Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.7 SOME PARTICULAR METHODS OF PROOF 1.7.1 Proof by induction The proof of the binomial expansion given in subsection 1.5.2 and ...

PRELIMINARY ALGEBRA This is precisely the original assumption, but withNreplaced byN+ 1. To complete the proof we only have to v ...

1.7 SOME PARTICULAR METHODS OF PROOF The essence of the method is to exploit the fact that mathematics is required to be self-co ...

PRELIMINARY ALGEBRA
The prime integerspiare labelled in ascending order, thusp 1 =1,p 2 =2,p 5 =7,etc. Show that there is no la ...

1.7 SOME PARTICULAR METHODS OF PROOF AIFFBAis true if and only ifBis trueor B⇐⇒ A, AandBnecessarily imply each other B⇐⇒ A. Alth ...

PRELIMINARY ALGEBRA 1.8 Exercises Polynomial equations 1.1 Continue the investigationof equation (1.7), namely g(x)=4x^3 +3x^2 − ...

1.8 EXERCISES (a) the sum of the sines ofπ/3andπ/6, (b) the sine of the sum ofπ/3andπ/4. 1.8 The following exercises are based o ...

PRELIMINARY ALGEBRA 1.16 Express the following in partial fraction form: (a) 2 x^3 − 5 x+1 x^2 − 2 x− 8 , (b) x^2 +x− 1 x^2 +x− ...

1.9 HINTS AND ANSWERS 1.27 Establish the values ofkfor which the binomial coefficientpCkis divisible byp whenpis a prime number. ...

PRELIMINARY ALGEBRA 1.11 Show that the equation is equivalent to sin(5θ/2) sin(θ)sin(θ/2) = 0. Solutions are− 4 π/ 5 ,− 2 π/ 5 , ...

2 Preliminary calculus This chapter is concerned with the formalism of probably the most widely used mathematical technique in t ...

PRELIMINARY CALCULUS A P x f(x) x+∆x f(x+∆x) ∆f θ ∆x Figure 2.1 The graph of a functionf(x) showing that the gradient or slope o ...

2.1 DIFFERENTIATION approximate the change in the value of the function, ∆f, that results from a small change ∆xinxby ∆f≈ df(x) ...

PRELIMINARY CALCULUS
Find from first principles the derivative with respect toxoff(x)=x^2. Using the definition (2.1), f′(x) = ...

2.1 DIFFERENTIATION separation is not unique. (In the given example, possible alternative break-ups would beu(x)=x^2 ,v(x)=xsinx ...

PRELIMINARY CALCULUS and using (2.6), we obtain, as before omitting the argument, df dx =u d dx (vw)+ du dx vw. Using (2.6) agai ...

2.1 DIFFERENTIATION
Find the derivative with respect toxoff(t)=2at,wherex=at^2. We could of course substitute fortand then diff ...

PRELIMINARY CALCULUS
Finddy/dxifx^3 − 3 xy+y^3 =2. Differentiating each term in the equation with respect toxwe obtain d dx (x^ ...

2.1 DIFFERENTIATION The pattern emerging is clear and strongly suggests that the results generalise to f(n)= ∑n r=0 n! r!(n−r)! ...

PRELIMINARY CALCULUS Q A B C f(x) x S Figure 2.2 A graph of a function,f(x), showing how differentiation corre- sponds to findin ...