Basic symbols and algebra notations. Background mathematics review

Transcription

1 Basic symbols and algebra notations Background mathematics review David Miller

2 Basic symbols and algebra notations Basic mathematical symbols Background mathematics review David Miller

3 Elementary arithmetic symbols Equals Addition or plus 3 5 Subtraction, minus or less 3 1 or Multiplication or / Division 63 6/3 or ( numerator) dividend ( demonimator) divisor quotient

4 Relational symbols Is equivalent to x x/ y y or 1 Is approximately equal to Is proportional to a x x Is greater than 3 Is greater than or equal to 1 x 1 Is less than 3 Is less than or equal to Is much greater than Is much less than 11 x

5 Greek characters used as symbols Upper case Lower case Name Roman equiv. alpha a a beta b b gamma g g delta d d epsilon e e zeta z z eta (e) h theta th q iota i i kappa k k lambda l l mu m m Upper case Lower case Name Roman equiv. Keyboard Keyboard nu n n xi x x omicron (o) o pi p p rho r r sigma s s tau t t upsilon u u phi ph f chi ch c psi psy y omega o w

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7 Basic symbols and algebra notations Basic mathematical operations Background mathematics review David Miller

8 Conventions for multiplication For multiplying numbers We explicitly use the multiplication sign 3 6 For multiplying variables We can use the multiplication sign But where there is no confusion We drop it ab c might be simply replaced by ab c

9 Use of parentheses and brackets When we want to group numbers or variables We can use parentheses (or brackets) (34) 714 For such grouping, we can alternatively use square brackets or curly brackets When used this way, there is no difference in the mathematical meaning of these brackets

10 Associative property Operations are associative if it does not matter how we group them e.g., addition of numbers is associative a bcabc e.g., multiplication of numbers is associative abcabc But division of numbers is not associative 8/4 / / 1 but 8/ 4/ 8/ 4

11 Distributive property Property where terms within parentheses can be distributed to remove the parentheses abcabac Here, multiplication is said to be distributive over addition Many other conceivable operations are not distributive, however E.g., addition is not distributive over multiplication

12 Commutative property Property where the order can be switched round e.g., addition of numbers is commutative a bba e.g., multiplication of numbers is commutative abba But e.g., subtraction is not commutative e.g., division is not commutative 6/3 3/6 1

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14 Basic symbols and algebra notations Algebra notation and functions Background mathematics review David Miller

15 Parentheses and functions A function is something that relates or maps One set of values Such as an input variable or argument x To another set of values which we could think of as an output For example, the function 1 f x x 4 f x x

16 Parentheses and functions Conventionally, we say f of x when we read Here obviously f x is not f times x Most commonly Only parentheses are used around the argument x not square [ ] or curly { } brackets f x f x x

17 Parentheses and functions For a few very commonly used functions Such as the trigonometric functions The parentheses are optionally omitted when the argument is simple sin instead of sin Note, incidentally, sin sin sin 1 1

18 Parentheses and functions For a few very commonly used functions Such as the trigonometric functions The parentheses are optionally omitted when the argument is simple cos instead of cos Note, incidentally cos cos cos 1 1

19 Sine, cosine, and tangent r or hypotenuse angle x, base or adjacent side y, height or opposite side Defined using a right-angled triangle sin y r cos Natural units for angles in mathematics are radians radians in a circle 1 radian ~ 57.3 degrees x r tan tan y x sin cos

20 r or hypotenuse angle x, base or adjacent side y, height or opposite side Cosecant, secant, and cotangent Cosecant r cosec csc y Secant sec r x 1 cos 1 sin Cotangent x 1 cos cotan cot y tan sin

21 Inverse sine function 1 The inverse sine function sin a or arcsine function Pronounced arc-sine works backwards to give the angle from the sine value If a sin then 1 arcsin a asin a sin a Note sin a does not mean 1 1/sina The -1 here means inverse function not reciprocal sin 1 a a

22 sin and cos functions sin 1 However sin sinsin sin Not sinsin Similarly cos cos Only trigonometric functions and their close relatives commonly use this notation 0 1 cos 0

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24 Powers, logs, exponentials and complex numbers Background mathematics review David Miller

25 Powers, logs, exponentials and complex numbers Powers Background mathematics review David Miller

26 Powers Remember from elementary algebra 3 which is to the power 3 The 3 here can also be called the exponent the power to which the number is raised

27 Powers Multiplying by the same number raises the power 3 4 Dividing by the same number lowers the power 3 / 1 Following this logic / 1 0 and / 1 Generalizing any number to the power zero is 1 0 x 1

28 Powers Continuing 0 1 1/ / and so on for further negative powers / 4 Generalizing, for any number x and any power a x a 1 x a

29 Reciprocal 5 1 x x 1 is called the reciprocal of x -5 It becomes arbitrarily large in magnitude as x goes towards zero Loosely, it explodes at the origin Rigorously, it is singular at the origin Negative powers generally have this property x

30 Squares and square roots Multiplying a number x by itself xx x is called taking the square Because it gives the area of a square of side x x Area x x x x

31 Squares and square roots For some number x the number x that, when multiplied by itself gives x is called the square root of x x x x e.g., 4 is the radical sign Also x x 1/ x x

32 Square root Note that If x x x So also x x x is the square root of 4 So also - is the square root of 4 Conventionally, we presume we mean the positive square root unless otherwise stated But we always have both positive and negative versions of the square root

33 r x y Distance and Pythagoras s theorem Pythagoras s theorem gives r x y or equivalently r x y where we always take the positive square root so r is a distance and is always positive

34 Quadratics and roots 3 x x For a quadratic equation of form ax bx c 0 the solutions or roots are b b 4ac x a For x x 0, a 1, b 1 and c So x or 1 Note x x x x Example of a parabola x

35 Powers of powers To raise a power to a power Multiply the powers, e.g., 3 6 Generalizing a b c a bc

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37 Powers, logs, exponentials and complex numbers Logarithms and exponentials Background mathematics review David Miller

38 Powers and logarithms The inverse operation of raising to a power is taking the logarithm (log for short) With logarithms, we need to specify the base of the logarithm Our example was to the power 3 is 8 The logarithm to the base of 8 is log 8 3

39 Powers and logarithms Generalizing a chain of n numbers g multiplied together is n ggg g and n is the log to the base g of g n n log g g n

40 Powers and logarithms Though we created these ideas using integer numbers of multiplications n we can generalize to non-integers For some arbitrary (real, positive, and non-zero) number b, we can write a log g b which means a g b

41 Powers and logarithms We note, for example, that Note that in multiplying we have added the exponents a b Generalizing g g a b g Equivalently, if we write a b A g so log g A a and B g so log g B b Then a b log gablog g( g g ) a b log g g ab log g A log multiplying numbers is equivalent to adding their logarithms g B

42 Bases for logarithms When logarithms are used for calculations Typically base 10 is used Base 10 logarithms are often used by engineers in expressing power ratios in practice using decibels (abbreviated db) which are 10 times the logarithm of the ratio

43 Bases for logarithms E.g., for an amplifier with an output power P out that is 100 times larger than the input power P in Gain (in db)=10 log 10 (P out /P in ) i.e., Gain (in db)=10 log 10 (100)=10x=0dB E.g., for an amplifier with a gain of Noting that log Gain of times (in db)=10 log 10 3dB

