CHALTE C Control & Dynamical Systems Key Points, Vector Calculus Math1C - Spring 2010 Jerry Marsden and Eric Rains Control and Dynamical Systems and Mathematics, Caltech For computing resources: www.cds.caltech.edu/~marsden Contents 1 The Geometry of Euclidean Space 7 1.1 Vectors in 2- and 3-Dimensional Space .... 8 1.2 The Inner Product, Length, and Distance .. 12 1.3 Matrices, Determinants, and the Cross Product .................. 14 1.4 Cylindrical and Spherical Coordinates .... 18 1.5 n-dimensional Euclidean Space ......... 20 2 Differentiation 24 2.1 Functions, Graphs, and Level Surfaces .... 25 2.2 Limits and Continuity .............. 27 2.3 Differentiation .................. 30 2.4 Introduction to Paths .............. 34 2.5 Properties of the Derivative .......... 36 2.6 Gradients and Directional Derivatives .... 38 3 Higher-Order Derivatives; Maxima and Minima 40 3.1 Iterated Partial Derivatives ........... 41 3.2 Taylor's Theorem ................. 43 3.3 Extrema of Real Valued Functions ...... 45 3.4 Constrained Extrema and Lagrange Multipliers ..................... 50 3.5 The Implicit Function Theorem ........ 53 4 Vector Valued Functions 57 4.1 Acceleration and Newton's Second Law ... 58 3 4.2 Arc Length .................... 60 4.3 Vector Fields ................... 62 4.4 Divergence and Curl ............... 64 5 Double and Triple Integrals 68 5.1 Introduction .................... 69 5.2 The Double Integral over a Rectangle .... 72 5.3 The Double Integral Over More General Re- gions ........................ 75 5.4 Changing the Order of Integration ...... 77 5.5 The Triple Integral ................ 79 6 The Change of Variables Formula and Ap- plications 81 2 2 6.1 The Geometry of Maps from R to R .... 82 6.2 The Change of Variables Theorem ...... 84 6.3 Applications of Double and Triple Integrals . 89 4 6.4 Improper Integrals ................ 93 7 Integrals over Curves and Surfaces 96 7.1 The Path Integral ................ 97 7.2 Line Integrals ................... 99 7.3 Parametrized Surfaces .............. 102 7.4 Area of a Surface ................. 104 7.5 Integrals of Scalar Functions over Surfaces . 107 7.6 Surface Integrals of Vector Functions ..... 111 7.7 Applications: Differential Geometry, Physics, Forms of Life ................... 117 8 The Integral Theorems of Vector Analy- sis 119 8.1 Green's Theorem ................. 120 8.2 Stokes' Theorem ................. 125 8.3 Conservative Fields ............... 128 5 8.4 Gauss' Theorem ................. 130 8.5 Applications: Physics, Engineering & Differ- ential Equations ................. 136 8.6 Differential Forms ................ 140 6 Chapter 1 The Geometry of Euclidean Space Chapter 1 1.1 Vectors in 2- and 3-Dimensional Space Key Points in this Section 1. Addition and scalar multiplication for three-tuples are defined by (a1; a2; a3) + (b1; b2; b3) = (a1 + b1; a2 + b2; a3 + b2) and α(a1; a2; a3) = (αa1; αa2; αa3): There are similar definitions for pairs of real numbers (just leave off the third component). 8 Chapter 1 2. A vector (in the plane or space) is a directed line segment with a specified tail (with the default being the origin) and an arrow at its head. 3. Vectors are added by the parallelogram law and scalar multiplication by α stretches the vector by this amount (in the opposite direction if α is negative). 4. If a vector has its tail at the origin, the coordinates of its tip are its components. 5. Addition and scalar multiplication of vectors (geometric) corresponds to the same operations on the components (algebraic). 6. Standard Bases: Unit vectors i, j, k along the x, y, and z-axes. 7. A vector a (a1; a2; a3) is written a = a1i + a2j + a3k: 9 Chapter 1 8. The vector joining two points P = (x; y; z) and P0 = (x0; y0; z0) −−! is the vector PP0, represented as an arrow from P to P0, and has components −−! PP0 = (x − x0; y − y0; z − z0): 9. The equation of the line through the point a (regarded as a vector from the origin) in the direction of the vector v (regarded as a vector based at the point a) is `(t) = a + tv; where t ranges over all real numbers. 10. The equations of the straight line through the points P1 = (x1; y1; z1) and P2 = (x2; y2; z2) are x = x1 + t(x1 − x2) y = y1 + t(y1 − y2) z = z1 + t(z1 − z2) 10 Chapter 1 11. The plane through the origin containing the vectors v and w consists of all points of the form sv + tw where s and t range over all real numbers. 