Mathematical Techniques 1 (SPA4121)
Module Overview
Dr. Will Sutherland — 19 Sept 2016
1
Overview
The following details summarise the course. N.B. You are expected to attend lectures and tutorials.
Attendance will be monitored.
Lecturer: Dr. Will Sutherland, Room 511, Physics Building ([email protected]).
Office Hours: Currently, Tues 3:40–4:40 and Fri 11:30–12:30 .
See my personal webpage for updates.
Pre-requisites: Familiarity with A-Level Mathematics will be assumed.
Assessment: Assessment will be via ten weekly homeworks (adding up to 20% of your final mark),
and an exam next May/June lasting 2hr 30min (80% of your final mark). Exam rules are available
from the departmental secretaries, and more details are available in the Student Handbook.
Lectures: 33
Ancillary teaching: There will be 10 exercise classes, 2 hrs each. For the exercise classes, students
are divided into four groups (see MySIS) and each group will be assigned to one class per week,
on one of Monday, Tues, Thurs or Fri afternoons. Note these start on Thursday 29th September.
Homework deadlines: Homework question sheets will be handed out at each Tuesday lecture,
and also posted on the QMPlus page. You must hand in solutions on the Thursday (9 days later).
There will be ten homeworks, with the hand-in deadlines in weeks 2–6 and 8–12 inclusive.
Synopsis: This course covers various techniques of mathematics, mostly calculus, required in the
study of the physical sciences: differentiation, partial differentiation, integration, multiple integration, centroids and moments, series, polar coordinates, complex numbers, and an introduction to
Fourier methods. The course structure includes both lectures and self-paced programmed learning,
with assessment by course work and the summer exam.
1
Reading List: All the required material will be covered in the lectures, but you may find one of
the following textbooks useful while taking this course.
• K. F. Riley, M. P. Hobson, and S. J. Bence, “Mathematical Methods for Physicists and
Engineers”, Cambridge (2006) 3rd Edition. ISBN 0-521-67971-0.
• K. A. Stroud, “Engineering Mathematics”, Palgrave MacMillan (2001) 6th Edition. ISBN
978-1-4039-4246-3.
Course web-site:
The module web page is on QMPlus: from the front page, enter the module code SPA4121 into
the Search box. Or, the direct URL for this module is
http://qmplus.qmul.ac.uk/course/view.php?id=3070
This QMPlus page provides details of the course (including these notes), scanned lecture notes,
homework questions, tutorial questions, past papers, revision notes and other material.
NOTE: If an equation given in the lectures, then you should assume that you will be expected to
know it, even if it is not included in the following study aid ! This includes standard integrals and
derivatives.
The remainder of this note is a short study aid.
2
Notation and Trigonometric formulas
This section serves as a reminder of some of the notation and formulae used throughout this course.
Table 1: The list of trigonometric functions, their inverse, and their reciprocal functions.
Name
sine
cosine
tangent
cosecant
secant
cotangent
Trig. Function
sin(x)
cos(x)
tan(x)
cosec(x)
sec(x)
cot(x)
Inverse Trig.
arcsin(x) or
arccos(x) or
arctan(x) or
arccosec(x) or
arcsec(x) or
arccot(x) or
Function
sin−1 (x)
cos−1 (x)
tan−1 (x)
cosec−1 (x)
sec−1 (x)
cot−1 (x)
1/(Trig Function)
cosec(x)
sec(x)
cot(x)
sin(x)
cos(x)
tan(x)
Remember that sin2 x means (sin x)2 and sin3 x means (sin x)3 etc for positive powers; BUT,
sin−1 (x) means arcsin(x), not (sin x)−1 = cosec(x).
This is somewhat “illogical”, but is standard mathematical notation which usually minimises the
number of brackets to write.
We may also use exp(x) which means ex ; this is used to avoid long fiddly superscripts if x is replaced
by a long expression.
It’s also useful to remember the addition 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 derive many others from them: replacing B with (−B),
and the fact that sin(−B) = − sin B and 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
Adding pairs of the above and rearranging, we get the “product rules”
sin A sin B =
cos A cos B =
sin A cos B =
1
[cos(A − B) − cos(A + B)]
2
1
[cos(A − B) + cos(A + B)]
2
1
[sin(A − B) + sin(A + B)]
2
Special cases (set B = A in appropriate cases above, and rearrange)
sin 2A = 2 sin A cos A
cos 2A = cos2 A − sin2 A
= 1 − 2 sin2 A = 2 cos2 A − 1
1
sin2 A =
[1 − cos 2A]
2
1
cos2 A =
[1 + cos 2A]
2
These above are often useful when needing to integrate functions like sin2 x or sin 2x cos 3x. It’s
also possible to iterate these, e.g. sin 3A = sin(2A + A) = 3 sin A − 4 sin3 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/2 in
above, then check it makes sense.
