Mathematical Techniques 1 (SPA4121) Module Overview

Transcription

1 Mathematical Techniques (SPA4) Moule Overview Dr. Jeanne Wilson - September 08 Overview The following etails summarise the course. N.B. You are expecte to atten lectures an tutorials. Attenance will be monitore. Lecturer: Dr. Jeanne Wilson, Room 45, G O Jones Builing (j.r.wilson@qmul.ac.uk). Office Hours: Currently, Tuesay 3pm an Friay 3pm If you cannot fin me, please to arrange a meeting. I o not work Wenesays an usually work from home on Monays. Pre-requisites: Familiarity with A-Level Mathematics will be assume. Assessment: The course total mark will be forme from 0% in-course assessment (break-own: 3 marke homework assessments an two in-class tests) an 80% from the final exam next May/June lasting hr 30min. The in-course assessments are esigne to help you gauge your progress an unerstaning throughout the course an to provie formative feeback. Exam rules are available from the epartmental office, an more etails are available in the Stuent Hanbook. Only nonprogrammable calculators are permitte in the exam an tests. Lectures: There will be 33 lectures in total. You are expecte to atten lectures in person - attenance will be monitore. Lectures will be recore on Q-review an will closely follow the lecture notes poste to this page. However, teaching will involve extensive use of blackboars where material is not always capture with sufficient resolution in Q-review. Ancillary teaching: There will be exercise classes, hrs each. For the exercise classes, stuents are ivie into four groups (see MySIS) an each group will be assigne to one class per week, on either Tuesay or Thursay afternoons ( 4pm). Note these start on Tuesay 5th September an attenance will be monitore. You are encourage to work in small groups uring the tutorials. Homework ealines: The homework ealines will be shown on QMplus - please see the Assesse Coursework section. For each assessment you shoul ownloa your personalise coversheet which shoul be securely attache to your work. It is goo practise to write your name an stuent ID on each page that you submit too. The nominal ealines, which may be ajuste if require, are at 4pm on Friay of week 3,8 an

2 We will eneavour to return marke scripts an worke solutions to you within weeks of submission. In-Class tests: The in-class tests will both be 45 minutes long an carrie out in the assigne teaching rooms, but uner exam conitions (no notes, no speaking, only permitte non-programable calculators) uring weeks 6 an. Self Stuy: In aition to the marke assessments, for each topic, a selection of example exam questions will be poste to QMplus for you to work through in your own time. Solutions will be provie weeks later. You shoul aim to spen about 0 hours per week on each 5 creit moule, incluing lectures an tutorial classes (ie. For MT that is 5 hours per week rereaing lecture notes, text books an working through examples). Synopsis: This course covers various techniques of mathematics, mostly calculus, require in the stuy of the physical sciences: vectors an scalars, vector components, vector aition an multiplication, ifferentiation, partial ifferentiation, integration, multiple integration, centrois an moments, series, polar coorinates, complex numbers, an an introuction to Fourier methos. The course structure inclues both lectures an self-pace learning, with assessment by course work an the summer exam. Reaing List: All the require material will be covere in the lectures, but you may fin one of the following textbooks useful while taking this course. K. F. Riley, M. P. Hobson, an S. J. Bence, Mathematical Methos for Physicists an Engineers, Cambrige (006) 3 r Eition. ISBN K. A. Strou, Engineering Mathematics, Palgrave MacMillan (00) 6 th Eition. ISBN Course web-site: The moule web page is on QMPlus: from the front page, enter the moule coe SPA4 into the Search box. Or, the irect URL for this moule is This QMPlus page provies etails of the course (incluing these notes), scanne lecture notes, homework questions, tutorial questions, past papers, revision notes an other material. NOTE: If an equation is given in the lectures, then you shoul assume that you will be expecte to know it, even if it is not inclue in the following stuy ai! This inclues stanar integrals an erivatives. The remainer of this note is a short stuy ai.

