Mathematical Formula Handbook - 國立臺灣大學
Mathematical Formula Handbook
ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Bibliography; Physical Constants1. Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Arithmetic and Geometric progressions; Convergence of series: the ratio test;Convergence of series: the comparison test; Binomial expansion; Taylor and Maclaurin Series;Power series with real variables; Integer series; Plane wave expansion2. Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Scalar product; Equation of a line; Equation of a plane; Vector product; Scalar triple product;Vector triple product; Non-orthogonal basis; Summation convention3. Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Unit matrices; Products; Transpose matrices; Inverse matrices; Determinants; 2 2 matrices;Product rules; Orthogonal matrices; Solving sets of linear simultaneous equations; Hermitian matrices;Eigenvalues and eigenvectors; Commutators; Hermitian algebra; Pauli spin matrices4. Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Notation; Identities; Grad, Div, Curl and the Laplacian; Transformation of integrals5. Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Complex numbers; De Moivre’s theorem; Power series for complex variables.6. Trigonometric Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Relations between sides and angles of any plane triangle;Relations between sides and angles of any spherical triangle7. Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Relations of the functions; Inverse functions8. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Standard forms; Standard substitutions; Integration by parts; Differentiation of an integral;Dirac δ-‘function’; Reduction formulae11. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Diffusion (conduction) equation; Wave equation; Legendre’s equation; Bessel’s equation;Laplace’s equation; Spherical harmonics12. Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713. Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Taylor series for two variables; Stationary points; Changing variables: the chain rule;Changing variables in surface and volume integrals – Jacobians14. Fourier Series and Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Fourier series; Fourier series for other ranges; Fourier series for odd and even functions;Complex form of Fourier series; Discrete Fourier series; Fourier transforms; Convolution theorem;Parseval’s theorem; Fourier transforms in two dimensions; Fourier transforms in three dimensions15. Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316. Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Finding the zeros of equations; Numerical integration of differential equations;Central difference notation; Approximating to derivatives; Interpolation: Everett’s formula;Numerical evaluation of definite integrals17. Treatment of Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Range method; Combination of errors18. Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mean and Variance; Probability distributions; Weighted sums of random variables;Statistics of a data sample x 1 , . . . , xn ; Regression (least squares fitting)
IntroductionThis Mathematical Formaulae handbook has been prepared in response to a request from the Physics ConsultativeCommittee, with the hope that it will be useful to those studying physics. It is to some extent modelled on a similardocument issued by the Department of Engineering, but obviously reflects the particular interests of physicists.There was discussion as to whether it should also include physical formulae such as Maxwell’s equations, etc., buta decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainlybecause, in its present form, clean copies can be made available to candidates in exams.There has been wide consultation among the staff about the contents of this document, but inevitably some userswill seek in vain for a formula they feel strongly should be included. Please send suggestions for amendments tothe Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. TheSecretary will also be grateful to be informed of any (equally inevitable) errors which are found.This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently byDr Dave Green, using the TEX typesetting package.Version 1.5 December 2005.BibliographyAbramowitz, M. & Stegun, I.A., Handbook of Mathematical Functions, Dover, 1965.Gradshteyn, I.S. & Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, 1980.Jahnke, E. & Emde, F., Tables of Functions, Dover, 1986.Nordling, C. & Österman, J., Physics Handbook, Chartwell-Bratt, Bromley, 1980.Speigel, M.R., Mathematical Handbook of Formulas and Tables.