44 Changing bases of logarithms Suppose log10 b a i.e., b 10 a Now, by definition log10 10 So a log 10 log 10 a b So logb log10a log10log10b Generalizing, and dropping parentheses and log b log d log b c c d

45 Bases for logarithms Sometimes log means log 10 e.g., on a calculator keyboard Another common base is base (i.e., log ) e.g., in computer science because of binary numbers Fundamental physical science and mathematics almost always uses logs to the base e e e is the base of the natural logarithms

46 Notations with e Logs to base e are called natural logarithms log e (sometimes just log ) or ln letter l for logarithm and letter n for natural To avoid confusion with other uses of e e.g., for the charge on an electron And to avoid superscript characters we use the exponential notation x exp x e Also means this can be referred to as the exponential function

47 Exponential and logarithm Exponential function For larger negative arguments Gets closer and closer ( asymptotes ) to the x axis For larger positive arguments Grows faster and faster Logarithm For smaller positive arguments arbitrarily large and negative expx ln x x x

48 Exponentials and logarithms Note all the following formulas Which follow from the discussions above a b ab exp exp exp 1 exp exp ln a exp a a ln 1/ a ln a a ln expa a ab a b ln ln ln

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50 Powers, logs, exponentials and complex numbers Imaginary and complex numbers Background mathematics review David Miller

51 Square root of minus one In ordinary real numbers no number multiplied by itself gives a negative result Equivalently 4 4 There is no (real) square root of a negative number If, however, we choose to define an entity that we call the square root of minus one We can write square roots of negative numbers We obtain a very useful algebra

52 Square root of minus one Define i 1 so i 1 and i 1 Also, common engineering notation is j 1 Any number proportional to i is called an imaginary number e.g., 4i, 3.74i, i Common to put the i after numbers, but before variables or constants Can write the square root of any negative number using i ii

53 Complex numbers A number that can be written g a ib where a and b are both real numbers is called a complex number a is called the real part of g a Re g b is called the imaginary part of g b Im g

54 Complex conjugate and modulus The complex conjugate has the sign of the imaginary part reversed And is indicated by a superscript * g a ib Multiplying g by g gives a positive number Called the modulus squared of g g g g gg The (positive) square root of this is called the modulus of g g g

55 Important complex number identities Note for the modulus squared g i.e., gg aibaib g a b and for the reciprocal 1 1 c id g c id cid cid a iabibai b i.e., 1 c d g i c id c d c d Still a sum of real and imaginary parts c id c d a b c d i c d c d

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57 Powers, logs, exponentials and complex numbers Euler s formula and the complex plane Background mathematics review David Miller

58 Euler s formula Euler s formula is the remarkable result exp i cos isin A major practical algebraic reason for use of complex numbers in engineering Exponentials much easier to manipulate than sines and cosines

59 Some results from Euler s formula Using Euler s formula expi cos isin Note that expi expi cos isin so expi expi Also expi expi expi i so cos isin cos isin cos isin exp0 1 cos isincos icossin i sin cos sin 1

60 Complex exponential or polar form r a b a b For any complex number g a ib we can write a ib a b g g g i g a b a b which we can write in the form g g cos isin so any complex number can be written in the form g g expi

61 Complex plane Propose a complex plane horizontal real axis vertical imaginary axis Then any complex number b g sin g aib g expi is a point on this plane Sometimes called an Argand diagram Sometimes is called the argument of this polar version of a complex number, arg g imaginary axis g. g exp( i ) g a b real axis a g cos

62 Multiplication in polar representation In the polar representation g g expi To multiply two numbers Multiply the moduli and add the angles I.e., with h h expi exp gh g exp i h i g h expi

63 nth roots of unity Note that the number exp i/ n when raised to the nth power is 1 (unity) i n i exp exp n exp i 1 n n Many different complex numbers when raised to the nth power can give 1 But this specific one is conventionally called the nth root of unity n 1 exp i/ n

64 Algebraic results for complex numbers All the following useful algebraic identities are easily proved from the complex exponential form gh g h 1 1 gh g h g h g h

65 Sine and cosine addition formulas Sine and cosine sum and difference formulas are easily deduced from the complex exponential form e.g., expi cos isin exp i exp i cos isin cos sin isincos Equating real parts gives cos cos sin Equating imaginary parts gives sin sincos

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67 Coordinates and vectors Background mathematics review David Miller

68 Coordinates and vectors Coordinate axes and vectors Background mathematics review David Miller

69 y Coordinate axes and vectors z x Ordinary geometry Three axes x, y, and z All at right angles Cartesian axes (from René Descartes) Lines or directions at right angles are also called orthogonal

70 y Coordinate axes and vectors z x Right-handed axes Using your right hand Thumb x Index ( first ) finger y Middle finger z No matter how you now rotate your whole hand the axes remain righthanded

71 y Coordinate axes and vectors z x If you use your left hand Thumb x Index ( first ) finger y Middle finger z give left-handed axes No rotation of this entire set of left-handed axes will ever make it right-handed We use right hand axes unless otherwise stated

72 z y P y origin. z P Coordinate axes and vectors For some point P in space. P The corresponding (x projections onto the P, y P, z P ) coordinate axes give x Cartesian coordinates P x x P, y P, and z P, relative to the origin of the axes Sometimes written (x P, y P, z P )

73 Coordinate axes and vectors G A vector is something with a magnitude such as a length and a direction Usually written in bold font e.g., G Sometimes G or G And shown as an arrow With length and direction

74 Coordinate axes and vectors A r B A vector could be the distance and direction you need to walk to get from A to B

75 Coordinate axes and vectors F A vector could be A force how hard you are pushing and what direction you are pushing

76 Coordinate axes and vectors v A vector could be A velocity how fast you are going (speed) e.g., the number on your car speedometer and what direction you are going in e.g., on a compass

77 r F v Coordinate axes and vectors An ordinary number which has no direction is called a scalar Distance how hard you push speed are all scalars Scalars are in ordinary fonts Usually italic in printing

78 y Coordinate axes and vectors G z G y k j i G G x x A vector has components along three orthogonal axes G x, G y, and G z We can also define vectors of unit length along each axis i unit vector along x j unit vector along y k unit vector along z z

79 y Coordinate axes and vectors G y Then we can write G=G x i+g y j+g z k G G y j G z G x i G x G z k x z

80 y Coordinate axes and vectors G y G G x i G y j G x x Then we can write G=G x i+g y j+g z k making the final vector up by adding its vector components G z k G z z

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82 Coordinates and vectors Operations with vectors Background mathematics review David Miller

83 Adding vectors G G + S S S G + S G To add vectors graphically connect them head to tail in any order

84 Adding vectors S z k G + S S y j G z k G y j G x i S x i To add vectors algebraically add them component by component G S G i G jg x x G S i G S j G S x y y S i S j x S z z k k y y z z k

85 Multiplying vectors Two kinds of multiplications or products for geometrical vectors Dot product ab Gives a scalar result Cross product ab Gives a vector result

86 Vector dot product angle a b One formula for the dot product is ab a b cos abcos Here the modulus sign means we take the length of the vector a a Note that a bba Also aa a So a aa

87 Vector dot product angle a b One formula for the dot product is ab a b cos abcos We can think of a b cos as The projection of vector b onto the direction of vector a Multiplied by the length of a or The projection of vector a onto the direction of vector b Multiplied by the length of b