3 Notation an Trigonometric formulas This section serves as a reminer of some of the notation an formulae use throughout this course. Table : The list of trigonometric functions, their inverse, an their reciprocal functions. Name Trig. Function Inverse Trig. Function /(Trig Function) sine sin(x) arcsin(x) or sin (x) cosec(x) cosine cos(x) arccos(x) or cos (x) sec(x) tangent tan(x) arctan(x) or tan (x) cot(x) cosecant cosec(x) arccosec(x) or cosec (x) sin(x) secant sec(x) arcsec(x) or sec (x) cos(x) cotangent cot(x) arccot(x) or cot (x) tan(x) Remember that sin x means (sin x) an sin 3 x means (sin x) 3 etc for positive powers; BUT, sin (x) means arcsin(x), not (sin x) = cosec(x). This is somewhat illogical, but is stanar mathematical notation which usually minimises the number of brackets to write. We may also use exp(x) which means e x ; this is use to avoi long fily superscripts if x is replace by a long expression. It s also useful to remember the aition formulae for sin & cos: sin(a + B) = sin A cos B + cos A sin B cos(a + B) = cos A cos B sin A sin B If you just memorise these two, you can erive many others from them: replacing B with ( B), an the fact that sin( B) = sin B an cos( B) = cos B, we get sin(a B) = sin A cos B cos A sin B cos(a B) = cos A cos B + sin A sin B Aing pairs of the above an rearranging, we get the prouct rules sin A sin B = [cos(a B) cos(a + B)] cos A cos B = [cos(a B) + cos(a + B)] sin A cos B = [sin(a B) + sin(a + B)]

4 Special cases (set B = A in appropriate cases above, an rearrange) sin A = sin A cos A cos A = cos A sin A = sin A = cos A sin A = [ cos A] cos A = [ + cos A] These above are often useful when neeing to integrate functions like sin x or sin x cos 3x. It s also possible to iterate these, e.g. sin 3A = sin(a + A) = 3 sin A 4 sin 3 A etc. To check you ve got the minus signs right, it can be useful to put in A = 0 or B = 0 or π/ in above, then check it makes sense. 3 Vectors an Scalars A scalar can be efine by a single number with appropriate units. (eg. spee) A vector is efine completely by its magnitue (with units) an irection. (eg. velocity) Vector components: v = (v x, v y, v z ) = aî + bĵ + cˆk where î, ĵ, ˆk are unit vectors in the x,y,z irections respectively. The magnitue, or length, of vector v can be written v = v = r = (x + y + z ). Vector aition is commutative an associative. C = A + B = ((a x + b x ), (a y + b y ), (a z + b z )) Any single vector OC can be replace by a sum of any number of vectors so long as they form a chain in the vector iagram. To multiply (or ivie) a vector by a scalar quantity, each component of is scale by that quantity. λ v = (λx, λy, λz) The scalar prouct of two vectors gives a scalar value that tells us how alike two vectors are. v. u = (v x, v y, v z ).(u x, u y, u z ) = v x u x + v y u y + v z u z = x,y,z v i u i v. u = v u cos θ Scalar proucts are commutative, associative an istributive over aition. The angle between two vectors can be obtaine from the scalar prouct. The vector prouct(or cross prouct) correspons to a new vector that is perpenicular to both the original vectors an therefore normal to the plane containing them. The three vectors u, v an u v form a right-hane set such that the irection of the prouct can be obtaine from the right han rule. The magnitue can be obtaine from:

