Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION dA ρ(φ) ρ(φ+dφ) ρdφ x y O B C Figure 2.9 Finding the area of a sectorOBCdefined by the curveρ(φ)and the radiiOB, ...

PRELIMINARY CALCULUS 2.2.12 Integral inequalities Consider the functionsf(x),φ 1 (x)andφ 2 (x) such thatφ 1 (x)≤f(x)≤φ 2 (x)for ...

2.2 INTEGRATION m f(x) a b x Figure 2.10 The mean valuemof a function.
Find the mean valuemof the functionf(x)=x^2 between the ...

PRELIMINARY CALCULUS ∆x ∆s ∆y y=f(x) x f(x) Figure 2.11 The distance moved along a curve, ∆s, corresponding to the small changes ...

2.2 INTEGRATION S y b V f(x) dx a x ds Figure 2.12 The surface and volume of revolution for the curvey=f(x).
Find the surface a ...

PRELIMINARY CALCULUS
Find the volume of a cone enclosed by the surface formed by rotating about thex-axis the liney=2xbetweenx= ...

2.3 EXERCISES 2.10 The functiony(x) is defined byy(x)=(1+xm)n. (a) Use the chain rule to show that the first derivative ofyisnmx ...

PRELIMINARY CALCULUS O C P Q ρ ρ r r+∆r c p p+∆p Figure 2.13 The coordinate system described in exercise 2.20. 2.20 A two-dimens ...

2.3 EXERCISES 2.26 Use the mean value theorem to establish bounds in the following cases. (a) For−ln(1−y), by considering lnxin ...

PRELIMINARY CALCULUS (c) [(x−a)/(b−x)]^1 /^2. 2.38 Determine whether the following integrals exist and, where they do, evaluate ...

2.4 HINTS AND ANSWERS 2.45 IfJris the integral ∫∞ 0 xrexp(−x^2 )dx show that (a) J 2 r+1=(r!)/2, (b)J 2 r=2−r(2r−1)(2r−3)···(5)( ...

PRELIMINARY CALCULUS πa 2 πa 2 a x y Figure 2.14 The solution to exercise 2.17. 2.21 (a) 2(2−9cos^2 x)sinx;(b)(2x−^3 − 3 x−^1 )s ...

3 Complex numbers and hyperbolic functions This chapter is concerned with the representation and manipulation of complex numbers ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS 1 1 2 2 3 3 4 4 5 z f(z) Figure 3.1 The functionf(z)=z^2 − 4 z+5. the first term is cal ...

3.2 MANIPULATION OF COMPLEX NUMBERS Rez Imz z=x+iy x y Figure 3.2 The Argand diagram. Our particular example of a quadratic equa ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS Rez Imz z 1 z 2 z 1 +z 2 Figure 3.3 The addition of two complex numbers. or in componen ...

3.2 MANIPULATION OF COMPLEX NUMBERS Rez Imz |z| x y argz Figure 3.4 The modulus and argument of a complex number. 3.2.2 Modulus ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS 3.2.3 Multiplication Complex numbers may be multiplied together and in general give a c ...

3.2 MANIPULATION OF COMPLEX NUMBERS Rez Imz iz −iz z −z Figure 3.5 Multiplication of a complex number by±1and±i. multiplyzby a c ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS Rez Imz z=x+iy x y −y z∗=x−iy Figure 3.6 The complex conjugate as a mirror image in the ...