(Schaum’s Outline Series, McGraw-Hill, 1968).Physical ConstantsBased on the “Review of Particle Properties”, Barnett et al., 1996, Physics Review D, 54, p1, and “The FundamentalPhysical Constants”, Cohen & Taylor, 1997, Physics Today, BG7. (The figures in parentheses give the 1-standarddeviation uncertainties in the last digits.)speed of light in a vacuumcpermeability of a vacuumµ0permittivity of a vacuum 0elementary chargeePlanck constanthh/2π h̄Avogadro constantNAunified atomic mass constantmumass of electronmemass of protonmpBohr magneton eh/4πmeµBmolar gas constantRBoltzmann constantkBStefan–Boltzmann constantσgravitational constantGOther dataacceleration of free fallg2·997 924 58 10 8 m s 14π 10 7 H m 1(by definition)(by definition)1/µ0 c2 8·854 187 817 . . . 10 12 F m 11·602 177 33(49) 10 19 C6·626 075 5(40) 10 34 J s1·054 572 66(63) 10 34 J s6·022 136 7(36) 10 23 mol 11·660 540 2(10) 10 27 kg9·109 389 7(54) 10 31 kg1·672 623 1(10) 10 27 kg9·274 015 4(31) 10 24 J T 18·314 510(70) J K 1 mol 11·380 658(12) 10 23 J K 15·670 51(19) 10 8 W m 2 K 46·672 59(85) 10 11 N m2 kg 29·806 65 m s 2(standard value at sea level)1
1. SeriesArithmetic and Geometric progressionsA.P.G.P.Sn a ( a d) ( a 2d) · · · [ a (n 1)d] 2Sn a ar ar · · · arn 11 rn, a1 r(These results also hold for complex series.)n[2a (n 1)d]2 aS 1 rfor r 1 Convergence of series: the ratio testSn u1 u2 u3 · · · unconverges asn iflimn un 1 1unConvergence of series: the comparison testIf each term in a series of positive terms is less than the corresponding term in a series known to be convergent,then the given series is also convergent.Binomial expansion(1 x)n 1 nx n(n 1) 2 n(n 1)(n 2) 3x x ···2!3! n If n is a positive integer the series terminates and is valid for all x: the term in x r is n Cr xr orwhere n Cr rn!is the number of different ways in which an unordered sample of r objects can be selected from a set ofr!(n r)!n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series isconvergent for x 1.Taylor and Maclaurin SeriesIf y( x) is well-behaved in the vicinity of x a then it has a Taylor series,dyu2 d2 yu3 d3 yy( x) y( a u) y( a) u ···dx2! dx23! dx3where u x a and the differential coefficients are evaluated at x a. A Maclaurin series is a Taylor series witha 0,x2 d2 yx3 d3 ydy ···y( x) y(0) x2dx2! dx3! dx3Power series with real variablesexln(1 x) 2x2xn ··· ···2!n!32xxnx · · · ( 1)n 1 · · ·x 23neix e ixx2x4x6 1 ···22!4!6!x3x5eix e ix x ···2i3!5!12 5x x3 x ···315x3x5x ···351.3 x51 x3 ···x 2 32.4 5 1 x cos x sin x tan x tan 1 x sin 1 x valid for all xvalid for 1 x 1valid for all values of xvalid for all values of xvalid for ππ x 22valid for 1 x 1valid for 1 x 1
Integer seriesN n1 1 2 3 ··· N NN ( N 1)2 n2 12 22 32 · · · N 2 1N ( N 1)(2N 1)6N n3 13 23 33 · · · N 3 [1 2 3 · · · N ] 2 1 1 1 N 2 ( N 1)24111( 1)n 1 1 · · · ln 2n234[see expansion of ln (1 x)]111π( 1)n 1 1 ··· 2n 135741 n2 1 1[see expansion of tan 1 x]111π2 ··· 49166N n(n 1)(n 2) 1.2.3 2.3.4 · · · N ( N 1)( N 2) 1N ( N 1)( N 2)( N 3)4This last result is a special case of the more general formula,N n(n 1)(n 2) . . . (n r) 1N ( N 1)( N 2) . . . ( N r)( N r 1).r 2Plane wave expansion exp(ikz) exp(ikr cos θ ) (2l 1)il jl (kr) Pl (cos θ),l 0where Pl (cos θ ) are Legendre polynomials (see section 11) and j l (kr) are spherical Bessel functions, defined byrπjl (ρ) J 1 (ρ), with Jl ( x) the Bessel function of order l (see section 11).2ρ l /22. Vector Algebra2If i, j, k are orthonormal vectors and A A x i A y j A z k then A A2x A2y A2z . [Orthonormal vectors orthogonal unit vectors.]Scalar productA · B A B cos θ where θ is the angle between the vectorsBx A x Bx A y B y A z Bz [ A x A y A z ] B y BzScalar multiplication is commutative: A · B B · A.Equation of a lineA point r ( x, y, z) lies on a line passing through a point a and parallel to vector b ifr a λbwith λ a real number.3