88 Vector dot product a b One formula for the dot product is ab a b cos abcos Note that for two vectors at right angles /90 and cos / 0 so the dot product is zero

89 Vector dot product j k i 0 The unit vectors along the coordinate directions are all orthogonal (at right angles) So all their dots products with one another are zero i j ik 0 jk 0 ji 0 ki 0 k j 0 Also, since these are unit length vectors, by definition ii 1 jj 1 kk 1

90 Vector dot product a b Since i j 0 ik 0 jk 0 ji 0 ki 0 k j 0 Forming the dot product algebraically ab axiayjazk bxibyjbzk gives ab axbx ayby azbz which is an equivalent formula for the dot product

91 Vector dot product G i The components of a vector can be found by taking the dot product with the unit vectors along the coordinate directions For example G i G i G y jg k i G x z x

92 Vector cross product For two vectors a a ia ja k x y z a b bxibyjbzk the vector cross product is abn a b sin nabsin b n is a unit vector with a direction given by the right hand screw rule

93 a b ab gives vector n away from you b a ab gives vector n towards you Right hand screw rule Imagine you have a corkscrew With an ordinary right-handed thread with its handle lined up along vector a Now rotate the handle so it lines up with vector b The direction, in or out, that the corkscrew moved is the direction of the vector n

94 Vector cross product a b Note that abba If we have to turn clockwise to go from a to b So the corkscrew goes in So n points inwards Then we have to turn anti-clockwise to go from b to a So the corkscrew goes out So n point outwards

95 Vector cross product An equivalent algebraic formula for the vector cross product is ab a ybz azby i azbx axbz j axby aybx k A short-hand way of writing this is ab ax a y a z b b b x y z which is the same as the determinant notation used with matrix algebra i j k

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97 Differential calculus Background mathematics review David Miller

98 Differential calculus First derivative Background mathematics review David Miller

99 First derivative For some function y x The (first) derivative is the slope gradient or rate of change of y as we change x If for some small infinitesimal change in x, called dx y changes by some small infinitesimal amount dy y the first derivative is written dy yx dx The ratio notation on the right is Leibniz notation x

100 First derivative The derivative at some specific point x 1 can be written dy yx1 dx The value of the derivative is the slope of the tangent line the dashed line in the figure at that point Equal in value to the tangent x 1 y y x dy dx x 1 x x 1 y x 3 dy dx x 3 x

101 First derivative y Looking at the slope yx x x yx1 yx1 x y x of the orange line as we reduce x the orange line slope becomes closer to the slope of the black tangent line 1 1 x x x x 1 x

102 First derivative In the limit as x becomes very small i.e., in the limit as x tends to zero lim x 0 this ratio becomes the (first) derivative dy dx x 1 lim x 0 x x yx1 yx1 x

103 Sign of first derivative If y increases as we increase x dy dx sloping up to the right 0 y dy dx 0 dy dx 0 If y decreases as we increase x dy dx sloping down to the right 0 x 1 x 3 x

104 Derivative of a power y x 1 d x dx n nx n1 dy dx x

105 Derivative of a power The derivative of a straight line is a constant The straight line has a constant slope 1 y x 1 dy dx

106 Derivative of a power The derivative does not depend on the height All these lines have the same slope y x 1 1 dy dx

107 Derivative of a power The derivative does not depend on the height All these lines have the same slope y x dy dx

108 Derivative of a power The derivative does not depend on the height All these lines have the same slope y x dy dx

109 Derivative of a power The derivative of a constant y is not changing with x y is zero dy dx

110 Derivative of an exponential 3 d dx exp x exp x 3 1 y exp x 1 dy expx dx

111 Derivative of a logarithm d dx ln x 1 x 1 1 y ln x dy dx 1 x

112 Derivatives of sine and cosine d sin dx x cos x y sinx 1 dy cosx dx 1 1 1

113 Derivatives of sine and cosine d dx cos x sin x y cosx 1 1 dy dx sin x 1 1

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115 Differential calculus Second derivative Background mathematics review David Miller

116 Second derivative The second derivative is The derivative of the derivative d y d dy y x dx dx dx The rate of change of the derivative or slope x

117 Second derivative The slope at x / is, for small dy y0 yx dx x/ x And similarly at x / dy yx y0 dx x So x/ d y d dy lim dx dx dx x 0 dy dx x dy dx x x/ x/ x

118 Second derivative The slope at x / is, for small dy y0 yx dx x And similarly at x / dy y x y dx x So x/ x/ d y d dy 0 lim dx dx dx x 0 x x y x y y y x x x x

119 Second derivative The slope at x / is, for small dy y0 yx dx x And similarly at x / dy y x y dx x So x/ x/ d y d dy 0 lim dx dx dx x 0 x 0 x y x y y x x

120 Sign of second derivative Going from a positive first derivative To a negative first derivative Gives a negative second derivative Going from a negative first derivative To a positive first derivative Gives a positive second derivative y d y 0 dx dy dy 0 dx 0 dx d y dx x 1 x x 3 x 4 0 x

121 Sign of second derivative Any region where the first derivative is decreasing with increasing x Has a negative second derivative Any region where the first derivative is increasing with increasing x Has a positive second derivative y d y dx 0 d y dx 0 x x x x x x x

122 Sign of second derivative Points where the derivative is neither increasing or decreasing i.e., second derivative is changing sign correspond to zero second derivative Known as inflection points y d dx y 0 x d y dx 0 x x

123 Curvature The second derivative can be thought of as the curvature of a function Large positive curvature

124 Curvature The second derivative can be thought of as the curvature of a function Small positive curvature

125 Curvature The second derivative can be thought of as the curvature of a function Large negative curvature

126 Curvature The second derivative can be thought of as the curvature of a function Small negative curvature

127 Curvature The value of the curvature does not depend on the height of the function All these curves have the same curvature

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129 Differential calculus Linearity and differentiation rules Background mathematics review David Miller

130 Linearity linear superposition For two functions ux and v x The derivative of the sum is the sum of the derivatives d ux vx du dv dx dx dx Example f x x ln x Split into ux x v x ln x So du 1 dx dv 1 dx x So du v 1 fx 1 dx x

131 Linearity multiplying by a constant For a function u x The derivative of a constant a times a function is a times the derivative d du au a dx dx Example f x a x ax Split into So So 1/ du dx a 1/ u x x 1 1/ 1 x x du a fxa dx x

132 Linearity An operation or function f x is linear if f y z f y f z and linear superposition or additivity condition f ax a f x multiplication by a constant (or formally homogeneity of degree one ) condition

133 Example of nonlinear operation The function does not represent a linear operation But f x x f yz yz y z yz f y f z y z So for this function f x y is not in general equal to f x f y

134 Product rule For two functions ux and v x The derivative of the product is d uvu dv v du dx dx dx Example f x x sin x Split into ux x v x sin x So du x dx dv cos x dx So duv fx x cos xxsin x dx

135 Quotient rule For two functions ux and v x The derivative of the ratio or quotient is du dv v u d u dx dx dx v v Example f Split into x 3 3 x 1 x u x x v x 1 x So So du 3x dx dv x dx d u 3 1 x 3x x x dx v 1 x

136 Quotient rule For two functions ux and v x The derivative of the ratio or quotient is du dv v u d u dx dx dx v v Example f Split into x 3 3 x 1 x u x x v x 1 x So So du 3x dx dv x dx d u 3 3x 1 x x x dx v 1 x