5 w = u v = u v sin θ Vector proucts o not commute, are not associative an are not istributive over aition. The cross prouct of two ientical vectors is zero. Applie to unit vectors we know: We can efine the cross-prouct as î î = ĵ ĵ = ˆk ˆk = 0 î ĵ = ˆk = ĵ î ĵ ˆk = î = ˆk ĵ ˆk î = ĵ = î ˆk u v = (u y v z u z v y )î + (u z v x u x v z )ĵ + (u x v y u y v x )ˆk which can also be expresse as the eterminant of the matrix (see later for matrices): u v = (u x, u y, u z ) (v x, v y, v z ) î ĵ ˆk = u x u y u z v x v y v z Direction cosines efine the angle that a vector makes with the axes of reference. For a vector OP = a.î + b.ĵ + c.ˆk with magnitue r = (a + b + c ) then the irection cosines are: Direction cosines obey the relation: 4 Matrices l = cos α = a r m = cos β = b r n = cos γ = c r cos α + cos β + cos γ = l + m + n = A matrix (plural matrices) is a set of numbers (or elements) arranges in rows an columns to form a rectangular array. A matrix having m rows an n columns is calle an m n matrix an is inicate by writing the array within brackets for example: We will come across an 3 3 matrices in this course. We can use suffixes, eg. a ij, to refer to the elements in a matrix: ( ) a b b a a 3 B =, A = a b b a a 3 a 3 a 3 a 33

6 The eterminant of a matrix is a single number that epens on the elements of the matrix. Determinants are only efine for square matrices. The eterminant of the matrix, B is efine as B = b b b b = b b b b whilst the eterminant of a 3 3 matrix A involves combining three matrix eterminants. This will be covere properly in MT. Any row or column can be use as the multipliers but here we simply give the prescription for using the first row (a, a, a 3 ): A = a a a 3 a a a 3 a 3 a 3 a 33 5 Differentiation = a (a a 33 a 3 a 3 ) a (a a 33 a 3 a 3 ) + a 3 (a a 3 a a 3 ) Notation: The following notation is use throughout the course to inicate the erivative of some function y with respect to some other variable x: y x, y. an the secon erivative is x ( ) y x = y x, = y. Similarly for the notation use to enote higher orer erivatives. Table lists a number of useful stanar erivatives. For y = f(x), where f(x) is a complicate function that can be simplifie by a substitution u = g(x), so that y = h(u) is easier to ifferentiate, one can use the chain rule: y x = y u u x. () For y = f(g(h(x))), where f(x) is a complicate function that can be simplifie as above. One can use the chain rule extension: The erivative of a prouct of two functions u an v of x is, The erivative of a quotient of two functions u an v of x is y x = y g h g h x. () u (uv) = v x x + uv x. (3) ( u ) = v u x u v x x v v. (4)

7 Table : Table of stanar erivatives. y = f(x) x n e x e kx a x y x nx n e x ke kx a x ln a ln x x log a x x ln a sin x cos x sin kx k cos kx (etc) cos x sin x tan x sec x cot x cosec x sec x sec x tan x cosec x cosec x cot x sinh x cosh x cosh x sinh x tanh x / cosh x sin x x cos x x tan x +x sinh x x + cosh x x tanh x x The raius of curvature of a function y is given by R = [ + ( ) ] 3/ y x. (5) y x For a function of two variables z = f(x, y), the first partial erivatives are z x, z y (6) an the secon partial erivatives of the same function are where z x, z y x, z y, z y x = z x y, (7) z x y. (8)

8 Remember that the curly signs remin us that we are ifferentiating with respect to one variable while keeping the other one constant. The total ifferential, δz, of a function z = f(x, y) is δz = z z δx + δy. (9) x y Rates of change. If both x an y are functions of time, t, then the total erivative enables us to calculate the rate of change of z with respect to t z t = z x x t + z y y t. (0) Change of variables. For some function z = f(x, y), where both x an y are functions of two other variables u an v, we can calculate z u = z x x u + z y y u, () z v = z x x v + z y y v, () in analogy with the transformation of variables for the rate of change (see above). 6 Integration The following integration rules are useful to remember, along with the other rules that are covere in lectures, an Table 3 lists a number of stanar integral results. When integrating f (x)f(x) x (3) we can recognize the solution as f (x)/ + C. When integrating we can recognize the solution as ln f(x) + C. f (x) x (4) f(x) When integrating a prouct of two functions by parts, recall u v x = uv v u x (5) x x which is erive from rearranging the prouct rule for ifferentiation. Integrals of the form f(x) x (6) g(x)

