Mathematical Tripos: IA Vector Calculus Contents 0 Introduction i 0.1 Schedule . i 0.2 Lectures . i 0.3 Printed Notes . ii 0.4 Examples Sheets . iii 0.5 Previous Comments, Comments on Comments and Style of Lectures. iii 0.6 Books ................................................... iv 0.7 Revision . iv 0.7.1 Vectors . iv 3 0.7.2 Cylindrical polar co-ordinates (ρ, φ, z) in R . ......................... vi 3 0.7.3 Spherical polar co-ordinates (r, θ, φ) in R ............................ vi 0.7.4 Determinants. vi 1 Partial Differentiation 1 1.1 Scalar Functions of One Variable . 1 1.1.1 Limits . 1 1.1.2 Continuity . 1 1.1.3 Differentiability . 1 1.1.4 Taylor’s Theorem . 2 1.2 Functions of Several Variables . 2 1.2.1 Limits . 3 1.2.2 Continuity . 3 1.3 Derivatives of Vector Functions of One Variable . 4 1.3.1 Example . 4 1.4 Partial Derivatives . 4 1.4.1 Definition . 5 1.4.2 Examples . 5 1.4.3 Directional Derivatives . 7 1.5 Differentiability . 7 1.5.1 Scalar Functions of Two Variables . 7 1.5.2 Functions of Several Variables . 8 1.5.3 Properties of Differentiable Functions . 9 1.6 The Chain Rule and Change of Variables . 9 m m 1.6.1 The Chain Rule for x : R → R and f : R → R ....................... 9 ` m m 1.6.2 The Chain Rule for x : R → R and f : R → R ...................... 11 ` m m n 1.6.3 The Chain Rule for x : R → R and f : R → R ...................... 12 Mathematical Tripos: IA Vector Calculus a c [email protected], Lent 2000 1.6.4 The Matrix Form of the Chain Rule . 12 1.6.5 Change of Variables . 12 1.7 The Gradient of a Scalar Field . 14 1.7.1 The Gradient is a Vector . 15 1.7.2 Directional Derivatives Revisited . 16 1.8 Partial Derivatives of Higher Order . 17 1.8.1 A Sufficient Condition for the Equality of Mixed Partial Derivatives . 18 1.8.2 Change of Variables: Example . 18 1.9 Taylor’s Theorem . 19 2 Curves and Surfaces 22 2.1 Curves . 22 2.1.1 The Length of a Simple Curve . 24 2.1.2 The Arc Length . 25 2.1.3 The Tangent . 26 2.1.4 The Osculating Plane . 26 2.1.5 The Serret-Frenet Equations . 27 2.2 Surfaces . 30 2.2.1 Representations . 30 2.2.2 Normals . 30 2.2.3 Definitions . 33 2.3 Critical Points or Stationary Points . 34 2.3.1 Definitions . 34 2.3.2 Functions of Two Variables . 35 2.3.3 Maximum, Minimum or Saddle? . 38 3 Line Integrals and Exact Differentials 41 3.1 The Riemann Integral . 41 3.2 Line Integrals . 42 3.2.1 Line Integrals of Scalar Functions . 42 3.2.2 Line Integrals of Vector Functions . 44 3.3 Differentials . 46 3.3.1 Interpretations . 47 3.3.2 Properties . 48 3.3.3 Exact Differentials . 48 3.3.4 Solution of Differential Equations . 51 3.4 Line Integrals of Exact Differentials . 52 3.4.1 Conservative Fields . 53 3.5 Non Simply Connected Domains . 55 Mathematical Tripos: IA Vector Calculus b c [email protected], Lent 2000 4 Multiple Integrals 56 4.1 Double Integrals (Area Integrals) . 56 4.1.1 Evaluation by Successive Integration . 56 4.2 Triple Integrals (Volume Integrals) . 59 4.2.1 Evaluation . 59 4.3 Jacobians and Change of Variables . 60 4.3.1 Jacobians . 61 4.3.2 Chain Rule for Jacobians . 62 4.3.3 Change of Variable in Multiple Integrals . 63 4.4 Surface Integrals . 69 4.4.1 Surfaces . 69 4.4.2 Evaluation of Surface Integrals . 71 4.4.3 Flux . 73 4.5 Unlectured Worked Exercise: Examples Sheet 2, Question 18. 73 5 Vector Differential Operators 75 5.1 Revision of Gradient . 75 5.1.1 Summary of Properties . 75 5.2 Divergence . 76 5.2.1 Divergence in Cartesian Co-ordinates . ..