137 Quotient rule For two functions ux and v x The derivative of the ratio or quotient is du dv v u d u dx dx dx v v Example f Split into x 3 x 1 x 3 u x x v x 1 x So So du 3x dx dv x dx d u 4 3x 1x x dx v 1 x

138 Quotient rule For two functions ux and v x The derivative of the ratio or quotient is du dv v u d u dx dx dx v v Example f Split into x 3 x 1 x 3 u x x v x 1 x So So du 3x dx dv x dx 4 d u 3x x dx v 1 x 1 x

139 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f gx df dg dx dg dx Example h x Split into g x 1 g x 1x x f y y

140 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f gx df dg dx dg dx Example h x 1x Split into g x1x f g g So dg df g x g dx dg So dh 1 x x 4x 1 x dx g x

141 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f gx df dg dx dg dx Example h x exp ax Split into gx ax f g exp g So dg df g expg dx a dg So dh expax a a expax dx g x

142 Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f gx df dg dx dg dx Example h x exp ax Split into g x ax f g exp g So dg df g ax expg dx dg So dh exp ax ax axexp ax dx g x

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144 Sums, products and power series Background mathematics review David Miller

145 Sums, products and power series Sum, factorial and product notations Background mathematics review David Miller

146 Summation notation If we want to add a set of numbers a 1, a, a 3, and a 4, we can write S a1a a3 a4 or we can use summation notation If the range of j is obvious S a j Here j is an index 4 S a a a j j j j1 j1,,3,4 j1,...,4 j plural of index indexes or indices

147 Example arithmetic series a1a a3 a4 For a set of numbers spaced by a constant amount, an arithmetic progression or sequence i.e., where the nth number is an a1 n1 d e.g., the numbers 4, 7, 10, and 13 i.e., a1 4, a 7, a3 10 and a4 13 with, therefore d 3 and m 4 terms in the progression gives the series (sum of the terms)

148 Example arithmetic series a1a a3 a4 For a set of numbers spaced by a constant amount, an arithmetic progression or sequence i.e., where the nth number is an a1 n1 d e.g., the numbers 4, 7, 10, and 13 i.e., a1 4, a 7, a3 10 and a4 13 with, therefore d 3 and m 4 terms in the progression gives the series (sum of the terms) 4 m aj a1 j 1d j1 j1

149 Example arithmetic series For a set of numbers spaced by a constant amount, an arithmetic progression or sequence i.e., where the nth number is an a1 n1 d e.g., the numbers 4, 7, 10, and 13 i.e., a1 4, a 7, a3 10 and a4 13 with, therefore d 3 and m 4 terms in the progression gives the series (sum of the terms) a1a a3 a4 m 1 1 j1 a j d m a a 1 4

150 Example arithmetic series For a set of numbers spaced by a constant amount, an arithmetic progression or sequence i.e., where the nth number is an a1 n1 d e.g., the numbers 4, 7, 10, and 13 i.e., a1 4, a 7, a3 10 and a4 13 with, therefore d 3 and m 4 terms in the progression gives the series (sum of the terms) m a1 am 413 a 1a a3 a4 a1 j1d m 4 34 j1

151 Example geometric series a1a a3 a4 With a constant ratio between successive terms, a geometric progression or sequence n 1 i.e., where the nth term is an ar 1 e.g., the numbers 3, 6, 1, and 4 i.e., a1 3, a 6, a3 1 and a4 4 with, therefore r and m 4 terms in the progression gives the series (sum of the terms)

152 Example geometric series a1a a3 a4 With a constant ratio between successive terms, a geometric progression or sequence n 1 i.e., where the nth term is an ar 1 e.g., the numbers 3, 6, 1, and 4 i.e., a1 3, a 6, a3 1 and a4 4 with, therefore r and m 4 terms in the progression gives the series (sum of the terms) 4 a m j j1 j1 j 1 ar 1

153 Example geometric series With a constant ratio between successive terms, a geometric progression or sequence n 1 i.e., where the nth term is an ar 1 e.g., the numbers 3, 6, 1, and 4 i.e., a1 3, a 6, a3 1 and a4 4 with, therefore r and m 4 terms in the progression gives the series (sum of the terms) 4 m m j1 1 r a1a a3 a4aj ar 1 a1 1 r j1 j1

154 Example geometric series With a constant ratio between successive terms, a geometric progression or sequence n 1 i.e., where the nth term is an ar 1 e.g., the numbers 3, 6, 1, and 4 i.e., a1 3, a 6, a3 1 and a4 4 with, therefore r and m 4 terms in the progression gives the series (sum of the terms) 4 m m j1 1r 116 a1a a3 a4aj ar 1 a r 1 j1 j1

155 Summation over multiple indexes We can extend the summation notation Suppose we have two lists of numbers a 1, a, a 3, a 4 and b 1, b, b 3 and we want to add up all the products R ab 1 1ab 1 ab 1 3 ab 1ab ab 3 ab 3 1ab 3 ab 3 3 abab ab

156 Summation over multiple indexes Then we can write R ab 1 1ab 1 ab 1 3 abab ab ab ab ab abab ab ab j k j1 k1 and, because order of addition does not matter in various equivalent notations R ab ab ab ab j k j k j k jk, j k j1 k1 k1 j1 j, k

157 Factorial notation Quite often, we need a convenient way of writing the product of successive integers e.g., 134 We can write this as 1 344! called four factorial and using the exclamation point! The notation is obvious for most other cases Note, though, that we choose 0! 1

158 Product notation Generally, when we want to write the product of various successive terms a a a a by analogy with the summation notation we can use the product notation a1aa3a 4 a For example, for all integers n 1 n n! p p1 4 j1 j

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160 Sums, products and power series Power series Background mathematics review David Miller

161 Analytic functions and power series For a very broad class of the functions in physics we presume they are analytic (except possibly at some singularities ) i.e., at least for some range of values of the The ellipsis means we omit writing some terms explicitly argument x near some point x o the function f x can be arbitrarily well approximated by a power series i.e., with x o =0for simplicity we have 3 f x ao a1xax a3x which may be an infinite series

162 Analytic functions and power series For a very broad class of the functions in physics we presume they are analytic (except possibly at some singularities ) i.e., at least for some range of values of the argument x near some point x o the function f x can be arbitrarily well approximated by a power series i.e., generally 3 o 1 o o 3 o f x a a xx a xx a xx

163 Maclaurin series Taking the simplest case of x o =0 first 3 f x ao a1xax a3x Obviously at x =0 f 0 ao so, trivially, ao f Now, df fx a1ax3a3x dx so df f 0 a1 dx 0 0

164 Maclaurin series Continuing d f f x a 3a3x! a 3! a3x dx so d f f 0! a dx 0 and 3 d f f 0 3! a 3 3 dx and so on 0

165 Maclaurin and Taylor series f f x x o Continuing gives the Maclaurin series n n xdf x d f x d f f x f 0 n 1! dx! dx n! dx Repeating the same procedure around x x o with 3 f x a a xx a xx a xx gives the Taylor series o 1 o o 3 o xx xx xx n n o df o d f o d f n 1! dx x! dx n! dx x x o o o

166 Example power series expansions x x x 3 x 3 x x expx1x! 3! 4 x x cos x 1! 4! sin x 3 5 x x x 3! 5!

167 Power series for approximations Maclaurin and Taylor series allow approximations for small ranges of the argument about a point Examples for small x Approximations to first order in x 1/ 1x 1x exp x 1 x sin x x 1x 1 x / tan x Note lowest order dependence on x for cos x is second order cos x 1 x / ln 1 x x x

