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KeyPoints_Slides

# KeyPoints_Slides - CALTECH Control Dynamical Systems Key...

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C ontrol & D ynamical S ystems C A L T E C H 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

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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

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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 6.1 The Geometry of Maps from R 2 to R 2 . . . . 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

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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

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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 ( a 1 , a 2 , a 3 ) + ( b 1 , b 2 , b 3 ) = ( a 1 + b 1 , a 2 + b 2 , a 3 + b 2 ) and α ( a 1 , a 2 , a 3 ) = ( αa 1 , αa 2 , αa 3 ) . 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 .

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KeyPoints_Slides - CALTECH Control Dynamical Systems Key...

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