# Useful Formulae

Equations are the magical invocations of millions of modern mathemagicians. Have you ever needed one fast Circles, Triangles, Spaceships, Mechanics, Trigonometry, Statistics, Areas, Liquids, Gases, Fireworks, Calculus, Fields, Dimensions, Constants, Circuits, Probabilities, Gears, More... Travelling mathemusician Matthew Watkins and wizardly illustrator Matt Tweed have compiled nearly three thousand years of arduous human cogitation into one extraordinary little book. Not for muggs.- useful FORMUL mathematical PHYSICAL Matthew Watkins Matt Tweed and illuminated by compiled by First
- Contents Introduction 1 Triangles 2 Twodimensional figures 4 Threedimensional figures 6 Coordinate g
- 1 INTRODUCTION This little book is an attempt to present the basic formul of mathematics and physics
- 2 TRIANGLES and their various centres A rightangled triangleobeys the Law of Pythagoras the square o
- 4 TWODIMENSIONALFIGURES areas and perimeters Formul for the perimeters and areas of various twodimen
- 6 THREEDIMENSIONALFIGURES volumes and surface areas Formul for the volumes and surface areas includi
- 8 COORDINATEGEOMETRY axes, lines, gradients and intersections A pair of axes imposed on the plane at
- 10 TRIGONOMETRY applied to right triangles A right triangle with sides of length a,b,c, and angle is
- 12 TRIGONOMETRICALIDENTITIES relating the six functions The definitions on the previous page lead to
- sinC 14 SPHERICALTRIGONOMETRY formulae for heaven and earth A spherical trianglehas internal angles
- 16 THEQUADRATICFORMULA discriminants and parabolas A quadratic equationis one of the form ax2 bx c
- ap ap a pq aq a Given some value a, we can define a squared and a cubedas follows a2 axa, a3 axa
- Given two numbers,aand b,then the three important averages or means,traditionally used in geometry,m
- 22 COMBINATIONSPERMUTATIONS ways of arranging things Suppose we have nitems and we wish to consider
- Statistical analysis allows us to process samples of observed data in order to reveal trends and mak
- 26 KEPLERSNEWTONSLAWS bodies in motion Johannes Kepler discovered three laws describing planetary mo
- 28 GRAVITYANDPROJECTILES featherless falling objects Newtons Law of Universal Gravitation states tha
- 30 ENERGY, WORKMOMENTUM conservation in action An object, mass m, moving in a straight line with vel
- 32 ROTATIONANDBALANCE whirling, gears and pulleys If an object of mass m on a string of length r is
- 34 SIMPLEHARMONICMOTION vibrating and oscillating phenomena Galileo discovered that the periodTof a
- 36 STRESS, STRAINANDHEAT expansion, contraction, tension and compression E When a mate
- 38 LIQUIDSANDGASSES temperature, pressure and flow Liquid flowing at velocity v though a pipe with a
- 40 SOUND harmonic wavelengths and passing sirens For a string fixed at two points distance L apart,
- 42 LIGHT refraction, lenses and relativity Light travelling from air into water is shown opposite. I
- 44 ELECTRICITYANDCHARGE tuning the circuit For a simple electrical circuit,the voltageEin volts acro
- 46 ELECTROMAGNETICFIELDS charge, flux and handedness There are strong parallels between gravitationa
- 48 CALCULUS differentiation and integration Calculus makes use of infinitesimalsand limitsto solve t
- 50 COMPLEXNUMBERS into the imaginary realm The familiar real numbersare contained within the larger
- 52 HIGHERDIMENSIONS beyond the familiar knot Many geometric forms in the twodimensional plane can be
- 55 54 glossary greater than greater than or equal to not equal to less than less
- 57 56 constants UNITS Mathematical Constants 3.14159265358979... e 2.718281828459045... 2 1.414213
- Riemanns zeta function Probably the most challenging and mysterious object of modern mathematics,in