168

169 Partial differentiation Background mathematics review David Miller

170 Partial differentiation Partial derivatives Background mathematics review David Miller

171 Partial derivative Suppose we have a function f(x,y) of two variables Then we can have the gradient of f as a function of x with y held constant f f which we write or sometimes just x x y the partial derivative of f with respect to x with y constant partial d f by d x - partial d

172 Partial derivative Suppose we have a function f(x,y) of two variables Then we can have the gradient of f as a function of y with x held constant f f which we write or sometimes just y y x the partial derivative of f with respect to y with x constant partial d f by d y - partial d

173 M Partial derivative f x, y We have a surface with height f x, y varying with the coordinates x and y At a specific position xo, y the height is f x, y o o o y o x o y x f x, y o o

174 Partial derivative f x, y M

175 Partial derivative f x, y Along the y direction at point xo, yo the rate of change of height with y slope at constant x is the slope of the orange line f y x x y y o M

176 Partial derivative f x, y M

177 M Partial derivative f x, y

178 M Partial derivative f x, y Along the x direction at point xo, yo the rate of change of height with x at constant y is the slope of the orange line x o slope x f x y

179 M Second partial derivatives We can also form second (and higher order) partial derivatives f f x x is a second derivative and is a measure of the curvature of the function in the x direction y f x, y x y

180 M Second partial derivatives We can also form second (and higher order) partial derivatives f f y y x is a second derivative and is a measure of the curvature of the function in the y direction f x, y x y

181 M Second partial derivatives f x, y We can also form second (and higher order) partial derivatives f f y y x is a second derivative and is a measure of the curvature of the function in the y direction x y f x f y 0 0

182 MM Second partial derivatives We can also form second (and higher order) partial derivatives f f y y x is a second derivative and is a measure of the curvature of the function in the y direction f x, y x y f x f y 0 0

183 Second partial derivatives f x, y We can also form second (and higher order) partial derivatives f f y y x is a second derivative and is a measure of the curvature of the function in the y direction saddle point x y f x f y 0 0 MS

184 W Cross derivative The cross derivative f f x y x y y x is a measure of how warped a surface is Note that, for ordinary smooth functions f f xy yx f x, y y x f xy 0

185 Cross derivative The cross derivative f f x y x y y x is a measure of how warped a surface is Note that, for ordinary smooth functions f f xy yx f x, y y x f xy 0 W

186

187 Partial differentiation Differentials Background mathematics review David Miller

188 Differential MP Look at one patch of a function f x, y x long in the x direction y long in the y direction How much does the height change, i.e., f, as we move from one corner to the other? f x f y x, y x y

189 Differential As we move by x along x f the height changes x x y As we move by y along y f the height changes y y x so the total change in height f f is f x y x y y x MP f x f y x, y f x x y y x f y y x

190 Differential In the limit as we make x and y very small i.e., infinitesimal f f f x y x y y x becomes f f df dx dy x y y x which is called a differential or sometimes an exact differential

191 Total derivative We suppose we know the slopes of the hill in the two coordinate directions We presume we also know how fast we are moving in the x and y directions So, in some small time t we will have moved amounts x and y in the x and y directions v x f x y f and y dx dy and vy dt dt dx dy x t and y t dt dt x

192 Total derivative So, using the differential idea we have or or, in the limit of small t we have the total derivative f f f x y x y y f dx f dy f t t x dt y dt y f f dx f dy t x dt y dt y df f dx f dy dt x dt y dt y x x x x

193 Changing coordinates for partial derivatives Suppose we want the slopes of a hill f(x,y) along the South-East (a) and North-East (b) directions instead of along the East (x) and North (y) directions but we only know the slopes along the East (x) and North (y) directions f f x and y y x North y North-East b East x a South-East

194 Changing coordinates for partial derivatives We do know that if we move South-East by one unit we move East by 1/ units since cos 451/ x 1 i.e., a b North y North-East b 1/ 45 1 East x a South-East

195 Changing coordinates for partial derivatives We do know that if we move South-East by one unit we move East by 1/ units since cos 451/ x 1 i.e., a b Similarly y 1 a b North y 1 45 North-East b 1 East x a South-East

196 Changing coordinates for partial derivatives We do know that if we move South-East by one unit we move East by 1/ units since cos 451/ x 1 i.e., a b Similarly y 1 x 1 a b b a North y 1 North-East b 45 1/ East x a South-East

197 Changing coordinates for partial derivatives We do know that if we move South-East by one unit we move East by 1/ units since cos 451/ x 1 i.e., a b Similarly y 1 x 1 y a b b b a a 1 North y 1 45 North-East b 1 East x a South-East

198 Changing coordinates for partial derivatives Suppose we make a small movement a along the a (South-East) direction and no movement along the b (North-East) direction x 1 Then x a a a b North y a North-East b x East x a South-East

199 Changing coordinates for partial derivatives Suppose we make a small movement a along the a (South-East) direction and no movement along the b (North-East) direction x 1 Then x a a a b and y 1 y a a a b y North y a North-East b East x a South-East

200 Changing coordinates for partial derivatives With these results x 1 y 1 x a a a y a a b a b the resulting change f in the value of the function f(x,y) from this movement along the a (South- East) direction is f x f y f a a x a y a y b x b

201 Changing coordinates for partial derivatives Starting from dividing by a gives taking the limit of small a and noting that this is all done at constant b x y Since and are just numbers a b a b we can move them to get f x f y f a a x y a b y a x b f f x f y a x a y a y b x b f f x f y a x a y a b y b x b f x f y f a a x a y b b y b x

202 Changing coordinates for partial derivatives Since f x f y f a a x a y b b y b holds for any function f(x,y) provided it is suitably differentiable we can write more generally x y a b a b x a y b y x which is a general way of changing the coordinates for a partial derivative x

203

204 Differential equations Background mathematics review David Miller

205 Differential equations First-order differential equations Background mathematics review David Miller

206 Differential equations in one variable A differential equation involves derivatives of some function E.g., dy dx ay where a is some real number The solution to this equation is y Aexpax where A is an arbitrary constant It is easy to verify solutions but it can be harder to find them

207 First order differential equations dy is a first order differential equation dx ay no derivatives higher than first order The solution y Aexpax has one undetermined constant A and is called a general solution Undetermined constants become fixed using boundary conditions

208 Boundary condition Suppose a ramp has a slope dy 0.4y dx The general solution is y Aexp 0.4x If we also know the boundary condition that at x 0, y 1.5 then, since exp(0)

209 Boundary condition 4 Suppose a ramp has a slope dy 0.4y dx The general solution is y Aexp 0.4x If we also know the boundary condition that at x 0, y 1.5 then, since exp(0) 1 y 1.5exp 0.4x

210 Boundary condition 4 With the same kind of ramp, i.e., dy 0.4y so y Aexp 0.4x dx suppose we know instead that dy at x 0, 0.6 dx dy Since 0.4Aexp0.4x, dx dy then Aexp00.4A dx x

211 Boundary condition 4 With the same kind of ramp, i.e., dy 0.4y so y Aexp 0.4x dx suppose we know instead that dy at x 0, 0.6 dx dy Since 0.4Aexp0.4x, dx dy then Aexp00.4A dx x 0 so A 0.6 /

