Further Pure 3 (OCR)
1. First Order Differential Equations.
In C4 you encountered simple first order equations (containing the first derivative only) that could either be differentiated directly or after separating the variables.
For equations of the form f(x) dy/dx + yg(x) = h(x) the variables cannot be separated but there is a trick available.
1. Divide by f(x) to get the standard form dy/dx + yp(x) = q(x)
- Find the simplest integral of p(x) call it l(x) (no constant)
- Let u(x) = el(x) this is the integrating factor.
- Multiply by the integrating factor to give d/dx (yu(x)) = q(x) u(x)
- Integrate the whole equation including a constant.
- Re-arrange to give y =
Not all first order equations can be solved with this trick. Sometimes a substitution will be given to transform the equation into something that can be integrated. Proceed as in C4.
2. Lines and Planes
From C4 the equation of a line in vector form is r = a + tb
Expressing this in Cartesian form gives
(x-a)/l = (y-b)/m = (z-c)/m where the line goes through the point a,b,c and has direction ( l )
( m )
( n )
So to test whether a point lies on a given line just substitute into the vector equation and see if it gives a consistent value for the parameter t.
He equation of a plane through a point a which contains non parallel lines in direction p and q is
r = a + sp + tq
Another equation for a plane is
n.(r – a) = 0 this is called the normal equation of a plane because n is the direction normal to the plane. a is any point in the plane.
Changing this to Cartesian form gives
ax + by +cz = d
where( a ) is the normal to the plane and d = a.n
( b )
( c )
Remember that the angle between two planes is equal to the angle between their normals. So if the normals are known the C4 dot product formula for cosθ can be used.
3. Linear Differential Equations
Remember that linear equations only have a term in y not y2 or above. For FP3 they are either first order or second order depending on whether contain just the first derivative or also the second derivative.
The solution of a linear differential equation consists of two parts – the particular integral – PI - (any solution of the equation) and the complimentary function – CF - (the solution with 0 in place of the terms which are independent of y).
The CF for a first order equation with constant coefficients is y = Ceλx. Sub this in (with the RHS = 0) and get the auxiliary equation which is solved to find the value of λ.
Similarly the CF for a second order equation with constant coefficients is Aeαx + Beβx. This gives a quadratic auxiliary equation in λ (RHS = 0) with solutions α and β.
If the auxiliary equation has a repeated root the the CF is (Cx + D)eλx.
Luckily you don’t have to do the general case with non-constant coefficients.
The particular integral must be found using common sense. So if the RHS is a polynomial of degree n then try a polynomial of degree n. If it is a sin or cos function (or both) try asinx + bcosx. If the PI turns out to be part of the CF then try xP(x) instead.
Finally apply any end conditions to determine the arbitrary constants.
4. The Vector Product
The vector (or cross) product of two vectors a and b has magnitude a b sinθ in the direction that is perpendicular to both vectors in the right hand sense. So unlike the dot product it is a vector. The order of the vectors does matter. a x b = - b x a
So i x i = j x j = k x k = 0 and i x j = j x k = k x i = 1 etc
This means that vectors in Cartesian components can be multiplied out.
The vector product is useful in 3D geometry whenever the sine of the angle is concerned such as the area of a triangle or the perpendicular distance from a point to a line.
In the previous section the Cartesian equation of a plane required knowledge of the normal to the plane. If three points in the plane (or two directions) are known the then vector product can be used to find the normal.
Finally the shortest distance between two non-intersecting lines is the component of the distance between any two points on the lines in the direction perpendicular to the lines. If the lines are r = a + tb and r = c + sd then the normal direction is defined by b x d and the length of the line joining the two points is a – c so the distance is
(a – c).(b x d) / ( b x d )
Don’t learn this just be able to do it.
5. Complex Numbers in Polar Form
A complex number can be written in polar form as z = r(cosθ + jsinθ) where z > 0 is the modulus and θ ( -π < θ =< π) is the argument.
In this form multiplication and division is easy. zw = z w
and arg (zw) = arg(z) + arg(w) + nπ where n ( -1, 0 or 1) is chosen so that the argument is in the normal range.
For division z / w = z / w and arg(z/w) = arg(z) - arg(w) + nπ
To find the quare root of a number in this form just root the modulus and half the argument (and ajust with +-π if necessary).
z can also be expressed in exponential form as z = rejθ = r(cosθ + jsinθ)
If z = x + jy then by definition ez = ex(cosy + jsiny)
In general ez = ex and arg(ez) = y + k(2π)
For addition and subtraction complex numbers and vectors are the same but for multiplication and division they are very different.