Equation of a planeA point r ( x, y, z) is on a plane if either(a) r · bd d , where d is the normal from the origin to the plane, oryzx(b) 1 where X, Y, Z are the intercepts on the axes.XYZVector productA B n A B sin θ, where θ is the angle between the vectors and n is a unit vector normal to the plane containingA and B in the direction for which A, B, n form a right-handed set of axes.A B in determinant formiAxBxjAyBykAzBzA B in matrix form 0 Az A yBx Az0 Ax By A y Ax0BzVector multiplication is not commutative: A B B A.Scalar triple productAxA B · C A · B C BxCxAyByCyAzBz A C · B,Czetc.Vector triple productA ( B C ) ( A · C ) B ( A · B)C,( A B) C ( A · C ) B ( B · C ) ANon-orthogonal basisA A1 e1 A2 e2 A3 e3A1 0 · Awhere 0 Similarly for A2 and A3 .e2 e3e1 · (e2 e3 )Summation conventiona ai eia·b ai bi( a b)i εi jk a j bkεi jkεklm δil δ jm δimδ jl4implies summation over i 1 . . . 3where ε123 1;εi jk εik j
3. Matrix AlgebraUnit matricesThe unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elementszero, i.e., ( I ) i j δi j . If A is a square matrix of order n, then AI I A A. Also I I 1 .I is sometimes written as In if the order needs to be stated explicitly.ProductsIf A is a (n l ) matrix and B is a (l m) then the product AB is defined byl Aik Bk j( AB)i j k 1In general AB 6 BA.Transpose matricesIf A is a matrix, then transpose matrix A T is such that ( A T )i j ( A) ji .Inverse matricesIf A is a square matrix with non-zero determinant, then its inverse A 1 is such that AA 1 A 1 A I.( A 1 )i j transpose of cofactor of A i j A where the cofactor of A i j is ( 1)i j times the determinant of the matrix A with the j-th row and i-th column deleted.DeterminantsIf A is a square matrix then the determinant of A, A ( det A) is defined by A i jk. A1i A2 j A3k . . .i, j,k,.where the number of the suffixes is equal to the order of the matrix.2 2 matricesIf A acbd then, A ad bcAT abcd A 1 1 A d c ba Product rules( AB . . . N ) T N T . . . B T A T( AB . . . N ) 1 N 1 . . . B 1 A 1 AB . . . N A B . . . N (if individual inverses exist)(if individual matrices are square)Orthogonal matricesAn orthogonal matrix Q is a square matrix whose columns q i form a set of orthonormal vectors. For any orthogonalmatrix Q,Q 1 Q T , Q 1,Q T is also orthogonal.5
Solving sets of linear simultaneous equationsIf A is square then Ax b has a unique solution x A 1 b if A 1 exists, i.e., if A 6 0.If A is square then Ax 0 has a non-trivial solution if and only if A 0.An over-constrained set of equations Ax b is one in which A has m rows and n columns, where m (the numberof equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes theerror Ax b ) is the solution of the n equations A T Ax A T b. If the columns of A are orthonormal vectors thenx A T b.Hermitian matricesThe Hermitian conjugate of A is A † ( A ) T , where A is a matrix each of whose components is the complexconjugate of the corresponding components of A. If A A † then A is called a Hermitian matrix.Eigenvalues and eigenvectorsThe n eigenvalues λ i and eigenvectors u i of an n n matrix A are the solutions of the equation Au λ u. Theeigenvalues are the zeros of the polynomial of degree n, Pn (λ ) A λ I . If A is Hermitian then the eigenvaluesλi are real and the eigenvectors u i are mutually orthogonal. A λ I 0 is called the characteristic equation of thematrix A.Tr A λi ,ialso A λi .iIf S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is thematrix whose columns are the normalized eigenvectors of A, thenU T SU ΛandS UΛU T.If x is an approximation to an eigenvector of A then x T Ax/( x T x) (Rayleigh’s quotient) is an approximation to thecorresponding eigenvalue.Commutators[ A, B][ A, B][ A, B]† AB BA [ B, A] [ B† , A† ][ A B, C ] [ A, C ] [ B, C ][ AB, C ] A[ B, C ] [ A, C ] B[ A, [ B, C ]] [ B, [C, A]] [C, [ A, B]] 0Hermitian algebrab† (b 1 , b 2 , . . .)Matrix formHermiticityb · A · c ( A · b) · cEigenvalues, λ realAu i λ(i) uiOrthogonalityCompletenessOperator formZψ Oφ Bra-ket form(Oψ) φui · u j 0b ui (ui · b)φ ψiψ i ψ j 0i Zψ i φhψ O φiO i i λ i i iOψi λ(i)ψiZiZ hi j i 0φ i i hi φiiRayleigh–RitzLowest eigenvalue6b · A · bλ0 b · bZλ0 Zψ Oψ ψ ψhψ O ψihψ ψi(i 6 j )