212 Boundary condition types Boundary conditions that specify the value of a function at some position or boundary are called Dirichlet boundary conditions Boundary conditions that specify the derivative or slope of a function at some position or boundary are called Neumann boundary conditions

213 Imaginary first derivative Note that the equation dy dx iby with b real has the general solution y Aexp ibx A cosbx isin bx Note that neither cosbx nor sinbx is a solution of this equation

214

215 Differential equations Second-order differential equations Background mathematics review David Miller

216 Second-order differential equations A second-order differential equation contains no derivatives higher than second order, e.g., first derivatives second derivatives but no higher derivatives such as third order dy dx d y dx 3 d y 3 dx

217 Simple harmonic oscillator equation One simple and important equation is Any of the functions is a solution If we think of t as time these all represent oscillations with angular frequency or frequency f / hence the name d y dt y exp( it) exp( it) cos( t) sin( t)

218 General solutions Since this is a second order equation the general solution needs two arbitrary constants Possible general solutions include where A, B, C, D, F, and are arbitrary constants Any of these solutions can be rewritten as any other one they are all equivalent sin y Acos t B t exp exp sin y C i t D i t y F t

219 Notation for time derivatives Since derivatives with respect to time arise in many situations there is a shorthand notation using dots above the variable being differentiated with respect to time with the number of dots indicating the order of differentiation da e.g., a and dt a da dt

220 Helmholtz equation With the same mathematics but with the second derivative in position z instead of time t d y d y k zor equivalently kz dz dz which describes oscillations in space rather than time an example of a wave equation Example amplitude of a standing wave on a string 0 S

221

222 Differential equations Partial differential equations Background mathematics review David Miller

223 Classical one-dimensional scalar wave equation The equation zt, 1 zt, z c t relates how strongly curved a function is (from the second spatial derivative) to the second time (or temporal) derivative We could write it more completely as zt, 1 zt, 0 z c t t z but usually we will not bother to do so 0

224 Classical one-dimensional scalar wave equation For this equation zt, 1 zt, 0 z c t we can easily check that any functions of the form f z ct or g z ct are solutions E.g., by the chain rule, at some time t o with s z ct o f zcto df s s df s z ds z ds szct o tt o szct s f z ct df s d f s d f s z z ds ds z s z ct s z ct tt ds o szct o o o o

225 Classical one-dimensional scalar wave equation Similarly, at some position z o with s z ct s f zcto df s df s c t ds t ds zz szoct o szoct s c t t ds ds z ds f z ct df s c d f s c d f s sz ct sz ct zzo sz ct o o o o So, at some position z o and time t o f 1 f d f s 1 c d f s z c t ds c ds as required for any f z ct to be a solution 0

226 Wave equation solutions forward waves The difference between the function f z ct 1 and the function f z ct is that f z ct is shifted to the right by ct t1 f z ct 1 f z ct ct t 1 The wave moves to the right with velocity c

227 Wave equation solutions backward waves The difference between the function gz ct 1 and the function g z ct is that gz ct is shifted to the left by ct t1 g z ct g z ct 1 ct t 1 The wave moves to the left with velocity c

228 Signs and propagation directions It is not the absolute sign of z or ct in f z ct or in gz ct that matters only the relative sign of z and ct f ct z is still a wave going to the right though its shape is the opposite of f z ct f z ct f ct z

229 Monochromatic waves Often we are interested in waves oscillating at one specific (angular) frequency i.e., temporal behavior of the form T t exp( it), exp( it), cos( t), sin( t) or any combination of these Then writing z, t Z ztt, we have t leaving a wave equation for the spatial part dzz kzz0 where k dz c the Helmholtz wave equation

230 Wave equations and linearity The wave equation zt, 1 zt, 0 z c t and therefore also the Helmholtz equation dzz kzz0 dz are linear If 1 z, t and z, t are both solutions so also is any combination a1 z, t b z, t linear superposition

231 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

232 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

233 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

234 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

235 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

236 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

237 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

238 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

239 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

240 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

241 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

242 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

243 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

244 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

245 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

246 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

247 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

248 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

249 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

250 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

251 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k / L and c/ L

252

253 Differential equations Laplacian and gradient operators Background mathematics review David Miller

254 Laplacian operator The Laplacian operator can be defined for ordinary Cartesian coordinates x y z A convenient way to say it is del squared Sometimes it is written instead of though we will not use this notation

255 Wave equation in three dimensions We can propose a three-dimensional wave equation xyzt,,, xyzt,,, xyzt,,, 1 xyzt,,, x y z c t which we can write more compactly as 1 xyzt,,, xyzt,,, 0 c t or just as 1 0 c t With some simplifying assumptions it describes many acoustic and electromagnetic waves 0

256 Gradient operator The gradient of a scalar function f( x, y, z) is f f f f grad f i j k x y z For a function hx, y in two dimensions, such as the height of a hill we could define a two-dimensional gradient h h xyh i j x y giving the magnitude and the vector direction of the largest slope

257 Gradient notation Note that 1) though f has no bold font or other vector notation it is a vector quantity ) we do allow ourselves to put subscripts on it for clarity on how many and what coordinates we are considering as in xy to represent a two-dimensional gradient The symbol can be called del or nabla

258

259 Integral calculus Background mathematics review David Miller

260 Integral calculus Integration in one variable Background mathematics review David Miller

261 Area under a curve Integration can be thought of as the area under a curve Approximately divide area under the curve into rectangles of widthx Add up the areas, f x, i.e., The integral is the limit as we make the rectangles thinner and thinner j j f j x f x x 1 x f 1 3 x1 x x f xdx lim f jx x 0 j x

262 Area under a curve We can extend the idea even for negative values of f x Then we have some negative areas but we still add them up to get the integral The integral of a non-zero function can be zero for equal area magnitudes above and below the axis f x x 1 x x + x1 x f xdx lim f jx x 0 j x

263 Notation for integrals It is common to think of the integral sign and the associated infinitesimal dx as effectively being brackets enclosing everything that has to be integrated (the integrand ) Alternatively we can regard everything with the same variable as the infinitesimal (here, x) as being the integrand x x x x 1 1 f x dx dx f x

264 Limits and integrals An integral with definite lower and upper bounds for the integration is called a definite integral Sometimes formally an indefinite integral without defined limits is useful as in an analytic result but then the result is arbitrary within an additional constant of integration (here C) e.g., e.g., x x 1 f x dx x dx x C

265 Relation between integration and differentiation Integration and differentiation are inverse operations of one another Specifically b df dx f b f a a dx which is known as the fundamental theorem of calculus Sometimes (but not often) the integral is called the antiderivative

266

267 Integral calculus Volume integration Background mathematics review David Miller

268 Volume integration If the volume of the bricks is V then adding up all these small volumes each labeled by some index j would give the total volume In the limit of very small bricks we would get the integral that gives us the total volume V dv lim V V 0 V j

269 Volume integral notation Various notations are used for the volume of the infinitesimal bricks all of which can be confusing using dv is not very specific about integration variables and V is the total volume also dr does not have the right dimensions (not meters cubed) d 3 r can be viewed as having the correct dimensions but, like dr, seems to imply a vector and the volume is actually scalar dv dr d V V V 3 r

270 Volume integral of a quantity We can integrate some quantity that is a function of position r such as the density r (kg/m 3 ) to get the total mass m tot (kg) in the volume m tot V 3 d r r

