7 Polar and parametric curves 57 INTUITIVE INFINITESIMAL CALCULUS 7.1 Polar coordinates . 57 7.2 Calculus in polar coordinates . 57 Viktor Blåsjö 7.3 Parametrisation . 58 © 2020 intellectualmathematics.com 7.4 Calculus of parametric curves . 59 7.5 Reference summary . 59 8 Vectors 61 8.1 Vectors . 61 CONTENTS 8.2 Scalar product . 61 8.3 Vector product . 63 8.4 Geometry of vector curves . 64 8.5 Reference summary . 64 1 Differentiation 3 1.1 Infinitesimals . .3 9 Multivariable differential calculus 67 1.2 The derivative . .4 9.1 Functions of several variables . 67 1.3 Derivatives of polynomials . .5 9.2 Tangent planes . 69 1.4 Derivatives of elementary functions . .6 9.3 Unconstrained optimisation . 69 1.5 Basic differentiation rules . .7 9.4 Gradients . 70 1.6 The chain rule . .7 9.5 Constrained optimisation . 71 1.7 Limits . .7 9.6 Multivariable chain rule . 72 1.8 Reference summary . .8 9.7 Reference summary . 73 2 Applications of differentiation 11 10 Multivariable integral calculus 76 2.1 Maxima and minima . 11 10.1 Multiple integrals . 76 2.2 Concavity . 12 10.2 Polar coordinates . 76 2.3 Tangent lines . 12 10.3 Cylindrical coordinates . 77 2.4 Conservation laws . 13 10.4 Spherical coordinates . 77 2.5 Differential equations . 14 10.5 Surface area . 78 2.6 Direction fields . 16 10.6 Reference summary . 78 2.7 Reference summary . 17 11 Vector calculus 81 3 Integration 19 11.1 Vector fields . 81 3.1 Integrals . 19 11.2 Divergence . 82 3.2 Relation between differentiation and integration . 20 11.3 Line integrals . 83 3.3 Evaluating integrals . 21 11.4 Circulation . 84 3.4 Change of variables . 22 11.5 Curl . 85 3.5 Integration by parts . 22 11.6 Electrostatics and magnetostatics . 85 3.6 Partial fractions . 23 11.7 Electrodynamics . 86 3.7 Reference summary . 24 11.8 Reference summary . 87 12 Further problems 90 4 Applications of integration 30 12.1 Newton’s moon test . 90 4.1 Volume . 30 12.2 The rainbow . 90 4.2 Arc length . 30 12.3 Addiction modelling . 91 4.3 Center of mass . 30 12.4 Estimating n!......................... 91 4.4 Energy and work . 32 12.5 Wallis’s product for ¼ ..................... 92 4.5 Logarithms redux . 33 12.6 Power series by interpolation . 92 4.6 Reference summary . 34 12.7 Path of quickest descent . 93 12.8 Isoperimetric problem . 93 5 Power series 36 12.9 Isoperimetric problem II . 94 5.1 The idea of power series . 36 5.2 The geometric series . 37 13 Further topics 96 5.3 The binomial series . 38 13.1Curvature............................ 96 5.4 Divergence of series . 39 13.2 Evolutes and involutes . 96 5.5 Reference summary . 40 13.3 Fourier series . 97 13.4 Hypercomplex numbers . 98 6 Differential equations 42 13.5 Calculus of variations . 98 6.1 Separation of variables . 42 6.2 Statics . 44 A Precalculus review 100 6.3 Dynamics . 45 A.1 Coordinates . 100 6.4 Second-order linear differential equations . 46 A.2 Functions . 100 6.5 Second-order differential equations: complex case . 48 A.3 Trigonometric functions . 101 6.6 Phase plane analysis . 50 A.4 Logarithms . 102 6.7 Reference summary . 52 A.5 Exponential functions . 103 1 A.6 Complex numbers . 103 A.7 Reference summary . 105 B Linear algebra 111 B.1 Introduction . 111 B.2 Matrices . 111 B.3 Linear transformations . 111 B.4 Gaussian elimination . 112 B.5 Inverse matrices . 113 B.6 Determinants . 113 B.7 Eigenvectors and eigenvalues . 114 B.8 Diagonalisation . 116 B.9 Data............................... 118 B.10 Reference summary . 119 C Notation reference table 125 2 1.1.2. Explain why. Hint: a tangent line may be considered a 1D IFFERENTIATION line that cuts a curve in “two successive points,” i.e., as the limit of a secant line as the two points of intersection are brought closer and closer together: § 1.1. Infinitesimals § 1.1.1. Lecture worksheet The basic idea of the calculus is to analyse functions by means of their behaviour on a “micro” level. Curves can be very com- plicated when taken as a whole but if you zoom in far enough they all look straight, and if you slice a complicated area thin enough the slices will pretty much be rectangles. Lines and rectangles are very basic to work with, so at the micro level ev- erything is easy. Here is an example: 1.1.3. Why is it called a “tangent” line? Hint: the dance “tango” and the adjective “tangible” share the same