY OF THE DOT PRODUCT 13 Figure 1.9: Vectors x and y in R2 and the angle θ between them vertices are 0, x, and y. Two of the sides of this triangle have lengths ∥x∥and ∥y∥, and the length of the third side is the length of the arrow −→ yx = x −y. See Figure 1.9. Thus, by the law of cosines, ∥x −y∥2 = ∥x∥... | Multivariable_Calculus_Shimamoto_Page_25_Chunk3243 |
14 CHAPTER 1. Rn If y = b1u2 + b2u2 is another element of P, then: x · y = (a1u1 + a2u2) · (b1u1 + b2u2) = a1b1u1 · u1 + (a1b2 + a2b1)u1 · u2 + a2b2u2 · u2 = a1b1 + a2b2. Thus the dot product in Rn agrees with the result we would expect in terms of the newly created internal coordinates in P. In particular, ∥x∥2 = x · ... | Multivariable_Calculus_Shimamoto_Page_26_Chunk3244 |
1.6. DETERMINANTS 15 1.6 Determinants The determinant is a function that assigns a real number to an n by n matrix. There’s a separate function for each n. We shall focus almost exclusively on the cases n = 2 and n = 3, since those are the cases we really need later. The determinant is not defined for matrices in which... | Multivariable_Calculus_Shimamoto_Page_27_Chunk3245 |
16 CHAPTER 1. Rn Meanwhile: | Multivariable_Calculus_Shimamoto_Page_28_Chunk3246 |
1.6. DETERMINANTS 17 For 3 by 3 matrices, the determinant is defined in terms of the 2 by 2 case: det a11 a12 a13 a21 a22 a23 a31 a32 a33 = a11 · det a22 a23 a32 a33 −a12 · det a21 a23 a31 a33 + a13 · det a21 a22 a31 a32 . (1.14) The signs in the sum alternate, and the pattern is that the terms run alon... | Multivariable_Calculus_Shimamoto_Page_29_Chunk3247 |
18 CHAPTER 1. Rn 1.7 Exercises for Chapter 1 Section 1 Vector arithmetic In Exercises 1.1–1.4, let x and y be the vectors x = (1, 2, 3) and y = (4, −5, 6) in R3. Also, 0 denotes the zero vector, 0 = (0, 0, 0). 1.1. Find x + y, 2x, and 2x −3y. 1.2. Find −→ yx and y + −→ yx. 1.3. If x + y + z = 0, find z. 1.4. If x −2y +... | Multivariable_Calculus_Shimamoto_Page_30_Chunk3248 |
1.7. EXERCISES FOR CHAPTER 1 19 (c) Find the set of all points x = (x1, x2, x3) in R3 such that T(x) = 0. In Exercises 3.3–3.6, find the matrix of the given linear transformation T : R2 →R2 with respect to the standard bases. 3.3. T is the counterclockwise rotation by π 2 about the origin. 3.4. T is the reflection in t... | Multivariable_Calculus_Shimamoto_Page_31_Chunk3249 |
20 CHAPTER 1. Rn Section 4 Matrix multiplication In Exercises 4.1–4.6, find the indicated matrix products. 4.1. AB and BA, where A = 1 −1 1 1 and B = 2 4 −1 3 4.2. AB and BA, where A = 1 2 3 4 and B = 1 0 0 1 4.3. AB and BA, where A = 0 0 1 1 0 0 0 1 0 and B = 0 1 0 0 0 1 1 0 0 4.4. AB and B... | Multivariable_Calculus_Shimamoto_Page_32_Chunk3250 |
1.7. EXERCISES FOR CHAPTER 1 21 4.9. Use matrices to prove that r ◦ρθ ◦r = ρ−θ. 4.10. (a) Compute the product A = RθSR−θ. (b) By thinking about the corresponding composition of linear transformations, give a ge- ometric description of the linear transformation T : R2 →R2 that is represented with respect to the standard... | Multivariable_Calculus_Shimamoto_Page_33_Chunk3251 |
22 CHAPTER 1. Rn 6.3. det 1 −2 3 −4 5 −6 7 −8 9 6.4. det 1 0 0 2 3 0 4 5 6 6.5. Find the area of the parallelogram in R2 determined by x = (4, 0) and y = (1, 3). 6.6. Find the area of the parallelogram in R2 determined by x = (−2, −3) and y = (−3, 2). 6.7. Let A = 1 −2 2 −4 . Use the product rule for... | Multivariable_Calculus_Shimamoto_Page_34_Chunk3252 |