271 Reducing to one-dimensional integrals To evaluate volume integrals we would like to reduce them to a set of nested one-dimensional integrals This can be done at least for simple volumes such as a cuboid a volume with rectangular faces x L y L z L x x y y z z x x y y x x c L c L c L c L c L c L dv dzdydx z dydx z y dx z y x V L L L L L L V x y z x y x c c c c c c This is not always possible for other volumes

272 Surface integrals We can also perform integrals over surfaces by dividing a surface S into patches of area A and similarly taking the limit of small patches A da lim A A0 S j We can use similar notations da dr d r S S V with similar confusions where r is position on the surface Total surface S of the box Patch of area A on the surface

273

274 Matrices Background mathematics review David Miller

275 Matrices Matrix notation Background mathematics review David Miller

276 Matrix notation A matrix is, first of all, a rectangular array of numbers An M N matrix has M rows (here ) Rows are horizontal N columns (here 3) Columns are vertical The array is enclosed in square brackets Aˆ This is a 3 rectangular matrix

277 Symbol for a matrix As a symbol for a matrix we could just use a capital letter, like A Here, we need to distinguish matrices and other linear operators from numbers and simple variables so we put a hat over a symbol  representing a matrix which distinguishes a matrix symbol when we write it by hand Aˆ

278 Rectangular and square matrices Because all matrices are, by definition, rectangular when we say a matrix is rectangular we almost always mean it is not a square matrix one with equal numbers of rows and columns Bˆ i 0.5i 1.5 This is a square matrix

279 Rectangular and square matrices The numbers or elements in a matrix can be real, imaginary, or complex The elements are indexed in rowcolumn order B 1 is the element (value -0.5i) in the first row and second column We often use the same letter, here B, for the matrix and for its elements or the lower case version, e.g., b 1 Bˆ i 0.5i 1.5 This is a square matrix

280 Diagonal elements The leading diagonal of a matrix or just the diagonal is the diagonal from top left to bottom right Elements on the diagonal here those with value 1.5 are called diagonal elements Elements not on the diagonal here those with value 0.5i and -0.5i are called off-diagonal elements Bˆ i 0.5i 1.5 This is a square matrix

281 Vectors In the matrix algebra version of vectors which are matrices of size 1 in one of their directions we must specify whether a vector is a row vector a matrix with one row or a column vector a matrix with one column c 4,,5,7 3i 5i d 4 i 76i

282 Transpose An important manipulation for matrices and vectors is the transpose denoted by a superscript T a reflection about a diagonal line from top left to bottom right for a matrix Algebraically ˆT A Aˆ mn nm Aˆ ˆ T A

283 Transpose An important manipulation for matrices and vectors is the transpose denoted by a superscript T a reflection about a diagonal line from top left to bottom right for a matrix Algebraically ˆT B Bˆ mn nm Bˆ Bˆ T i 0.5i i 0.5i 1.5

284 Transpose An important manipulation for matrices and vectors is the transpose denoted by a superscript T a reflection about a diagonal line from top left to bottom right for a matrix or at 45 for a vector c T c 4,,5,

285 Transpose An important manipulation for matrices and vectors is the transpose denoted by a superscript T a reflection about a diagonal line from top left to bottom right for a matrix or at 45 for a vector d 3i 5i 4 i 76i T d [ 3i 5i 4 i 7 6 i]

286 Hermitian transpose or adjoint Another common manipulation is the Hermitian adjoint, Hermitian transpose, or conjugate transpose denoted by a superscript pronounced dagger a reflection about a diagonal line from top left to bottom right for a matrix or at 45 for a vector and taking the complex conjugate of all the elements Dˆ i 0.3i 1.5 ˆ i D 0.5i 1.5 Dˆ mn Dˆnm

287 Hermitian transpose or adjoint 3i Another common manipulation is the 5i d Hermitian adjoint, Hermitian 4 i transpose, or conjugate transpose 76i denoted by a superscript pronounced dagger d 3i 5i 4i 76i a reflection about a diagonal line from top left to bottom right for a matrix or at 45 for a vector and taking the complex conjugate of all the elements

288 Hermitian matrix A matrix is said to be Hermitian if it is equal to its own Hermitian adjoint i.e., Bˆ Bˆ or, element by element B ˆ B ˆ nm nm Bˆ i 0.5i 1.5 ˆ i B 0.5i 1.5 Bˆ

289

290 Matrices Matrix algebra Background mathematics review David Miller

291 Adding and subtracting matrices If two matrices are the same size i.e., the same numbers of rows and columns we can add or subtract them by adding or subtracting the individual matrix elements one by one Fˆ 1 i 1 3i Kˆ Fˆ Gˆ Gˆ 5 4i 6 7 8i

292 Adding and subtracting matrices If two matrices are the same size i.e., the same numbers of rows and columns we can add or subtract them by adding or subtracting the individual matrix elements one by one Fˆ 1 i 1 3i Gˆ 5 4i 6 7 8i Kˆ Fˆ Gˆ 15 i4i 6 173i8i

293 Adding and subtracting matrices If two matrices are the same size i.e., the same numbers of rows and columns we can add or subtract them by adding or subtracting the individual matrix elements one by one Fˆ 1 i 1 3i Gˆ 5 4i 6 7 8i Kˆ Fˆ Gˆ 15 i4i i 8i 6 5i 4 85i

294 Multiplying a vector by a matrix Suppose we want to multiply a column vector by a matrix The number of rows in the vector must match the number of columns in the matrix This is generally true for matrix-matrix multiplication The number of rows in the matrix on the right must match the number of columns in the matrix on the left matrix vector 7 8 9

295 Multiplying a vector by a matrix First we put the vector sideways on top of the matrix then multiply element by element and add to get the first element of the resulting vector Move down and repeat for the next row matrix vector

296 Multiplying a vector by a matrix First we put the vector sideways on top of the matrix then multiply element by element and add to get the first element of the resulting vector Move down and repeat for the next row We can also write this multiplication with a sum over the repeated index matrix = vector d  c d A c m mn n n 7 8 9

297 Multiplying a matrix by a matrix To multiply a matrix by a matrix repeat this operation for each column of the matrix on the right working from left to right Write down the resulting columns in the resulting matrix also working from left to right Summation notation sums over the repeated index ˆR ˆB Â Rmp Bmn Anp n

298 Vector vector products An inner product of a row and a column vector collapses two vectors to a number analogous to geometrical vector dot product An outer product of a column and a row vector generates a square matrix f c d f cndn n ˆF d c F d c mp m p

299 Matrix algebra properties Matrix algebra, like normal algebra is associative and has distributive properties but matrix multiplication is not in general commutative as is easily proved by example CB ˆ ˆ A ˆ Cˆ BA ˆˆ ˆ ˆ Aˆ B Cˆ AB ˆ AC ˆ ˆ BA ˆˆ AB ˆ ˆ in general

300 Multiplying a matrix by a number Multiplying a matrix by a number means we multiply every element of the matrix by that number Also, we can take out a common factor from every element multiplying the matrix by that factor Such results are easily proved in summation notation e.g., for matrix vector multiplication where B mn = A mn d A c A c m mn n mn n n n n A c B c mn n mn n n

301 Multiplying a matrix by a number Since number multiplication is commutative we can move simple factors around arbitrarily in matrix products e.g., for a number and a vector c ABc ˆ ˆ AˆBc ˆ AB ˆ ˆc This result is also easily proved using summation notation