Latin root. 1.1.1. The area of a circle is equal to that of a triangle with its dy radius as height and circumference as base. 1.1.4. Argue that dx is the slope of the graph. (a) Explain how this follows from the figure below. § 1.1.2. Problems 1.1.5. Explain how the result of problem 1.1.1 can also be ob- tained by considering the area as made up of infinitesi- mally thin concentric rings instead of “pizza slices.” = 1.1.6. † Isaac Newton’s Philosophiae Naturalis Principia Math- ematica (1687) is arguably the most important scientific work of all time. The very first proposition in this work is Kepler’s law of equal areas. The law says that planets sweep out equal areas in equal times: = ∆t ∆t (b) Explain why this is equivalent to the school formula A ¼r 2. Æ In the context of the calculus we utilise this idea by system- Newton’s proof uses nothing but very simple infinitesi- atically dividing the x-axis into “infinitely small” or “infinitesi- mal geometry. mal” pieces, which we call dx (“d” for “difference”). Here I have drawn such a dx and the associated change dy in the value of C c the function: b B ds dy S A dx In an infinitely small period of time the planet has moved from A to B. If we let an equal amount of time pass Since dx is so small, the curve may be considered to coincide again then the planet would continue to c if it was not with the hypotenuse ds on this interval. Of course in the figure for the gravity of the sun, which intervenes and deflects this is not quite so, but the figure is only schematic: in reality the planet to C. Since the time it takes for the planet to dx is infinitely small, so the hypotenuse ds really does coincide move from B to C is infinitely small, the gravitational pull with the curve exactly, we must imagine. has no time to change direction from its initial direction BS, thus causing cC to be parallel to BS. In fact, if we extend the hypotenuse segment ds we get the tan- gent line to the curve. (a) Conclude the proof of the law. 3 § 1.2. The derivative x.” This is what we do for example when we speak of so- and-so many “miles per hour.” It can be confusing that § 1.2.1. Lecture worksheet concepts like “per hour” or “per step in the x-direction” occurs in the description of something that is supposedly dy The idea that dx is the slope of the graph of y(x) is very useful. instantaneous and not at all ongoing for hours. It has many faces besides the geometrical one: The confusion can be illustrated with the following sce- dy • Geometrically, dx is the slope of the graph of y. nario. A police officer stops a car. dy OFFICER: The speed limit here is 60 km/h and you were • Verbally, dx is the rate of change of y. going 80. • Algebraically, DRIVER: 80 km/h? That’s impossible. I have only been dy y(x dx) y(x) driving for ten minutes. Å ¡ . dx Æ dx OFFICER: No, it doesn’t mean that you have been driving for an hour. It means that if you kept going at that speed • Physically, the rate of change of distance is velocity; the for an hour you would cover 80 km. rate of change of velocity is acceleration. DRIVER: Certainly not. If I kept going like that I would 1.2.1. Sketch the graphs of the distance covered, the speed, and soon smash right into that building there at the end of the acceleration of a sprinter during a 100 meter race, the street. and explain how your graphs agree with the above char- acterisations of these quantities. (a) How can the officer better explain what a speed of 80 km/h really means? Of course the rate of change of y(x) is generally different for dif- ferent values of x. We use y0(x) to denote the function whose Imagine an electric train travelling frictionlessly on an value is the rate of change of y(x). We call y0(x) the derivative infinite, straight railroad. The train is running its engine of y(x). Thus y(x) is the primitive function, meaning the start- at various rates, speeding up and slowing down accord- ing point, while y0(x) is merely derived from it. For example, ingly. Then at a certain point it turns off the engine. Of y0(0) 3 and y0(1) 1 means that the function is at first ris- course the train keeps moving inertially. Æ Æ¡ ing quite steeply but is later coming back down, albeit at a less (b) Explain how this image captures both the “instan- rapid rate. taneous” and the “per hour” aspect of.