Appendix C: Some Mathematical Relations

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Appendix C

Some Mathematical Relations

C.1 SOME ALGEBRAIC RELATIONS

C.1.1 Expansions

(a ± b)² = a²± 2ab + b²

(a ± b)³ = a³ ± 3a²b + 3ab² ± b³

(a ± b)⁴ = a⁴ ± 4a³ b + 6a²b² ± 4ab³ + b⁴

Here the binomial co-efficient equation is called ‘factorial r’.

C.1.2 Factors

a² – b² = (a + b)(a – b)

a³ ± b³ = (a ± b)(a² ∓ ab + b²)

aⁿ – bⁿ = (a – b)(a^{n –}¹ + a^{n –}² b + ··· + b^{n –}¹)

aⁿ + bⁿ = (a + b)(a^{n –}¹– a^{n –}² b + ··· + b^{n –}¹), for n odd.

C.1.3 Sum of Numbers

C.1.4 Binomial Series for x² < 1

(1 ± x)^–1 = 1 ∓ x + x² ∓ x³ + x⁴ ···

(1 ± x)^–2 = 1 ∓ 2x + 3x² ∓ 4x³ + 5x⁴ ···

C.1.5 Exponential Series

C.1.6 Logarithmic Series

C.2 SOME TRIGONOMETRIC RELATIONS

C.2.1 Fundamental Identities

sin²α + cos²α = 1

1 + tan²α = sec²α

1 + cot² α = cosec²α

sin (α ± β) = sin α cos β ± cos α sin β

cos (α ± β) = cos α cos β ∓ sin α sin β

sin2 α = 2sin α cos α

cos 2α = cos²α – sin²α = 2cos²α – 1 = 1 – 2sin²α

sin3α = 3sin α – 4sin³α

cos 3α = 4 cos³α – 3 cosα

C.2.2 Relations among Inverse Functions

C.2.3 Hyperbolic Functions

sinh (x ± y) = sinh x cosh y ± cosh x sinh y

cosh(x ± y) = cosh x cosh y ± sinh x sinh y

C.2.4 Connection between Circular and Hyperbolic Functions

e^ix = cos x + i sin x,

e^–^ix = cos x – i sin x

tan x = − i tanh ix

C.2.5 Trigonometric Series

C.3 COORDINATE SYSTEMS

Cartesian Coordinate System

Figure C.1 Cartesian coordinate system. The differential volume is dV = dx dy dz
Spherical Coordinate System

Figure C.2 Spherical coordinate system. The differential volume is dV = r² sin θ dr dθ dϕ

Unit vectors e_x, e_y, e_z in cartesian coordinate system and e_r, e_θ, e_ϕ in spherical coordinate system are related as

The components of vector A = (A_x, A_y, A_z) and A = (A_r, A_θ, A_ϕ) are related through the transformation

or through inverse transformation

C.4 SOME VECTOR RELATIONS

C.4.1 Cartesian Coordinates (x, y, z)

C.4.2 Spherical Polar Coordinates (r, θ, ϕ)

C.4.3 Relations Involving the ∇ Operator

Here A, B, C denote vectors; ϕ, ψ denote scalars.

∇(ϕ ψ) = ϕ∇ ψ + ψ∇ϕ (C. 15a)

∇ × (A + B) = ∇ × A + ∇ × B (C.15b)

∇ · (ϕ A) = A · ∇ϕ + ϕ∇ · A (C.15c)

∇ × (ϕA) = ϕ∇ × A – A × ∇ϕ (C.15d)

∇ · (A × B) = B · ∇ × A – A · ∇ × B (C.15e)

∇ × (A × B) = (B · ∇) A – B∇ · A – (A · ∇) B + A∇ · B (C.15f)

∇(A · B) = (B · ∇)A + (A · ∇)B + B × ∇ × A + A × ∇ × B (C.15g)

∇ × ∇ × A = ∇(∇ · A) – ∇²A (C.15h)

C.4.4 Other Vector Relations

A × (B × C) = B(A · C) – C(A · B) (C.16a)

If r denotes the position vector, then

∇ · r = 3 (C.17a)

∇ × r = 0 (C.17b)

∇ rⁿ = nr^{n –} ²r (C.17c)

∇²rⁿ = n(n + 1)r^{n –} ² (C.17d)

(Gauss) Divergence Theorem: The normal surface integral of a vector function F over the boundary of a closed surface (of arbitrary shape) is equal to the volume integral of the divergence of F taken over the enclosed volume. In mathematical form it is expressed as

(Stokes) Curl Theorem: If a vector function F and its first derivative are continuous, the line integral of F around a closed curve C is equal to the normal surface integral of Curl F over an open surface bounded by C. In mathematical form it is expressed as

C.5 SOME CALCULUS RELATIONS

C.5.1 Derivatives

If U = U(x), V = V(x), and b = constant

C.5.2 Integrals

C.6 SOME DEFINITE INTEGRALS

C.7 AN IMPORTANT INTEGRAL

If the electronic charge density in a state in an atom is ρ(r) (e.g. in s-state of an electron), which is spherically symmetric, then the self-potential energy of the charge cloud is

Let us consider the work done in building up this charge distribution. In the process of built up, let us consider that a sphere of charge of radius r is already built. The potential at the surface of this sphere is

Then the work done in bringing more charge from infinity to form a spherical shell of thickness dr and density ρ(r) at the surface of sphere of radius r already present,

The total self-potential energy for the entire charge distribution is

Let us take ρ(r) = e⁻^2αr, so

Integrating by parts, the second integral gives

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Table of Contents for Appendix C: Some Mathematical Relations

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