9 Table 3: Table of stanar integrals. Derivative Integral x (xn ) = nx n x (ex ) = e x x (ekx ) = ke kx x (ax ) = a x ln a x (ln x) = x x (log a x) = x ln a x (sin x) = cos x x n x = xn+ n+ + C [for n ] e x x = e x + C e kx x = ekx k + C a x x = ax ln a + C x x (sin kx) = k cos kx sin kx x = k x (cos x) = sin x x (tan x) = sec x x (cot x) = cosec x x x x x (sec x) = sec x tan x (sinh x) = cosh x (cosh x) = sinh x (tanh x) = cosh x x (sin x) = x x (cos x) = x x (tan x) = x (sinh x) = +x x (cosh x) = x x (tanh x) = x x = ln x + C x ln a x = log a x + C sin x x = cos x + C cos kx + C (etc) cos x x = sin x + C sec x x = tan x + C cosec x x = cot x + C sec x tan x x = sec x + C sinh x x = cosh x + C cosh x x = sinh x + C +x cosh x x = tanh x + C x x = sin x + C x x = cos x + C x = tan x + C +x x = +x sinh x + C x = x cosh x + C x = tanh x + C x where the quotient f(x) g(x) can be separate into partial fractions, can be re-written in terms of h (x) h(x) x + k (x) x +... (7) k(x) where the solution is of the form ln h(x) + ln k(x) C. It is useful to remember the following, when trying to express a quotient in terms of partial fractions: Factors of (ax + b) result in partial fractions of the form A ax+b. Factors of (ax + b) result in partial fractions of the form A (ax+b) + Factors of (ax + b) 3 result in partial fractions of the form A (ax+b) + B (ax+b). Factors of ax + bx + c result in partial fractions of the form Ax+B ax +bx+c. B (ax+b) + C (ax+b) 3.

10 A few other general tips for integrating: For terms like sin x an cos x, use the ouble-angle formulae. For higher powers, try things like sin 3 x = ( cos x) sin x, or try the ouble-angle formula more than once. For terms like x sin x an x sin x an xe x, try integration by parts. For terms like / ( x ) an /( + x ), try the inverse-trig results above. For terms like /(ax + bx + c), try partial fractions as above. Unlike ifferentation where there is a well-efine proceure, there is no guarantee rule for complicate integrals; the above are just a hint for what to try first. 6. Applications of integration A efinite integral of a function f(x), represents the area boune by the x axis an the curve f(x) between the two limits specifie in the integral (remember that area below the x axis counts as negative; an swapping the two limits changes the sign of the answer). This is one of many applications of integration. Remember that an integral is not always an area: e.g. in a -D integral it can be graphically shown as a volume. In general, an integral is the limit of an infinite sum when some object is slice into a large number of tiny pieces, then summe. Some useful formulae that are use in this course are liste below. The arc-length along a curve, s is just x + y assuming that one works in the limit where x tens to zero. Integrating this we obtain x ( ) y s = + x (8) x x θ (x ) ( ) y s = + θ (9) θ θ to eal with integrals over x, or some parametric variable θ, respectively. θ The concept of the moment of inertia of a mass m is use in this course. If a mass element m is rotating about some axis, at a istance r from that axis, then the moment of inertia I of that mass is efie as I = r m. If we integrate over both sies, we obtain I = m m r m (0) an if we know the ensity (mass per unit area) of the mass, we can perform this integral in terms of area, or r when we nee to. We can calculate the centre of gravity (x, y, z) of an object by noting that x m = x m () (an similarly for y an z). As x is a constant, an in this course, we only consier objects of uniform ensity, we can simplify this equation by writing it as x A x = () A