If complex numbers are represented by translations of a plane then multiplication by a complex number has then effect of a spiral enlargement with a scale factor of s and a rotation arg(s).
In an Argand diagram the triangles represented by the points 0, t and st and 0, 1 and s are similar.
6. De Moivre’s Theorem
In order to make printed maths easier to read it is usual to write ef(x) as exp(f(x) especially if f(x) is complicated.
De Moivre proved that
(ejθ)n = ejnθ
or(exp(jθ))n = exp(jnθ)
or(cosθ + jsinθ)n = cos(nθ) + jsin(nθ)
These results can be used to evaluate powers of complex numbers and prove various trig identities for multiple angles.
The nth roots of unity are 1, ω, ω2, …… wn-1 where ω = exp(j2π/n)
It can be shown by considering the sum of the roots of a polynomial equation of the form zn – 1 = 0 that for n> 1 then the sum of the nth roots of 1 = 0.
To find the nth roots of a complex number just express it in exponential form and Bob’s your uncle!
7. Further Trigonometry
For complex numbers:
Re(az) = a.Re(z) Im(az) = aIm(z)
Re(z + w) = Re(z) + Re(w) Im(z + w) = Im(z) + Im(w)
z + z* = 2Re(z)z – z* = 2Im(z)
Based on these, de Moivre and the binomial expansion multiple angle formula can be easily derived!!
(cosθ + jsinθ)3 = cos3θ + 3C1.cos2θ.jsinθ + 3C2.cosθ.j2sin2θ + j3sin3θ
= cos3θ - 3cosθsin2θ + j(3cos2θsinθ - sin3θ)
taking the real part
cos3θ= cos3θ - 3cosθ(1 – cos2θ)
= 4cos3θ - 3cosθ
and sin3θ= 3sinθ – 4sin3θ
Some polynomial equations can be solved in trig form using this approach. You would be led through them.
If z = cosθ + jsinθ then 1/z = cosθ – jsinθ
So 2cosθ = z + 1/z and 2jsinθ = z - 1/z
Using de Moivre
2cossθ = zs + 1/zs and 2jsinsθ = zs - 1/zs
these can be used to give powers of sin and cos in terms of multiple angles as required in integration.
e.g ∫ sin5x dx
start by expressing sin5x in terms of multiple angles using the above.
(2jsinx)5 = (z – 1/z)5 = (z5 – 5z3 + 10z - 1/z5 + 5/z3 -1/z5)
= (z5 – 1/z5) + ……..
giving
sin5x = (1/16)(sin5x - 5sin3x + 10sinx)
which is easily integrated.
Series can have complex numbers as their parameter.
The infinite series ∑azs converges to a/(1 – z) provided z < 1. This defines a region for convergence rather than a range. Similarly with the binomial expansion.
8. Calculus with Complex Numbers
If c is complex and x is real then d/dx(ecx) = c.ecx
When solving second order equations with constant coefficients the roots may be complex. Stick with it , put in the end conditions and assuming these are real then the solution will turn out to be real as well. Alternatively assume that the solution is real and replace the constants which some of which APPEAR to be complex with real ones.
In general when integrating or differentiating (maybe finding a particular integral) then if the function is a sin or cos function, especially if it is of the form eaxsin(bx), then let it be the real or imaginary part of ecx and proceed as usual. Take the corresponding part of the solution as the answer.
9. Groups
A set is a collection of similar objects, for our purposes numbers. A set is comprised of a number of elements e.g. A = {1,3,5,7} is the set of the first 4 odd natural numbers. The symbol ε means is a member of so 3 ε A
The set of real numbers is denoted by R, natural numbers by N and integers by Z. Complex numbers are denoted by C and rational numbers by Q.
Superscripts can be used to further define sets. Z+ is the set of positive integers. 0 can be excluded from a set with e.g. Z – {0}
A binary operation ○ on a set S is a rule which assigns to each ordered pair of elements x,y in S exactly one element denoted by x ○ y. ○ is pronounced blob.
If the result of the operation is always in the set S then the operation is said to be closed within the set.
A binary operation on a set is commutative if a ○ b = b ○ a for all elements in the set. Otherwise it is non commutative.
If (a ○ b) ○ c = a ○ (b ○c) then the operation is associative. Otherwise it is non associative.