Part II Vector-valued functions of one variable 23 | Multivariable_Calculus_Shimamoto_Page_35_Chunk3253 |
Chapter 2 Paths and curves This chapter is concerned with curves in Rn. While we may have an intuitive sense of what a curve is, at least in R2 or R3, the formal description here is somewhat indirect in that, rather than requiring a curve to have a defining equation, we describe it by how it is swept out, like the trac... | Multivariable_Calculus_Shimamoto_Page_37_Chunk3254 |
26 CHAPTER 2. PATHS AND CURVES Figure 2.1: A parametrization α: [a, b] →Rn of a curve C Example 2.1. Circles in R2: x2 + y2 = a2, where a is the radius of the circle. The equation of the circle can be rewritten as | Multivariable_Calculus_Shimamoto_Page_38_Chunk3255 |
2.1. PARAMETRIZATIONS 27 Example 2.3. Helices in R3. A helix winds around a circular cylinder, say of radius a. If the axis of the cylinder is the z-axis, then the x and y-coordinates along the helix satisfy the equation of the circle as in Example 2.1, while the z-coordinate changes at a constant rate. Thus we set x =... | Multivariable_Calculus_Shimamoto_Page_39_Chunk3256 |
28 CHAPTER 2. PATHS AND CURVES Example 2.5. Find a parametrization of the line in R3 that passes through the points a = (1, 2, 3) and b = (4, 5, 6). The line passes through the point a = (1, 2, 3) and is parallel to v = −→ ab = b −a = (4, 5, 6) − (1, 2, 3) = (3, 3, 3). The setup is indicated in Figure 2.6. Therefore on... | Multivariable_Calculus_Shimamoto_Page_40_Chunk3257 |
2.2. VELOCITY, ACCELERATION, SPEED, ARCLENGTH 29 change of distance ds dt as the speed. Unfortunately, these terms should be defined more carefully, and, to get everything in the right logical order, it seems best to define the speed first. To define what we expect ds dt to be, we make the intuitive approximation that ... | Multivariable_Calculus_Shimamoto_Page_41_Chunk3258 |
30 CHAPTER 2. PATHS AND CURVES 2.3 Integrals with respect to arclength Let C be a curve in Rn, and let f : C →R be a real-valued function defined on C. To formulate the definition of the integral of f over C, we adapt the usual approach of first-year calculus for integrating a function over an interval, namely: • Chop ... | Multivariable_Calculus_Shimamoto_Page_42_Chunk3259 |
2.4. THE GEOMETRY OF CURVES: TANGENT AND NORMAL VECTORS 31 Example 2.9. Consider again the portion of the helix parametrized by α(t) = (cos t, sin t, t), 0 ≤t ≤4π. Find R C (x + y + z) ds. Here, f(x, y, z) = x + y + z, so f(α(t)) = f(cos t, sin t, t) = cos t + sin t + t. In other words, we read off the components of th... | Multivariable_Calculus_Shimamoto_Page_43_Chunk3260 |
32 CHAPTER 2. PATHS AND CURVES Figure 2.10: The unit tangent vector T(t), translated to start at α(t)1 The remaining two curve-related coordinate directions are orthogonal to this first one. In R3, there is a whole plane of orthogonal possibilities, but one of the possibilities turns out to be naturally distinguished. ... | Multivariable_Calculus_Shimamoto_Page_44_Chunk3261 |
2.5. THE CROSS PRODUCT 33 Definition. If T′(t) ̸= 0, then: N(t) = 1 ∥T′(t)∥T′(t) is called the principal normal vector. See Figure 2.11. Figure 2.11: The unit tangent and principal normal vectors, T(t) and N(t) Example 2.13. For the helix parametrized by α(t) = (cos t, sin t, t), find the unit tangent T(t) and the prin... | Multivariable_Calculus_Shimamoto_Page_45_Chunk3262 |