11 Chapter 1 1.2 The Inner Product, Length, and Distance Key Points in this Section 1. The inner product of the vectors a = (a1; a2; a3) and b = (b1; b2; b3) is defined as a · b = (a1b1 + a2b2 + a3b3); this inner product is sometimes denoted ha; bi. 2. The length or norm of a = (a1; a2; a3) is q p 2 2 2 kak = a · a = a1 + a2 + a3: 3. To normalize a nonzero vector a, form the unit vector a : kak −−! 4. The distance between two points P and Q is kPQk. 12 Chapter 1 5. The angle θ between two vectors a and b satisfies a · b = kakkbk cos θ: 6. The Cauchy-Schwarz Inequality: ja · bj ≤ kakkbk: 7. The orthogonal projection of the vector v on the nonzero vector a is a · v p = a: kak2 Note that this is unchanged is a is multiplied by any nonzero scalar. 8. Triangle Inequality: ka + bk ≤ kak + kbk: 9. If an object has a constant velocity vector v, then after t units of time, the object is moved by the displacement vector d = tv. 13 Chapter 1 1.3 Matrices, Determinants, and the Cross Product Key Points in this Section 1. Matrices are arrays of numbers, such as the 2 × 2 matrix 1 3 −1 4 and the general 3 × 3 matrix 2 3 a11 a12 a13 4a21 a22 a235 a31 a32 a33 2. The determinant of a 2 × 2 matrix is a11 a12 = a11a22 − a21a12: a21 a22 14 Chapter 1 3. The determinant of a 3 × 3 matrix is a11 a12 a13 a22 a23 a21 a23 a21 a22 a21 a22 a23 = a11 − a12 + a13 a32 a33 a31 a33 a31 a32 a31 a32 a33 4. Determinants may be expanded along any column or any row using the following checkerboard pattern 2+ − +3 4− + −5 + − + 5. Any multiple of one row can be added to another row with out changing the determinant. Same for columns, but you cannot mix rows and columns. 6. The cross product of the vectors a and b is the vector i j k a × b = a1 a2 a3 b1 b2 b3 15 Chapter 1 7. The length of a × b is ka × bk = kakkbk sin θ; where θ is the angle (with 0 ≤ θ ≤ π) between the vectors a and b, and equals the area of the parallelogram spanned by these vectors. 8. The triple product a1 a2 a3 a · (b × c) = b1 b2 b3 c1 c2 c3 is the volume of the parallelogram spanned by the three vectors a, b, and c. 9. The equation of the plane through the point (x0; y0; z0) and normal to the vector n = Ai + Bj + Ck is A(x − x0) + B(y − y0) + C(z − z0) = 0: 16 Chapter 1 that is, Ax + By + Cz + D = 0; where D = −(Ax0 + By0 + Cz0). 10. The distance from the point (x1; y1; z1) to the plane Ax + By + Cz + D = 0 is jAx + By + Cz + D Distance = p1 1 1 : A2 + B2 + C2 17 Chapter 1 1.4 Cylindrical and Spherical Coordinates Key Points in this Section 1. The polar coordinates (r; θ) of a point (x; y) in the xy-plane are determined by x = r cos θ and y = r sin θ: 3 2. The cylindrical coordinates (r; θ; z) of a point (x; y; z) in R are determined by x = r cos θ; y = r sin θ; and z = z: 3 3. The spherical coordinates (ρ, θ; φ) of a point (x; y; z) in R are determined by x = ρ sin φ cos θ; y = ρ sin φ sin θ; and z = ρ cos φ. 4. The equations of geometric objects can sometimes be eas- iest to describe using one of these coordinate systems. 18 Chapter 1 For example, a cylinder is described by r = constant and a sphere by ρ = constant. 19 Chapter 1 1.5 n-dimensional Euclidean Space Key Points in this Section n 1. Euclidean n-space, denoted R , consists of n-tuples of real numbers: x = (x1; x2; : : : ; xn). 2. Addition and scalar multiplication of n-tuples is defines as we did with 2- and 3-tuples: (x1; x2; : : : ; xn) + (y1; y2; : : : ; yn) = (x1 + y1; x2 + y2; : : : ; xn + yn) α(x1; x2; : : : ; xn) = (αx1; αx2; : : : ; αxn) 3. The inner, or dot product is defined by x · y = x1y1 + x2y2 + · + xnyn; 2 3 and satisfies properties as with vectors in R and R . 20 Chapter 1 4. In particular, the Cauchy-Schwarz and triangle inequal- ities hold: jx · yj ≤ kxkkyk and kx + yk ≤ kxk + kyk: 5. An n × n matrix is a square array of numbers with n rows and n columns. For instance, a 4 × 4 matrix has the form 2 3 a11 a12 a13 a14 6a21 a22 a23 a247 6 7 4a31 a32 a33 a345 a41 a42 a43 a44 6. The determinant of a 4×4 matrix may be expanded along any row or column with a pattern of alternating +'s and 21 Chapter 1 −'s, as in the three by three case. For example, a11 a12 a13 a14 a22 a23 a24 a21 a23 a24 a21 a22 a23 a24 = a11 a32 a33 a34 − a12 a31 a33 a34 a31 a32 a33 a34 a42 a43 a44 a41 a43 a44 a41 a42 a43 a44 a21 a22 a24 a21 a22 a23 + a13 a31 a32 a34 − a14 a31 a32 a33 a41 a42 a44 a41 a42 a43 7.