Engineering Mathematics: YouTube Workbook Christopher C. Tisdell Contents How to use this workbook 8 About the author 9 Acknowledgments 10 1 Partial derivatives & applications 11 1.1 Partial derivatives & partial differential equations 12 1.2 Partial derivatives & chain rule 14 1.3 Taylor polynomial approximations: two variables 16 1.4 Error estimation 18 1.5 Differentiate under integral signs: Leibniz rule 20 2 Some max/min problems for multivariable functions 22 2.1 How to determine & classify critical points 23 2.2 More on determining & classifying critical points 25 2.3 The method of Lagrange multipliers 27 2.4 Another example on Lagrange multipliers 29 2.5 More on Lagrange multipliers: 2 constraints 31 3 A glimpse at vector calculus 33 3.1 Vector functions of one variable 34 3.2 The gradient field of a function 36 3.3 The divergence of a vector field 38 3.4 The curl of a vector field 40 3.5 Introduction to line integrals 42 3.6 More on line integrals 44 3.7 Fundamental theorem of line integrals 46 3.8 Flux in the plane + line integrals 48 4 Double integrals and applications 50 4.1 How to integrate over rectangles 51 4.2 Double integrals over general regions 53 4.3 How to reverse the order of integration 55 4.4 How to determine area of 2D shapes 57 4.5 Double integrals in polar co-ordinates 59 4.6 More on integration & polar co-ordinates 61 4.7 Calculation of the centroid 63 4.8 How to calculate the mass of thin plates 65 5 Ordinary differential equations 67 5.1 Separable differential equations 68 5.2 Linear, first-order differential equations 70 5.3 Homogeneous, first-order ODEs 72 5.4 2nd-order linear ordinary differential equations 74 5.5 Nonhomogeneous differential equations 76 5.6 Variation of constants / parameters 78 6 Matrices and quadratic forms 80 6.1 Quadratic forms 81 7 Laplace transforms and applications 83 7.1 Introduction to the Laplace transform 84 7.2 Laplace transforms + the first shifting theorem 86 7.3 Laplace transforms + the 2nd shifting theorem 88 7.4 Laplace transforms + differential equations 90 8 Fourier series 93 8.1 Introduction to Fourier series 94 8.2 Odd + even functions + Fourier series 96 8.3 More on Fourier series 98 8.4 Applications of Fourier series to ODEs 100 9 PDEs & separation of variables 102 9.1 Deriving the heat equation 103 9.2 Heat equation & separation of variables 105 9.3 Heat equation & Fourier series 107 9.4 Wave equation and Fourier series 109 Bibliography 111 This workbook is designed to be used in conjunction with the author’s free online video tutorials. Inside this workbook each chapter is divided into learning modules (subsections), each having its own dedicated video tutorial. View the online video via the hyperlink located at the top of the page of each learning module, with workbook and