Pauli spin matricesσx 01 1,0σ xσ y iσ z ,σy 0iσ yσ z iσ x , i,0σz σ zσ x iσ y , 100 1 σ xσ x σ yσ y σ zσ z I4. Vector CalculusNotationφ is a scalar function of a set of position coordinates. In Cartesian coordinatesφ φ( x, y, z); in cylindrical polar coordinates φ φ(ρ, ϕ, z); in sphericalpolar coordinates φ φ(r, θ , ϕ); in cases with radial symmetry φ φ(r).A is a vector function whose components are scalar functions of the positioncoordinates: in Cartesian coordinates A iA x jA y kA z , where A x , A y , A zare independent functions of x, y, z. x j k In Cartesian coordinates (‘del’) i y x y z zgrad φ φ,div A · A,curl A AIdentitiesgrad(φ1 φ2 ) grad φ1 grad φ2grad(φ1φ2 ) φ1 grad φ2 φ2 grad φ1curl( A A ) curl A1 curl A2div(φ A) φ div A (grad φ) · A,div( A1 A2 ) div A1 div A2curl(φ A) φ curl A (grad φ) Adiv( A1 A2 ) A2 · curl A1 A1 · curl A2curl( A1 A2 ) A1 div A2 A2 div A1 ( A2 · grad) A1 ( A1 · grad) A2div(curl A) 0,curl(grad φ) 0curl(curl A) grad(div A) div(grad A) grad(div A) 2 Agrad( A1 · A2 ) A1 (curl A2 ) ( A1 · grad) A2 A2 (curl A1 ) ( A2 · grad) A17
Grad, Div, Curl and the LaplacianCartesian CoordinatesConversion toCartesianCoordinatesCylindrical Coordinatesx ρ cos ϕVector AAxi A y j Az kGradient φ φ φ φi j k x y z ·AijLaplacian1 φ1 φ b φbbr θ ϕ rr θr sin θ ϕ1 Aθ sin θ1 (r 2 Ar ) 2 rr sin θ θr1 Aϕ r sin θ ϕ1 Aϕ A z1 (ρ Aρ ) ρ ρρ ϕ z11b ϕbbρzρρ ρ ϕ zAρ ρ Aϕ A zk 2φ 2φ 2φ 2 22 x y z 2φb AϕϕbArbr Aθθ1 φ φ φb b bρϕz ρρ ϕ z x y zAx A y AzCurl Ax r cos ϕ sin θ y r sin ϕ sin θz r cos θz zb Aϕϕb AzbAρ ρz A y A z A x x y zDivergencey ρ sin ϕSpherical Coordinates1 ρ ρ11 b1bbrθϕrr2 sin θ r sin θ r θ ϕArrAθrAϕ sin θ 11 φ2 φr 2sin θ r θr2 rr sin θ θ φ1 2φ 2φρ 2 2 2 ρρ ϕ z Transformation of integralsL the distance along some curve ‘C’ in space and is measured from some fixed point.S a surface areaτ a volume contained by a specified surfacebt the unit tangent to C at the point Pbn the unit outward pointing normalA some vector functiondL the vector element of curve ( bt dL)dS the vector element of surface ( bn dS)ZThenCA · bt dL and when A φZCZZ( φ) · dL CCA · dLdφGauss’s Theorem (Divergence Theorem)When S defines a closed region having a volume τalsoZ( · A) dτ Z( φ) dτ ττ8ZZSS(A · bn) dS φ dSZSA · dSZτ( A) dτ ZS(bn A) dS 2φ12r sin θ ϕ22
Stokes’s TheoremWhen C is closed and bounds the open surface S,ZalsoZSS( A) · dS ZC(bn φ) dS ZCA · dLφ dLGreen’s TheoremZψ φ · dS S Z · (ψ φ) dτZτ τGreen’s Second TheoremZτ ψ 2φ ( ψ) · ( φ) dτ(ψ 2φ φ 2 ψ) dτ ZS[ψ( φ) φ( ψ)] · dS5. Complex VariablesComplex numbersThe complex number z x iy r(cos θ i sin θ ) r ei(θ 2nπ), where i2 1 and n is an arbitrary integer. Thereal quantity r is the modulus of z and the angle θ is the argument of z. The complex conjugate of z is z x iy 2r(cos θ i sin θ ) r e iθ ; zz z x2 y2De Moivre’s theorem(cos θ i sin θ )n einθ cos nθ i sin nθPower series for complex variables.ezz33!z2cos z 1 2!z2ln(1 z) z 2sin zz2zn ··· ···2!n!z5 ···5!z4 ···4!z3 ···3 1 z z convergent for all finite zconvergent for all finite zconvergent for all finite zprincipal value of ln (1 z)This last series converges both on and within the circle z 1 except at the point z 1.z3z5 ···35This last series converges both on and with
Equation of a plane A point r (x, y, z)is on a plane if either (a) r bd jdj, where d is the normal from the origin to the plane, or (b) x X y Y z Z 1 where X,Y, Z are the intercepts on the axes. Vector product A B n jAjjBjsin , where is the angle between the vectors and n is a unit vector normal to the plane containing A and B in the direction for which A, B, n form a right-handed set .