3
Differentiation
Notation: The following notation is used throughout the course to indicate the derivative of some
function y with respect to some other variable x:
dy
,
dx
y0.
and the second derivative is
d
dx
dy
dx
d2 y
,
dx2
= y 00 .
=
Similarly for the notation used to denote higher order derivatives. Table 2 lists a number of useful
standard derivatives.
• For y = f (x), where f (x) is a complicated function that can be simplified by a substitution
u = g(x), so that y = h(u) is easier to differentiate, one can use the chain rule:
dy
dy du
=
.
dx
du dx
(1)
• For y = f (g(h(x))), where f (x) is a complicated function that can be simplified as above.
One can use the chain rule extension:
dy
dy dg dh
=
.
dx
dg dh dx
(2)
• The derivative of a product of two functions u and v of x is,
d
du
dv
(uv) = v
+u .
dx
dx
dx
(3)
Table 2: Table of standard derivatives.
y = f (x)
xn
ex
ekx
ax
ln x
loga x
sin x
sin kx
cos x
tan x
cot x
sec x
cosec x
sinh x
cosh x
tanh x
sin−1 x
cos−1 x
tan−1 x
sinh−1 x
cosh−1 x
tanh−1 x
dy
dx
nxn−1
ex
kekx
ax ln a
1
x
1
x ln a
cos x
k cos kx (etc)
− sin x
sec2 x
−cosec2 x
sec x tan x
−cosec x cot x
cosh x
sinh x
1/ cosh2 x
√ 1
1−x2
√ −1
1−x2
1
1+x2
√ 1
x2 +1
√ 1
x2 −1
1
1−x2
• The derivative of a quotient of two functions u and v of x is
− u dv
d u v du
= dx 2 dx .
dx v
v
• The radius of curvature of a function y is given by
2 3/2
1 + dy
dx
.
R=
d2 y
(4)
(5)
dx2
• For a function of two variables z = f (x, y), the first partial derivatives are
∂z ∂z
,
∂x ∂y
(6)
• and the second partial derivatives of the same function are
∂2z ∂2z ∂2z ∂2z
,
,
,
,
∂x2 ∂y∂x ∂y2 ∂x∂y
(7)
where
∂2z
∂2z
=
.
∂y∂x
∂x∂y
(8)
Remember that the curly ∂ signs remind us that we are differentiating with respect to one
variable while keeping the other one constant.
• The total differential, δz, of a function z = f (x, y) is
∂z
∂z
δx +
δy.
∂x
∂y
δz =
(9)
• Rates of change. If both x and y are functions of time, t, then the total derivative enables us
to calculate the rate of change of z with respect to t
dz
∂z dx ∂z dy
=
+
.
dt
∂x dt
∂y dt
(10)
• Change of variables. For some function z = f (x, y), where both x and y are functions of two
other variables u and v, we can calculate
∂z
∂z ∂x
∂z ∂y
=
+
,
∂u
∂x ∂u ∂y ∂u
∂z
∂z ∂x ∂z ∂y
=
+
,
∂v
∂x ∂v ∂y ∂v
(11)
(12)
in analogy with the transformation of variables for the rate of change (see above).
4
Integration
The following integration rules are useful to remember, along with the other rules that are covered
in lectures, and Table 3 lists a number of standard integral results.
• When integrating
Z
f 0 (x)f (x) dx
(13)
we can recognize the solution as f 2 (x)/2 + C.
• When integrating
Z
f 0 (x)
dx
f (x)
(14)
we can recognize the solution as ln |f (x)| + C.
• When integrating a product of two functions by parts, recall
Z
Z
dv
du
u dx = uv − v dx
dx
dx
which is derived from rearranging the product rule for differentiation.
(15)
Table 3: Table of standard integrals.