302 Hermitian adjoint of a product The Hermitian adjoint of a product is the reversed product of the Hermitian adjoints ˆ ˆ ˆ ˆ AB B A We can prove this using summation notation Suppose Rˆ AB ˆ ˆ so that Rmp AmnBnp and so ˆ ˆ ˆ * AB ( R ) pm Rmp ( AmnBnp ) A n n mnbnp pm ( Aˆ ) ( Bˆ ) ( Bˆ ) ( Aˆ ) Bˆ Aˆ n nm pn n pn nm pm n

303 Inverse of a matrix For ordinary algebra, the reciprocal or inverse of a number or variable x is which has the obvious property 1 For a matrix, if it has an inverse it has the property where Î is called the identity matrix which is the diagonal matrix with 1 for all diagonal elements and zeros for all other elements 1/ x 1 x x  1 or x x Aˆ Aˆ Iˆ x 1 1

304 Identity matrix For example, the 3x3 identity matrix is The identity matrix in a given multiplication has to be the right size so we do not typically bother to state the size of the identity matrix For any matrix  we can write AI ˆˆ IA ˆˆ Aˆ Like the number 1 in ordinary algebra the identity matrix is almost trivial but is very important Iˆ

305

306 Matrix eigenequations Background mathematics review David Miller

307 Matrix eigenequations Linear equations and matrices Background mathematics review David Miller

308 Linear equations and matrices Suppose we have equations for two straight lines Note, if you are used to the form we can rewrite this as so these are equivalent We can rewrite these equations as the one matrix equation or, with b 1 instead of x and b instead of y in summation form A x A y c A x A y c 1 y mx c y ( A / A ) x( c / A ) ˆ x A y A11 A1 x c1 A A y c 1 n1 A b c mn n m

309 Linear equation solutions With the linear equations in matrix ˆ x A11 A1 x c1 form A y A1 A y c we can formally solve them if we know the inverse  1 1 ˆ multiplying by  1ˆ ˆ c x 1 1 A A A y c Since ˆ1 A Aˆ Iˆ we have the solution x the intersection point of ˆ c 1 1 ( x, y) A the lines y c Solving linear equations and inverting a matrix are the same operation

310 Determinant If the determinant of a matrix is not zero then the matrix has an inverse and if a matrix has an inverse, the determinant of the matrix is not zero A nonzero determinant is a necessary and sufficient condition for a matrix to be invertible

311 Determinant of a matrix The determinant of a matrix  is written in one of two notations There are two complete formulas for calculating it Leibniz s formula Laplace s formula and many numerical techniques to calculate it we will not give these general formulas or methods here det Aˆ A A A N A A A 1 N A A A N1 N NN

312 Determinant of a x matrix For a x matrix det A A A A A A A A A ˆ we add the product on the leading diagonal and subtract the product on the other diagonal

313 Determinant of a 3x3 matrix For a 3x3 matrix, we have det A A A 11 A11 A1 A A11 A 13 A33 A3A3 Aˆ A A A A A A A A A A A A A A A A A A A A A A A A A A A 3 33 A A A A 1 3 A A A A 11 1 A 1 A 31 3 A A A

314 General form of determinant If we multiply out the 3x3 determinant expression ˆ det A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A we see that each term, e.g., A1 A3A31 contains a different element from each row and the elements in each term are never from the same column but we always have one element from each row and each column in each term

315 General form of the determinant We see that this form ˆ det A A A A A A A A A A A A A A A A A A A contains every possible term with one element from each row all from different columns this is a general property of determinants

316 General form of the determinant To understand how to construct a determinant it only remains to find the sign of the terms To do so count the number of adjacent row (or column) swaps required to get all the elements in the term onto the leading diagonal if that number is even, the sign is + if that number is odd, the sign is -

317 Sign of determinant terms For the term A A A 11 A A A 1 A A A 1 13 A11A3A A A A we have to perform 1 row swap 1 is an odd number so the sign of this term in the determinant is negative 11 A A A A A A 3

318 Sign of determinant terms For the term A1 A3A31 A11 A1 A13 A11 A1 A13 A31 A A 1 31 A A A A A A we have to perform row swaps is an even number so the sign of this term in the determinant is positive This is actually Leibniz s determinant formula A A A A 3 A A A A A A A A 3

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320 Matrix eigenequations Eigenvalues and eigenvectors Background mathematics review David Miller

321 Matrix eigenequation An equation of the form Âd d where d is a vector, is a number, and  is a square matrix is called an eigenequation with eigenvalue and eigenvector d If there are solutions they may only exist for specific values of

322 Solving an eigenequation We can rewrite Âd d as Ad ˆ Id ˆ where we have introduced the identity matrix Î which we can always do because Îd d So A ˆ I ˆ d 0 (strictly, the 0 here is a vector with elements 0) so, writing Bˆ Aˆ Iˆ we have Bd ˆ 0

323 Solving an eigenequation Now, for Bd ˆ 0 to have any solutions for any non-zero vector d the matrix ˆB cannot have an inverse if it did have an inverse ˆB 1 ˆ1ˆ ˆ ˆ1 B Bd Id d B 0 but any (finite) matrix multiplying a zero vector must give a zero vector so there is no non-zero solution d Hence, by reductio ad absurdum, ˆB has no inverse

324 Solving an eigenequation The fact that Bˆ Aˆ Iˆ has no inverse means from the properties of the determinant This equation will allow us to construct a secular equation whose solutions will give the eigenvalues From those we will deduce the corresponding eigenvectors d det Aˆ Iˆ 0

325 Solving an eigenequation Suppose we want to find the eigenvalues and eigenvectors if they exist of the matrix ˆ i A 0.5i 1.5 So we write the determinant condition for finding eigenvalues ˆ ˆ i 1 0 det AI det 0 0.5i

326 Solving an eigenequation Now i i 0 det det 0.5i i i det 0.5i 1.5 So our secular equation becomes, from det Aˆ Iˆ 0 i i 1.5 (0.5 ) i.e., 3 0

327 Solving an eigenequation Solving this quadratic equation 3 0 gives roots 1 1 and Now that we know the eigenvalues we substitute them back into the eigenequation and deduce the corresponding eigenvectors

328 Solving an eigenequation Our eigenequation Âd d is, explicitly i d1 d1 0.5i 1.5 d d where now, for a given eigenvalue we are trying to find d 1 and d so we know the corresponding eigenvector Rewriting gives i d i 1.5 d 0

329 Solving an eigenequation Now evaluating i d i 1.5 d 0 for a specific eigenvalue say, the first one, 1 1 gives id i 0.5 d 0 or, as linear equations 0.5d 0.5id id 0.5d 0 1

330 Solving an eigenequation From either one of these equations 0.5d10.5id 0 0.5id1 0.5d 0 we can now deduce the eigenvector Either equation gives us d id1 We are free to choose one of the elements say, choose d1 1 which gives the eigenvector v 1 v Using and similar mathematics gives the other eigenvector v v 1 1 i 1 i

331 Solving eigenequations For larger matrices with eigensolutions e.g., N N we have correspondingly higher order polynomial secular equations which can have N eigenvalues and eigenvectors e.g., a 33 matrix can have 3 eigenvalues and eigenvectors Note eigenvectors can be multiplied by any constant and still be eigenvectors

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