a handbook of essential mathematical formulae

[...]... inverse matrix , A 1 , exists 1 and is given by A 1 = det A adj A, where adj A is the transposed matrix of cofactors The inverse matrix has the property AA−1 = A 1 A = I If det A = 0 then A is said to be singular A is called an orthogonal matrix if AT = A 1 For a 2 × 2 matrix a b c d −1 = 1 ad − bc d −b −c a For an n × n matrix det A = det AT and det (kA) = kn det A The rank of a matrix, A, is the largest... a1 2 x2 + a1 3 x3 = h1 a2 1 x1 + a2 2 x2 + a2 3 x3 = h2 a3 1 x1 + a3 2 x2 + a3 3 x3 = h3 ∆1 = h1 a1 2 a1 3 h2 a2 2 a2 3 , h3 a3 2 a3 3 ∆= a1 1 a1 2 a1 3 a2 1 a2 2 a2 3 a3 1 a3 2 a3 3 a1 1 h1 a1 3 a2 1 h2 a2 3 , a3 1 h3 a3 3 ∆2 = ∆3 = a1 1 a1 2 h1 a2 1 a2 2 h2 a3 1 a3 2 h3 The solution is x1 = ∆1 /∆, 2.2 x2 = ∆2 /∆, x3 = ∆3 /∆ Matrices The m × n matrix is written as  a1 1  a2 1  A=   a1 2 a2 2 ··· ··· a1 n a2 n am1 am2 · · · amn... of straight lines if af 2 + bg2 + ch2 = 2f gh + abc a parabola if h2 = ab an ellipse if h2 < ab a hyperbola if h2 > ab a rectangular hyperbola if a + b = 0 1.9 Mensuration Circle, radius r: perimeter is 2πr, area is πr 2 For a segment of angular width θ (radians), arc length is rθ and area is 1 r 2 θ 2 Ellipse, axes 2a and 2b: perimeter is approximately 2π (a2 + b2 ) /2, area is πab Cylinder , radius... a1 ˆ + a2 ˆ + a3 k, ı j ˆ b = b1ˆ + b2ˆ + b3 k ı j a. b = a1 b1 + a2 b2 + a3 b3 a b = Magnitude: |a| = a = ˆ ˆ k ı j ˆ a1 a2 a3 b1 b2 b3 = −b a a2 + a2 + a2 1 2 3 Unit vector in the direction of the vector a is ˆ = a 1 a= |a| a1 a2 a3 b1 b2 b3 c1 c2 c3 a (b×c) = (a b) c = = [a b c] a × (b×c) = (a. c) b − (a. b) c (a b) × c = (a. c) b − (b.c) a (a b) (c×d) = a. c a. d b.c b.d (a b) × (c×d) = [a b c] c − [a b... second moments of a plane area A about an axis are given respectively by r 2 dA rdA and A A where r is the distance from the axis of the element dA 28 The moment of inertia, I, of a body, of density ρ and volume V , about an axis is given by r 2 ρdV V where r is the distance from the axis of the element dV Parallel axes theorem If IG is the moment of inertia about an axis through the centroid and I is the... the moment of inertia about a parallel axis distance d away, then I = IG + md2 Table of moments of inertia 1 Uniform Body mass m Axis M .of I Perpendicular to bar through one end Perpendicular to bar through centroid ma2 3 4ma2 3 2 Rectangular lamina sides 2a and 2b Parallel to side 2b through centroid Perpendicular to plane through centroid ma2 3 m (a2 + b2 ) 3 3 Rectangular solid edges 2a and 2b, depth... a3 3 a1 1 a1 2 · · · a1 n a2 1 a2 2 · · · a2 n n×n:∆= an1 an2 · · · ann 2×2 : a c a1 1 a2 1 a3 1 a2 1 a2 2 a3 1 a3 2 The minor , αij , of the element aij is the (n − 1)th order determinant formed from ∆ by omitting the row and the column containing aij The cofactor , Aij , of the element aij is given by Aij = (−1)i+j αij The value of the n × n determinant is ∆ = ai1 Ai1 + ai2 Ai2 + + ain Ain (expansion... 4ax Parametric equation: x = at2 , y = 2at 2 Ellipse (ǫ < 1); foci at ( a , 0), directrices x = a/ ǫ Major axis of length 2a, minor axis of length 2b 1 x2 y 2 Cartesian equation: 2 + 2 = 1 with b = a 1 − ǫ2 2 a b Parametric equation: x = a cos θ, y = b sin θ 3 Hyperbola (ǫ > 1); foci at ( a , 0), directrices at x = a/ ǫ 1 x2 y 2 Cartesian equation: 2 − 2 = 1 with b = a ǫ2 − 1 2 a b Parametric equation:... inequality If x > −1 then (1 + x)n ≥ 1 + nx Arithmetic mean 1 2 (x + y) Geometric mean 1 2 (x + y) ≥ √ xy √ xy Triangle inequality |x + y| ≤ |x| + |y| |x| − |y| ≤ ||x| − |y|| ≤ |x − y| Cauchy Schwarz inequality |u.v| ≤ u v Minkowski inequality u + v ≤ u + v 9 Chapter 2 Determinants and Matrices 2.1 Determinants b = ad − bc d a1 2 a1 3 a2 1 a2 3 a2 2 a2 3 +a1 3 a1 2 3×3 : a2 2 a2 3 = a1 1 a3 1 a3 3 a3 2 a3 3 a3 2 a3 3 a1 1... = f (x)dx (add arbitrary constant where necessary) x √ x2 ± a2 −x √ a2 − x2 √ √ x2 ± a2 a2 − x2 eax (a cos bx − b sin bx) eax cos bx eax (a sin bx + b cos bx) eax sin bx 4.3 a2 ln |x + x2 ± a2 | (4.31) 2 a2 x a2 − x2 + sin−1 (4.32) 2 a x 2 x 2 x2 ± a2 ± eax (a cos bx + b sin bx) a2 + b2 eax (a sin bx − b cos bx) a2 + b2 (4.33) (4.34) Definite integrals Wallis’s formulae (reduction formulae also hold