34 CHAPTER 2. PATHS AND CURVES Our goal for the moment is this: given vectors v and w in R3, find a vector, written v × w, that is orthogonal to both v and w. We concentrate initially not so much on what v × w actually is but rather on requiring it to have a certain critical property. Namely, whatever v × w is, we insi... | Multivariable_Calculus_Shimamoto_Page_46_Chunk3263 |
2.5. THE CROSS PRODUCT 35 Similarly, c2 and c3 are determined by looking at j · (v × w) and k · (v × w), respectively. Definition. Given vectors v and w in R3, their cross product is defined by: v × w = det v2 v3 w2 w3 , −det v1 v3 w1 w3 , det v1 v2 w1 w2 . One can check that this is the same as: v × w = de... | Multivariable_Calculus_Shimamoto_Page_47_Chunk3264 |
36 CHAPTER 2. PATHS AND CURVES The basic properties of the cross product are summarized below. 1. w × v = −v × w (Justification. Interchanging rows in (×) changes the sign of the determinant.) 2. v × v = 0 (Justification. Equal rows in (×) implies the determinant is zero.) 3. (The length of the cross product.) ∥v × w∥=... | Multivariable_Calculus_Shimamoto_Page_48_Chunk3265 |
2.5. THE CROSS PRODUCT 37 Figure 2.14: The parallelepiped determined by u, v, w in R3 This follows from property (P), which we can use to reverse course and take what we’ve learned about dot and cross products to tell us about determinants. We leave the details for the exercises (namely, Exercise 5.13), though the rele... | Multivariable_Calculus_Shimamoto_Page_49_Chunk3266 |
38 CHAPTER 2. PATHS AND CURVES fingers of your right hand from i to j, your thumb points in the direction of k. This is not so much a fact as it is an agreement: whenever we draw R3, we agree to orient the positive x, y, and z-axes so that the right-hand rule for (i, j, k) is satisfied, as in Figure 2.15. As we continu... | Multivariable_Calculus_Shimamoto_Page_50_Chunk3267 |
2.6. THE GEOMETRY OF SPACE CURVES: FRENET VECTORS 39 As a cross product, B(t) is orthogonal to T(t) and N(t). Moreover, its length is ∥B(t)∥= ∥T(t)∥∥N(t)∥sin θ = 1 · 1 · sin π 2 = 1. Thus (T(t), N(t), B(t)) is a collection of orthogonal unit vectors. The vectors are called the Frenet vectors of α and are illustrated in... | Multivariable_Calculus_Shimamoto_Page_51_Chunk3268 |
40 CHAPTER 2. PATHS AND CURVES 2.7 Curvature and torsion We use the Frenet vectors (T, N, B) to make two geometric measurements: • curvature, the rate of turning, and • torsion, the rate of wobbling. First, curvature measures how fast the tangent direction is changing, which is represented by ∥T′(t)∥, the magnitude of ... | Multivariable_Calculus_Shimamoto_Page_52_Chunk3269 |
2.7. CURVATURE AND TORSION 41 Example 2.22 (The torsion of a helix). To find the torsion of the helix α(t) = (a cos t, a sin t, bt), we again piggyback on the calculations of Example 2.20. For instance, using the result that B(t) = 1 √ a2+b2 (b sin t, −b cos t, a), we obtain B′(t) = 1 √ a2+b2 (b cos t, b sin t, 0). To ... | Multivariable_Calculus_Shimamoto_Page_53_Chunk3270 |
42 CHAPTER 2. PATHS AND CURVES 2.8 The Frenet-Serret formulas We saw in Examples 2.21 and 2.22 that, for the helix α(t) = (a cos t, a sin t, bt), a > 0, the curvature and torsion are given by: κ = a a2 + b2 and τ = b a2 + b2 , respectively. For example, if α(t) = (2 cos t, 2 sin t, t), i.e., a = 2 and b = 1, then κ = 2... | Multivariable_Calculus_Shimamoto_Page_54_Chunk3271 |