11 for plane objects (in -imensions), or alternatively x = x V V (3) for objects that exten into 3-imensions (with similar equations for y an z). Note that the factors of ensity rop out as we assume that the object is of uniform ensity. If we revolve a lamina (a planar shape) about the x axis, then the volume element of this object is given by V = πy x. So the centroi position (x, y) is given by the equations y = 0, by symmetry (4) xy x x = y (5) x If we revolve a lamina about the y axis, then the volume element of this object is given by V = πxyx. So the centroi position (x, y) is given by the equations y = xy x xy x (6) x = 0, by symmetry (7) 7 Series The sum of the first n terms of an arithmetic series is n i=0 a + i. The sum of the first n terms of a geometric series is n i=0 ari. The Taylor Series expansion f(x) about a point a is f(x) = f(a) + f (a)(x a) + f (a)! (x a) + f (a) (x a) (8) 3! where f(a) is the function f(x) evaluate at x = a, likewise for the erivatives of f. The Maclaurin Series expansion for a function f(x) is f(x) = f(0) + f (0)x + f (0)! x + f (0) x (9) 3! where f(0) is the function f(x) evaluate at x = 0, likewise for the erivatives of f. The Binomial Series expansion ( + x) n is ( + x) n = + nx + n(n ) x +! n(n )(n ) x (30) 3! The series converges for any value of n if x < ; or if n is a positive integer it becomes a finite series (with exactly n + terms) an therefore converges for any value of x. If neither of these applies (i.e. x > an n is not a positive integer) the series iverges. More generally if we have (a + x) n, it can be convenient to rewrite as a n [ + (x/a)] n, then use the above with y = x/a.

12 A few useful tests for convergence of a series covere in the lectures are liste below: lim n U n = 0, If this test is satisfie, then the series may or may not converge (test inconclusive). However, if this test is not satisfie, then the series efinitely oes not converge (it may iverge or oscillate). Comparison test: Test a series against one known to converge. If (for all large numbers n) the n th term in a series is smaller than the n th term in a series known to converge, then the series converges. D Alembert s ratio test: For a series U + U U n +..., look at the limit: U n+ R n = lim. (3) n U n If R n <, the series converges; if R n > the series iverges; if R n =, the test is inconclusive. L Hôpital s rule: if lim f(x) an lim g(x) are both zero or both infinite, then f(x) lim x g(x) = lim f (x) x g (x). (3) (if the latter limit exists). This is often useful if one obtains ineterminate results of 0 0 or when taking limits of f, g. 8 Complex Numbers Here efine i =. If z = a + ib, then Re(z) = a, an Im(z) = b. z = a + ib = re iθ = r(cos θ + i sin θ), where r = z = a + b, an tan θ = b a calle the moulus, an θ is calle the argument of z. ; here z is The prouct of two complex numbers z an z is z z = r r e i(θ +θ ). The quotient of two complex numbers z an z is z z = r r e i(θ θ ). The nth power is z n = r n e inθ. For nth roots, n z = n re i(θ+mπ)/n, where m = 0,,,... n gives the set of istinct solutions with an argument between zero an π. These roots form a regular n gon with the centroi at (0, 0). (Other values of m just give repeats of these with extra π s in the argument). Also note the following useful relations between complex exponentials an trigonometric functions sin(x) = eix e ix, i (33) cos(x) = eix + e ix, (34)

13 9 Fourier Series A perioic function f(x) can be written as a Fourier series of the form y(t) = A 0 + n= [ ( ) ( )] πn πn A n cos T t + B n sin T t (35) where the coefficients are A n = T B n = T A 0 t t t t t ( πn y(t) cos T t y(t) sin ( πn T t ) t, (36) ) t, (37) = y(t) t, (38) T t t = y(t) t, (39) t t t = y(t), (40) Note the funny-looking factor of in A 0 is arbitrary but is put in so the A n equation still works for n = 0. The Fourier series can be re-expresse in terms of the angular frequency ω or frequency f by noting that ω = π T = πf. 0 Fourier Integrals We can transform between space of a function y(x) an the reciprocal space (Fourier space) Y (u), where x = /u using Y (u) = y(x) = y(x) e iπux x, Y (u) e iπux u. (4) The Dirac Delta Function is given as δ(u u 0 ) = 0, for u u 0 (4) =, for u = u 0 (43) δ(u u 0 ) u =. (44)