Derivative
Integral
d
n
n−1
dx (x ) = nx
d
x
x
dx (e ) = e
R
d
kx
kx
dx (e ) = ke
d
x
x
dx (a ) = a ln a
d
1
dx (ln x) = x
1
d
dx (loga x) = x ln a
d
dx (sin x) = cos x
d
dx (sin kx) = k cos kx
d
dx (cos x) = − sin x
d
2
dx (tan x) = sec x
d
2
dx (cot x) = −cosec x
d
dx (sec x) = sec x tan x
d
dx (sinh x) = cosh x
d
dx (cosh x) = sinh x
1
d
dx (tanh x) = cosh2 x
1
d
−1
x) = √1−x
2
dx (sin
d
−1
−1
x) = √1−x2
dx (cos
d
1
−1 x) =
dx (tan
1+x2
−1
d
1
x) = √1+x
2
dx (sinh
−1
d
x) = √x12 −1
dx (cosh
−1
d
1
x) = 1−x
2
dx (tanh
xn dx =
R
xn+1
n+1
x
e dx
+ C [for n 6= −1]
= ex + C
R
ekx dx =
R
ax dx =
ekx
k +C
ax
ln a + C
1
x dx = ln |x| + C
1
x ln a dx = loga x + C
R
R
R
R
sin x dx = − cos x + C
sin kx dx = − k1 cos kx + C (etc)
R
cos x dx = sin x + C
R
sec2 x dx = tan x + C
R
cosec2 x dx = − cot x + C
R
sec x tan x dx = sec x + C
R
sinh x dx = cosh x + C
R
cosh x dx = sinh x + C
R 1
dx = tanh x + C
cosh2 x
R
√ 1
dx = sin−1 x + C
1−x2
R −1
√
dx = cos−1 x + C
1−x2
R 1
dx = tan−1 x + C
1+x2
R
√ 1
dx = sinh−1 x + C
1+x2
R
√ 1
dx = cosh−1 x + C
x2 −1
R 1
dx = tanh−1 x + C
1−x2
• Integrals of the form
Z
where the quotient
f (x)
g(x)
f (x)
dx
g(x)
(16)
can be separated into partial fractions, can be re-written in terms of
Z 0
Z 0
h (x)
k (x)
dx +
dx + . . .
(17)
h(x)
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)2 result in partial fractions of the form
A
(ax+b)
+
B
.
(ax+b)2
– Factors of (ax + b)3 result in partial fractions of the form
A
(ax+b)
+
B
(ax+b)2
– Factors of ax2 + bx + c result in partial fractions of the form
+
C
.
(ax+b)3
Ax+B
.
ax2 +bx+c
A few other general tips for integrating:
• For terms like sin2 x and cos2 x, use the double-angle formulae. For higher powers, try things
like sin3 x = (1 − cos2 x) sin x, or try the double-angle formula more than once.
• For terms like x sin x and x2 sin x and xex , try integration by parts.
p
• For terms like 1/ (1 − x2 ) and 1/(1 + x2 ), try the inverse-trig results above.
• For terms like 1/(ax2 + bx + c), try partial fractions as above.
Unlike differentation where there is a well-defined procedure, there is no guaranteed rule for complicated integrals; the above are just a hint for what to try first.
4.1
Applications of integration
A definite integral of a function f (x), represents the area bounded by the x axis and the curve
f (x) between the two limits specified in the integral (remember that area below the x−axis counts
as negative; and 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 2-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 sliced into
a large number of tiny pieces, then summed.
Some useful formulae that are used in this course are listed below.
p
• The arc-length along a curve, ds is just dx2 + dy2 assuming that one works in the limit
where dx tends to zero. Integrating this we obtain
s
2
Z x2
dy
s =
1+
dx
(18)
dx
x1
s
Z θ2 2 2
dx
dy
s =
+
dθ
(19)
dθ
dθ
θ1
to deal with integrals over x, or some parametric variable θ, respectively.
• The concept of the moment of inertia of a mass dm is used in this course. If a mass element
dm is rotating about some axis, at a distance r from that axis, then the moment of inertia dI
of that mass is defied as dI = r2 dm. If we integrate over both sides, we obtain
Z m2
I=
r2 dm
(20)
m1
and if we know the density (mass per unit area) of the mass, we can perform this integral in
terms of area, or r when we need to.
• We can calculate the centre of gravity (x, y, z) of an object by noting that
Z
Z
x dm = x dm
(21)
(and similarly for y and z). As x is a constant, and in this course, we only consider objects
of uniform density, we can simplify this equation by writing it as
R
x dA
x= R
(22)
dA
for plane objects (in 2-dimensions), or alternatively
R
x dV
x= R
dV
(23)
for objects that extend into 3-dimensions (with similar equations for y and z). Note that the
factors of density drop out as we assumed that the object is of uniform density.
• If we revolve a lamina (a planar shape) about the x axis, then the volume element of this
object is given by dV = πy 2 dx. So the centroid position (x, y) is given by the equations
y = 0, by symmetry
R 2
xy dx
x = R 2
y dx
(24)
(25)
• If we revolve a lamina about the y axis, then the volume element of this object is given by
dV = 2πxydx. So the centroid position (x, y) is given by the equations
R 2
xy dx
y = R
(26)
xy dx
x = 0, by symmetry
(27)
5
Series
P
• The sum of the first n terms of an arithmetic series is n−1
i=0 a + id.
P
i
• The sum of the first n terms of a geometric series is n−1
i=0 ar .