2.9. THE CLASSIFICATION OF SPACE CURVES 43 The same conclusion applies to rotations, though we don’t really have the tools to give a rigorous proof at this point. So we try an intuitive explanation. Suppose that α is rotated in R3 to obtain a new path β. The velocities α′(t) and β′(t) are related by the same rotation, ... | Multivariable_Calculus_Shimamoto_Page_55_Chunk3272 |
44 CHAPTER 2. PATHS AND CURVES Step 3. A calculation. We show that eα = β. Since eα has been obtained from α by translation and rotation, the theorem follows. The argument uses the Frenet-Serret formulas, which we repeat for convenience: T′ = κvN, N′ = −κvT + τvB, B′ = −τvN. We denote the Frenet vectors of eα by (eT, e... | Multivariable_Calculus_Shimamoto_Page_56_Chunk3273 |
2.10. EXERCISES FOR CHAPTER 2 45 2.10 Exercises for Chapter 2 Section 1 Parametrizations In Exercises 1.1–1.8, sketch the curve parametrized by the given path. Indicate on the curve the direction in which it is being traversed. 1.1. α: [0, 2] →R2, α(t) = (t, t2) 1.2. α: [0, 6π] →R2, α(t) = (t cos t, t sin t) 1.3. α: R ... | Multivariable_Calculus_Shimamoto_Page_57_Chunk3274 |
46 CHAPTER 2. PATHS AND CURVES Figure 2.17: The foot of the perpendicular from p to ℓ 1.18. Let ℓbe the line in R3 parametrized by α(t) = (1 + t, 1 + 2t, 1 + 3t), and let p = (0, 0, 4). Find the foot of the perpendicular dropped from p to ℓ. 1.19. Let ℓand m be lines in R3 parametrized by: α(t) = a + tv and β(t) = b + ... | Multivariable_Calculus_Shimamoto_Page_58_Chunk3275 |
2.10. EXERCISES FOR CHAPTER 2 47 Section 3 Integrals with respect to arclength In Exercises 3.1–3.2, evaluate the integral with respect to arclength for the curve C with the given parametrization α. 3.1. R C xyz ds, where α(t) = (t3, 3t2, 6t), 0 ≤t ≤1 3.2. R C z ds, where α(t) = (cos t + t sin t, sin t −t cos t, t), 0 ... | Multivariable_Calculus_Shimamoto_Page_59_Chunk3276 |
48 CHAPTER 2. PATHS AND CURVES 5.4. v = (2, 0, −1), w = (0, −2, 1) 5.5. Find a nonzero vector in R3 that points in a direction perpendicular to the plane that contains the origin and the points p = (1, 1, 1) and q = (2, 1, −3). 5.6. Find a nonzero vector in R3 that points in a direction perpendicular to the plane that ... | Multivariable_Calculus_Shimamoto_Page_60_Chunk3277 |
2.10. EXERCISES FOR CHAPTER 2 49 5.12. True or false: u × (v × w) = (u × v) × w for all u, v, w in R3. Either give a proof or find a counterexample. 5.13. Let u, v, and w be points in R3 such that 0, u, v, and w are not coplanar. Prove that the volume of the parallelepiped determined by u, v, and w is equal to det u ... | Multivariable_Calculus_Shimamoto_Page_61_Chunk3278 |
50 CHAPTER 2. PATHS AND CURVES Section 9 The classification of space curves 9.1. Consider the curve parametrized by: α(t) = (cos t, −sin t, t). (a) Find the speed v(t). (b) Find the unit tangent T(t). (c) Find the principal normal N(t). (d) Find the binormal B(t). (e) Find the curvature κ(t). (f) Find the torsion τ(t).... | Multivariable_Calculus_Shimamoto_Page_62_Chunk3279 |
2.10. EXERCISES FOR CHAPTER 2 51 In Exercises 9.4–9.8, α is a path in R3 with velocity v(t), speed v(t), acceleration a(t), and Frenet vectors T(t), N(t), and B(t). You may assume that v(t) ̸= 0 and T′(t) ̸= 0 for all t so that the Frenet vectors are defined. 9.4. Prove the following statements. (a) v = vT (b) a = v′T ... | Multivariable_Calculus_Shimamoto_Page_63_Chunk3280 |
52 CHAPTER 2. PATHS AND CURVES (c) Show that κ(t) ≥1 a for all t. 9.13. Let α be a path in R3 that has: • constant speed v = 1, • positive torsion τ(t), and • binormal vector B(t) = | Multivariable_Calculus_Shimamoto_Page_64_Chunk3281 |