For a set with a closed binary operation ○ if there exists an element e for all elements such that e ○ a = a ○ e = a then e is the identity element for S with operation ○
If there exists an element b such that b ○ a = a ○ b = e (an identity element) then b is the inverse of a in S with the operation ○.
A set G with a binary operation ○ is called a group if it has four properties called axioms:
- Closure
- Associativity
- Identity
- Inverse
It is denoted by (G, ○)
The number of elements in the group is its order, which can be infinite.
For any group a ○ x = a ○ y ↔ x = y and x ○ a = y ○ a ↔ x = y
This is the cancellation law.
For any group a ○ x = b ↔ x = a-1 ○ b and x ○ a = b ↔ x = b ○ a-1
If in a group (G, ○) a ○ b = b ○ a then the group is commutative or abelian.
Examples may be set involving modulo arithmetic
Modulo arithmetic is just like ordinary arithmetic except that the answer is always the remainder when the “real answer” is divided by the modulo being used.
So if we work in modulo 4 then 1 + 2 = 3 2 + 5 = 3 3 x 4 = 0 3 x 5 = 3
So the answer is always less than the modulo in which you are working.
Questions involving transformations can also be set. Suppose that the three vertices of a triangle are subject to a transformation X. It can be described by
(1 2 3) X (1 2 3)
(A B C) (A C B)
which shows what happens to the points ABC after the transformation. For example B starts in position 1 and ends up in position 3.
The following theorems are often required to solve the exam problems:
The identity element for a group is unique
For any element in a group its inverse is unique.
For a,b ε G if a ○ b = e (the identity element) then a = b-1 and b = a-1 and b ○ a = e
For a,b ε G then (a ○ b)-1 = b-1 ○ a-1
For a ε G (a-1)-1 = a
In a Group table each element appears just once in every row and in every column
10. Subgroups
a is an element of a group G. If n is a positive integer then an = aaaa (n times) and a-n = a-1a-1 … (n times). a0 = e by definition.
The usual laws of indices apply except that (ab)m is not = to ambm unless G is commutative.
An element a is said to have finite order if an = e for some positive n. The least value of n is the order of the element. If no such n exists then the order is infinite.
If every element in a group has the form an and a is in the group then the group is said to be cyclic. The element a is the generator for G. The notation in this case is G = (a).
If H is a subset of a group G with operation ○ such that H is a group with operation ○ then H is a subgroup of G. For every group there are two trivial subgroups e and the whole group G. The rest are called proper subgroups.
In general H is a subgroup of G if for a,b ε H → ab ε H (i.e the subset is closed)
and e ε H
and for a ε H → a-1 ε H
If the group is finite then only the first condition is required.
The subgroup H of a group G generated by H = {an:n ε Z) is said to be the subgroup generated by a. The order of the subgroup is the same as the order of the generator.
Lagrange’s Theorem. The order of a subgroup of a finite group G divides the order of G.
The theorem has the following corollaries:
The order of the element of a finite group divides the order of the group.
A group of prime order has no proper subgroups.
Every group of prime order p is cyclic
11. Isomorphisms of Groups
Groups are really functions in disguise. The input set is called the domain and the result of the operation results in the target set of elements. Together they make up the range. The element f(x) is the image of the element x. If for each target element there is only one element in the domain for which y = f(x) then the function is one-one.
f(ab) = f(a)f(b)
Two groups (G, ο) and (H, ●) are isomorphic if there exists a function f: G → H such that
the range of f is H
f is one-one
f(a ○ b) = f(a) ● f(b) for all a,b ε G
The function is called an isomorphism.
If e is the identity element of G then f(e) is the identity element of H
If a ε G then f(a-1) is the inverse of f(a) in H
The order of a ε G is the same as the order of f(a) in H
This can sometimes be used to show that groups are NOT isomorphic – just find the order of every element in the groups and show that they do not match.
To show groups are isomorphic find the required function to map between them (probably given) and show that it is one-one and f(ab) = f(a)f(b). However if both groups are cyclic with the same order then that is sufficient.
Every infinite cyclic group is isomorphic to (Z, +)
Every finite cyclic group of order n is isomorphic to (Zn, +)
The groups of order up to and including 7 are:
(Z2, +) of order 2
(Z3, +) of order 3
(Z4, +) and V of order 4
(Z5, +) of order 5
(Z6, +) and D3 order 6
(Z7, +) of order 7
That’s it folks. That last section is still a mystery to a mere engineer like me.
Richard Vincent
24th August 2008