Handbook of Mathematical Functions The Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [1] was the culmination of a quarter century of NBS work on core mathematical tools. Evaluating commonly occurring mathematical functions has been a fundamental need as long as mathematics has been applied to the solution of
Table 68: Shirt Laundry Formula 04: White (No Starch) Table 69: Shirt Laundry Formula 05: Colored (No Starch) Table 70: Shirt Laundry Formula 06: Delicates Table 71: Shirt Laundry Formula 07: Stain Treatment Table 72: Shirt Laundry Formula 08: Oxygen Bleach Table 73: Shirt Laundry Formula 09: Stain Soak Table 74: Shirt Laundry Formula 10 .
the empirical formula of a compound. Classic chemistry: finding the empirical formula The simplest type of formula – called the empirical formula – shows just the ratio of different atoms. For example, while the molecular formula for glucose is C 6 H 12 O 6, its empirical formula
A Note about Array formulas (not for Excel 365 / Excel 2021) Sometimes, you will need to enter a formula as array formula. In Excel 365/Excel 2021, all formulas are treated as Array formula, hence you need not enter any formula as Array formula. Only for older versions of Excel, you might need to enter a formula as Array formula.
The F1 FORMULA 1 logo, F1 logo, FORMULA 1, FORMULA ONE, F1, FIA FORMULA ONE WORLD CHAMPIONSHIP, GRAND PRIX and related marks are trade marks of Formula One Licensing BV, a . FORMULA 1 HEINEKEN DUTCH GRAND PRIX 2022 - Zandvoort Race History Chart. LAP 6 GAP TIME 1 1:16.350 16 1.051 1:16.213 . Race History Chart. LAP 11 GAP TIME 1 1:16.671 16 .
The F1 FORMULA 1 logo, F1 logo, FORMULA 1, FORMULA ONE, F1, FIA FORMULA ONE WORLD CHAMPIONSHIP, GRAND PRIX and related marks are trade marks of Formula One Licensing BV, a . FORMULA 1 HEINEKEN AUSTRALIAN GRAND PRIX 2022 - Melbourne Race History Chart. LAP 6 GAP TIME 16 2:24.953 . Race History Chart. LAP 11 GAP TIME 16 1:23.356 1 3.085 1:24 .
The F1 FORMULA 1 logo, F1 logo, FORMULA 1, FORMULA ONE, F1, FIA FORMULA ONE WORLD CHAMPIONSHIP, GRAND PRIX and related marks are trade marks of Formula One Licensing BV, a . FORMULA 1 HEINEKEN GRANDE PRÊMIO DE SÃO PAULO 2022 - São Paulo Sprint History Chart. LAP 6 GAP TIME 1 1:14.644 . Sprint History Chart. LAP 11 GAP TIME 1 1:15.456 63 .
The F1 FORMULA 1 logo, F1 logo, FORMULA 1, FORMULA ONE, F1, FIA FORMULA ONE WORLD CHAMPIONSHIP, GRAND PRIX and related marks are trade marks of Formula One Licensing BV, . FORMULA 1 HEINEKEN GRANDE PRÊMIO DO BRASIL 2019 - São Paulo Race History Chart. LAP 6 GAP TIME 33 1:13.493 . Race History Chart. LAP 11 GAP TIME 33 1:13.484 44 2.381 1: .
