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Table of Integrals

Ic+b in az+dc2-b2 cos az ,b+csin z )2 < c2z - 1 ln b+c sin z Idz 1 ln I c+b cosaz+dc2-b2 sin azb+c cos az - a,/m b+c cos az 1 b 2 < c 2

8 1 sin-' a x dx = x sin- ' a x + 4 9 1 xn s in-' ax dx = s i n - l a X zn d~n+l J 4+' dz5 0 1 a n- ' a x d x = x tan-' ax & l n ( 1 a2x2) 511 xn tan-' ax dx = ;;T;i an - ' ax -

5 2 1 eazdz = 531xeaZdz= (ax 1 5 4 1 x2eaZdx = (a2x2- 2ax + 2) (baz is ea( 'nb)z)sin- ' dx5 5 & = 1 n I l n a ~ l N o te le me nta ry : J e z 2 d x , e z 1 n ~ d ~ , J , J d ~ , J % d ~ , J

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Exponentials and Logarithms Equations and Their Solutions,J c yo e ty cy s yoect eCt 1y' cy bd c d c-by0b de-ct YOy X2y cos At and sin A tmy dy ky e x l t a n d exs t or t x l tYn+l ayn a nYOYn+l a,Jn 3 anyo 3

Vectors and Determinants Matrices and InversesA a l i + a 2 j + a 3 k Ax combination of colum ns bA A .A a: a: a: ( le n g th sq u ar e d) Solution A-lb if A-'A I

cos B Least squares Z T b B a l bl a2b2 a 3 b 3 J A J J B IA .B ( A BI (Schwarz inequality: cos B I 1 Ax Ax (A is an eigenvalue)IA B < IAl IBI (triangle inequality)IA x BI IAIIBJIsin B J (cross produ ct)

i j k i(a2bs a3ba)A x B a1 a2 a3 +j(a3bl alb3)

b l b2 b3 + k ( a 1 b z a 2 h )R i g h th a n d r u l e i x j = k , j x k = i , k x i = jParallelogram are a (alb2 a2bll IDetJn i a n g l e a re a i la 1b 2 a2bl ? I ~ e t lBox volume IA ( B x C) I Determinant

SI Units Symbols From To Multiply bylength meter m degrees radians .01745mass kilogram kg calories joules 4.1868time second s BTU joules 1055.1current ampere

Hz - 11s foot-pounds joules 1.3558frequency hertz feet meters .3048force newton N - kg*m/s2 miles km 1.609pressure pascal Pa ~ / m ~ feet /sec km/hr 1 0973energy, work joule Nom pounds kg .45359power watt W - J/s ounces kg .02835charge coulomb C - AWS gallons liters 3.785emperature kelvin K horsepower watts 745 .7Speed of light c 2.9979 x108 m/s Radius at Equator R 6378 km 3964 milesGravity G 6.6720 x 10-l1Nm2/kg2 Acceleration g 9.8067 m s2 32.174 ft s2

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Sums and Infinite Series Area - Volume - Length - Mass - Momentl + x + . . . + x n - ' = 1-21-2 Circle ?rr2 Ellipse ru b Wedge of circle r28 /2l + n x + ~ x 2 + . . . + x n = 1+ z) Cyl inder s ide 2 m h Volume r r 2 h She ll dV = 27rrh drl + 2 + . . . + n = $ n ( n + l ) k Sp here surface 47rr2 Volum e $7rr3 Shell dV = 4 a r 2 d r12 22 . 2 = n ln+ l ) l 2n+ l ) n3~ 7 Cone or pyramid Volume (base area) (heig ht)61+ + - . .+ I n n -+ a (harmonic) Length of curve $ d s = $ J l + (dy/dx)2 dx1 - L + L2 3 In 2 (a l te rna t ing) Area be tween curves $(v(x ) - w ( x ) ) d x~ - L $ - I . . . = c+=ZEL= ' Surface area of revolution J27rr d s(r = x or = y)3 5 6 n4 90-x = 1+ x + z 2 + (geometric: 1x1 < 1) Volume of revolution: Slices $ 7ryZdx Shells J27rxh d z1, -XI, - 1+ 22 + 3xZ+ . . = (A) Area of surface z z, ) 4- dx d y

_L = 1 - x + x 2 - . . .l + s (geometric for -2) Mass M = JJ p dA Moment My JJ ox dA3 3 dx -l n ( l + x ) = X - ~ + F - . - . = J G x = M,/M, = Ms/M Moment of Inertia Iy= JJ p ~ 2 d ~

sin x = x - x3/6 + x5/120 - (a l l x) Work W = F ( x ) d x = V(b) - V(a) Force F = dV/dzcos x = 1- x2 2 + x4124 - . . (al l x) Partial Derivatives of z = f x , )ex = l + x + $ + . - . ( e = 1 + 1 . + $ + . . . )e = cos x + i sin x (Euler 's formu la) Tangent plane z - o = ( g ) ( x- x o ) + ($?)(y - o)c o s h x = $ ( ex + e - % ) = I + + . . . Approximat ion A z ;3. ( % ) A X + (%)ays i n h x = i ( e ~ - ~ - ~ ) ~ + d + - . .3 N o m a 1 N = ( f z , f,, -1 or ( Fz ,Fy, Z )(cos + sin 8) = cos n + sin n6 G r a d i e n t V = i+ gjf ( x ) = f 0)+ f l ( 0 ) x+ f ( 0 ) $ + . (Taylor) Directional derivative: u = V f .u = f x u l + j V u 2Polar and Spherical Vector f i e l d F x , y , z ) = M i + N j + P kx = r c o s 0 a n d y = r s i n 8r = d w n d t a n 6 = y / xx + y = r(cos 0 + i sin 8) = reiArea $ ?r2d9 Length $ d m d 8

W ork J F a d R F l u x $ M d y - N d xa a aPDivergence of F = V .F = i +

c u r l of F = v x P = a/ay a / a zx = p sin d cos 6, y = p sin sin 6, z = p cos d I M N P IArea dA = dx dy = r d r d6 = J u dv Conservative F = V f = gradient of f if curl F =Volume r dr d6 z = p2 sin dp dd dB Green's Theorem f dx +N dy = s f ( g)x dy

I I

Stretching factor J = = xu xu~ u , w ) Divergence Theorem $$ F n S = $J$ iv F dVStokes' Theorem f F d R = $$ (cur l F) .n d SAn additional table of integrals is included just afte r the index.