Part III Real-valued functions 53 | Multivariable_Calculus_Shimamoto_Page_65_Chunk3282 |
Chapter 3 Real-valued functions: preliminaries Thus far, we have studied functions α for which the input is a real number t and the output is a vector α(t) = (x1(t), x2(t), . . . , xn(t)). The techniques from calculus that we used were basically familiar from first-year calculus. Now, we reverse the roles and consider ... | Multivariable_Calculus_Shimamoto_Page_67_Chunk3283 |
56 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES For instance, consider the cross-sections with the three coordinate planes. The yz-plane is where x = 0, so the equation of the intersection of the graph with this plane is z = p 02 + y2 = |y| and x = 0, a V -shaped curve. Similarly, the cross-section with the xz-plane... | Multivariable_Calculus_Shimamoto_Page_68_Chunk3284 |
3.1. GRAPHS AND LEVEL SETS 57 The information about horizontal cross-sections in Figure 3.2 can also be presented by projecting the cross-sections onto the xy-plane and presenting them like a topographical map. As we saw above, the cross-section with z = c is described by the points (x, y) such that x2 + y2 = c2, a cir... | Multivariable_Calculus_Shimamoto_Page_69_Chunk3285 |
58 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES The graph: For the graph, the level sets are taken out of the xy-plane and raised to height z = c. Furthermore, identifying the cross-sections with the coordinate planes provides some additional framework. Coordinate plane Equation of cross-section in that plane yz-pla... | Multivariable_Calculus_Shimamoto_Page_70_Chunk3286 |
3.2. MORE SURFACES IN R3 59 Figure 3.8: The unit sphere x2 + y2 + z2 = 1 (left) and the cylinder x2 + y2 = 1 (right) Example 3.5. A circular cylinder of radius a whose axis is the z-axis. Here, as long as (x, y) satisfies the condition for being on the circle of radius a, z can be anything. Thus: Equation of a circular... | Multivariable_Calculus_Shimamoto_Page_71_Chunk3287 |
60 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES Figure 3.10: The cooling tower x2 + y2 −z2 = 1 level set of f(x, y, z) = x2 + y2 corresponding to c = a2; and the cooling tower the level set of f(x, y, z) = x2 + y2 −z2 corresponding to c = 1. This suggests a way to visualize the behavior of functions of three variabl... | Multivariable_Calculus_Shimamoto_Page_72_Chunk3288 |
3.3. THE EQUATION OF A PLANE IN R3 61 In other words, n·(x−p) = 0. This says that x lies on the level set if and only if x−p is orthogonal to n. Geometrically, this is true if and only if x lies in the plane that both passes through p and is perpendicular to n. This is illustrated in Figure 3.11. We say that n is a nor... | Multivariable_Calculus_Shimamoto_Page_73_Chunk3289 |
62 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES The scalar multiple n = (1, −2, 1) is also a normal vector, and, since it’s a little simpler, it’s the one we use. Substituting into n · x = n · p gives: (1, −2, 1) · (x, y, z) = (1, −2, 1) · (1, 2, 3), x −2y + z = 1 −4 + 3, or x −2y + z = 0. Example 3.10. Are the plan... | Multivariable_Calculus_Shimamoto_Page_74_Chunk3290 |
3.4. OPEN SETS 63 Figure 3.14: Open balls in R and R2 Definition. A subset U of Rn is called open if, given any point a of U, there exists a positive real number r such that B(a, r) ⊂U. So to show that a set U is open, one starts with an arbitrary point a of U and finds a value of r so that the open ball about a of rad... | Multivariable_Calculus_Shimamoto_Page_75_Chunk3291 |