• The Taylor Series expansion f (x) about a point a is
f (x) = f (a) + f 0 (a)(x − a) +
f 000 (a)
f 00 (a)
(x − a)2 +
(x − a)3 + . . .
2!
3!
(28)
where f (a) is the function f (x) evaluated at x = a, likewise for the derivatives of f .
• The Maclaurin Series expansion for a function f (x) is
f (x) = f (0) + f 0 (0)x +
f 00 (0) 2 f 000 (0) 3
x +
x + ...
2!
3!
where f (0) is the function f (x) evaluated at x = 0, likewise for the derivatives of f .
(29)
• The Binomial Series expansion (1 + x)n is
(1 + x)n = 1 + nx +
n(n − 1) 2 n(n − 1)(n − 2) 3
x +
x + ...
2!
3!
(30)
The series converges for any value of n if |x| < 1; or if n is a positive integer it becomes a
finite series (with exactly n + 1 terms) and therefore converges for any value of x. If neither
of these applies (i.e. |x| > 1 and n is not a positive integer) the series diverges.
More generally if we have (a + x)n , it can be convenient to rewrite as an [1 + (x/a)]n , then
use the above with y = x/a.
• A few useful tests for convergence of a series covered in the lectures are listed below:
– limn→∞ Un = 0, If this test is satisfied, then the series may or may not converge (test
inconclusive). However, if this test is not satisfied, then the series definitely does not
converge (it may diverge or oscillate).
– Comparison test: Test a series against one known to converge. If (for all large numbers
n) the nth term in a series is smaller than the nth term in a series known to converge,
then the series converges.
– D’Alembert’s ratio test: For a series U1 + U2 + . . . + Un + . . ., look at the limit:
Un+1
.
n→∞ Un
Rn = lim
(31)
If Rn < 1, the series converges; if Rn > 1 the series diverges; if Rn = 1, the test is
inconclusive.
– L’Hôpital’s rule: if lim f (x) and lim g(x) are both zero or both infinite, then
f (x)
f 0 (x)
= lim 0
.
x→∞ g(x)
x→∞ g (x)
lim
(32)
(if the latter limit exists). This is often useful if one obtains indeterminate results of
∞
or ∞
when taking limits of f , g.
6
0
0
Complex Numbers
• Here define i =
√
−1.
• If z = a + ib, then Re(z) = a, and Im(z) = b.
• z = a + ib = reiθ = r(cos θ + i sin θ), where r = |z| =
called the modulus, and θ is called the argument of z.
√
a2 + b2 , and tan θ =
b
a
; here |z| is
• The product of two complex numbers z1 and z2 is z1 z2 = r1 r2 ei(θ1 +θ2 ) .
• The quotient of two complex numbers z1 and z2 is
z1
z2
=
r1 i(θ1 −θ2 )
.
r2 e
• The nth power is z n = rn einθ .
√
√
• For nth roots, n z = n rei(θ+2mπ)/n , where m = 0, 1, 2, . . . n − 1 gives the set of distinct
solutions with an argument between zero and 2π. These roots form a regular n−gon with the
centroid at (0, 0).
(Other values of m just give repeats of these with extra 2π’s in the argument).
Also note the following useful relations between complex exponentials and trigonometric functions
eix − e−ix
,
2i
eix + e−ix
cos(x) =
,
2
sin(x) =
7
(33)
(34)
Fourier Series
A periodic function f (x) can be written as a Fourier series of the form
y(t) =
∞ X
2πn
2πn
1
A0 +
An cos
t + Bn sin
t
2
T
T
(35)
n=1
where the coefficients are
An =
Bn =
A0
2
=
=
=
t2
2πn
y(t) cos
t dt,
T
t1
Z
2πn
2 t2
y(t) sin
t dt,
T t1
T
Z t2
12
y(t) dt,
2 T t1
Z t2
1
y(t) dt,
t2 − t1 t1
hy(t)i,
2
T
Z
(36)
(37)
(38)
(39)
(40)
Note the funny-looking factor of 12 in A0 is “arbitrary” but is put in so the An equation still works
for n = 0. The Fourier series can be re-expressed in terms of the angular frequency ω or frequency
f by noting that ω = 2π
T = 2πf .
8
Fourier Integrals
We can transform between space of a function y(x) and the reciprocal space (Fourier space) Y (u),
where x = 1/u using
Z ∞
Y (u) =
y(x) e−i2πux dx,
Z−∞
∞
y(x) =
Y (u) ei2πux du.
(41)
−∞
The Dirac Delta Function is given as
δ(u − u0 ) = 0, for u 6= u0
= ∞, for u = u0
Z
(42)
(43)
∞
δ(u − u0 ) du = 1.
−∞
(44)