64 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES Figure 3.16: An open quadrant ball about a includes points outside of U, namely, points in the upper half-plane. This is shown in Figure 3.17. Thus no value of r works for this choice of a. Figure 3.17: A nonopen quadrant Example 3.15. If U and V are open sets in Rn, p... | Multivariable_Calculus_Shimamoto_Page_76_Chunk3292 |
3.4. OPEN SETS 65 U3 = B((0, 0), 1 3), etc. Each Un is open, but the only point common to all of them is the origin. Thus U1 ∩U2 ∩· · · = {(0, 0)}. This is no longer an open set. There is also a notion of closed set, though the definition may not be what one would guess. Definition. A subset K of Rn is called closed if... | Multivariable_Calculus_Shimamoto_Page_77_Chunk3293 |
66 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES 3.5 Continuity We are ready for continuity. Intuitively, the idea is that a function f is continuous at a point a if limx→a f(x) = f(a). That is, as x gets close to a, f(x) gets close to f(a). We make this precise by expressing the requirement in terms of open balls. D... | Multivariable_Calculus_Shimamoto_Page_78_Chunk3294 |
3.5. CONTINUITY 67 At the same time, f(a) = f(c, d) = c, which agrees. Thus we expect intuitively that f is continuous at a. To prove this rigorously, let ϵ > 0 be given. We want to find a radius δ so that the open ball B | Multivariable_Calculus_Shimamoto_Page_79_Chunk3295 |
68 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES suppose that we approach along the x-axis. Then f(x, 0) = x·0 x2+02 = 0, so: lim (x,0)→(0,0) f(x, 0) = lim (x,0)→(0,0) 0 = 0. On the other hand, if we approach along the line y = x, then f(x, x) = x·x x2+x2 = 1 2, so: lim (x,x)→(0,0) f(x, x) = lim (x,x)→(0,0) 1 2 = 1 2... | Multivariable_Calculus_Shimamoto_Page_80_Chunk3296 |
3.6. SOME PROPERTIES OF CONTINUOUS FUNCTIONS 69 We get a consistent answer of 0, but this does not prove that the limit is 0. We have not exhausted all possible ways of approaching the origin, and being close to the origin is different from being close to it along any individual curve. Nevertheless, the evidence sugges... | Multivariable_Calculus_Shimamoto_Page_81_Chunk3297 |
70 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES Before proving this result, let’s apply it to some examples. Example 3.22. We showed earlier in Example 3.18 that the projection onto the x-axis f : R2 →R, f(x, y) = x, is continuous. Likewise, the projection onto the y-axis g: R2 →R, g(x, y) = y, is continuous. It the... | Multivariable_Calculus_Shimamoto_Page_82_Chunk3298 |
3.7. THE CAUCHY-SCHWARZ AND TRIANGLE INEQUALITIES 71 Proof. Let a be a point of U, and let ϵ > 0 be given. We want to find an open ball about a that is mapped by g ◦f inside the interval B(g(f(a)), ϵ) = | Multivariable_Calculus_Shimamoto_Page_83_Chunk3299 |
72 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES Proof. It’s equivalent to square both sides and prove that ∥v + w∥2 ≤(∥v∥+ ∥w∥)2. But: ∥v + w∥2 = (v + w) · (v + w) = v · v + v · w + w · v + w · w = ∥v∥2 + 2v · w + ∥w∥2. (3.5) Also, v · w ≤|v · w| ≤∥v∥∥w∥, where the second inequality is Cauchy-Schwarz. Substituting t... | Multivariable_Calculus_Shimamoto_Page_84_Chunk3300 |
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