Higgs production through gluon fusion at leading order


 Flora Long
 2 years ago
 Views:
Transcription
1 NIKHEF 57 Higgs production through gluon fusion at leading order Stan Bentvelsen, Eric Laenen, Patrick Motylinski Abstract The dominant Higgs production mechanism at the LHC involves gluon fusion via an intermediate topquark loop. The lowest order theoretical cross section is wellknown and used in many LHC studies to determine the experimental discovery sensitivity of the Higgs particle. Although the expression for this cross section can readily be found in the literature, the derivation of this result is usually not given and is not so easy to calculate. In this note we will go stepbystep through the calculation.
2 Introduction In this report we calculate in detail the cross section for the Higgs production process gg H. () at lowest order, assuming the Standard Model. This lowest order process proceeds through a top quark loop between the gluons and the Higgs particle, see Fig.. Although in principle all quarks should be included in the loop, in practice the restriction to just the top quark suffices because the Higgs couples about 35 times more strongly to the top than to the nextheaviest fermion, the bottom quark, leading to a relative suppression of the bottom contribution by a factor 35. The calculation of the matrix element in lowest order should be finite, even though the diagrams feature a loop, which is a common source of infinities. This must be so, because there is no fundamental ggh coupling in the Standard Model which could absorb such infinities. We shall indeed see that the diagrams give a finite answer. This report is organized as follows. We quote first the results of the calculations for the process in the next section, so that it is clear what the final result of the calculation should be. In section 3 we then list the diagrams and Feynman rules for the process, and in 4 we construct the matrix element. In section 5 we reduce the resulting loop integrals to the simplest (scalar) form, and in sections 6 and 8 we calculate these explicitly. In section 7 we put it all together and compute the cross section. For some of the cumbersome calculations we make use of the algebraic manipulation program form []; the relevant code is reproduced in the appendix. Results Before starting the calculation in detail, we first write down the expression for the gluon fusion process in lowest order. It reads [, 3]: ˆσ LO = σ M H δ ( ŝ M H), () σ = G F α s 88 π A(τ), (3) A(τ) = 3 τ [ + ( τ)f(τ)], (4) arcsin / τ τ [ f(τ) = ln + ] τ τ iπ τ <, (5) with M H the Higgs mass, G F Fermi s constant, α s the strong coupling constant defined at some as yet unspecified (see section 8) scale and with τ defined as τ = 4m t M H. (6) For the full cross section for protonproton collisions the gluon cross section must be folded with the density of gluons in the proton σ (pp H) = τ dx τ/x dx g(x, M H)g(x, M H) ˆσ LO,gg H (ŝ = x x S) (7)
3 where x and x are the momentum fractions of the gluons. The deltafunction in the leading order cross section σ LO () ensures that the Higgs particle is onshell and that the Higgs mass is related to the protonproton centreofmass S as: ŝ = x x S = M H. (8) In the heavy top mass limit, which corresponds to τ, the function ( A(τ) reduces to the constant. This can be seen by expanding the function arcsin / ) (τ) : arcsin / ( ) τ = + τ 3! ( τ) 3 + O ( τ) 5. (9) Inserted in A(τ) in EQ. (4) this yields to order /τ: A(τ) = 3 [ τ + ( τ) + ] ( ) + O = + O 3τ τ ( ). () τ The limit τ corresponds to the heavy Higgs mass limit, and the function A(τ) becomes in this case: A(τ) = 3 [ ln τ ] iπ () 3 Feynman rules and diagrams In lowest order there are two Feynman diagrams contributing to the process gg H, drawn in Fig. 3. The charge flow is indicated by the arrow on the fermion lines, and the direction of the momentum flow is indicated by the accompanying arrow. We use the momentum definitions as given in the Figure. Because gluons are massless, the following ν ν k l k k l + k l l k l + k q = k + k k l k q = k + k µ µ Figure : The two contributing Feynman diagrams with the notation as used in the text. The charge flow is denoted by the arrows on the spinor fields. The momentum flow is indicated by the separate arrows. kinematic relations hold: k = () k = (3) (k + k ) q = ŝ = M H. (4) 3
4 We use the indices i, j to label the color of the quarks in the fundamental SU(3) representation. The color of the gluons in the SU(3) adjoint representation are denoted by the roman indices a, b. The spinor indices are α and β, whereas the indices µ, ν denote the Lorentz group indices for the gluons. We need the Feynman rules to write down the matrix element of the two diagrams. The vertices of the topquark with the gluon and the topquark with the Higgs particle are given in the standard model by i β a, µ j α ig s γ µ βα [ta ] ji i β j α iyt δ ij δ αβ with the top Yukawa coupling, y t /, given by the Higgs vacuum expectation value v as y t / = m t /v 75/46. The propagator of the topquark is given by i α p j β i( p + m) βα δ ij p m + iɛ To construct the matrix element one should follow the lines against the direction of the arrows indicating the charge flow. 4 Matrix element It is straightforward to write down the matrix element M for the two diagrams. We collect the couplings, the imaginary factors i of the propagators, the color factors and the overall minussign for the fermion loop in front. The polarization of the external gluons is given by ɛ µ (λ, k) in spin state λ and with momentum k. The momentum in the loop is l, and the loopintegral is regularized by working in d = 4 ε dimensions: ( M = ( ig s ) i y ) t i 3 Tr [t a t b ] ( ) ɛ ν (λ, k )ɛ µ (λ, k ) (5) d d l Tr [( l+ k (π) d + m)γ µ ( l + m)γ ν ( l k + m)+ D D D 3 ( l+ k + m)γ ν ( l + m)γ µ ( l k + m)]. In this expression the contributions from the two diagrams are added. Note that for the crossed diagram the Lorentzlabels µ and ν are interchanged and the loop momentum l reversed sign. The signs of the momenta are chosen such that the propagator terms are identical for the two diagrams. The trace of the spinor indices is taken because the quark lines form a loop. The factor ( ) in the first line is inserted because of the fermion loop. The denominators of the propagators are D = l m (6) D = (l k ) m D 3 = (l + k ) m 4
5 The goal is now to calculate the matrix element M in Eq. (5) by performing the trace and calculate the integral over the loop momentum l. The trace has to be done in d dimensions and can be done with the algebraic manipulation program form []. The relevant form program, together with its output, is listed in the appendix. The result is : Tr [( l+ k + m)γ µ ( l + m)γ ν ( l k + m)+ (7) ( l+ k + m)γ ν ( l + m)γ µ ( l k + m)] = 8m [ k µ kν kν kµ + kµ lν k ν lµ + 4l µ l ν g µν k k g µν l l + g µν m ] This expression still needs to be integrated over the loop integral l, and this will lead to the main complexity of the Higgs production calculation. As it turns out, the terms linear to the loop integral l µ cancel in the final result and only the terms proportional to l µ l ν and g µν l l remain in the calculation. We will be doing them by reducing the integrals to simpler expressions, using the PassarinoVeltman reduction method. We describe this method for this case in the next section. The cross section is then obtained from standard techniques, i.e. square the matrix element, sum over final state spins, average over initial state spins, insert the correct flux factor and insert the one particle phase space. This we will do in section 7. 5 Reduction of loop integrals When inserting Eq. (7) into Eq. (5), the main calculatioal difficulty originates from the tensor integral over the loop momentum d d l l µ l ν C µν =, (8) (π) d D D D 3 with the propagator terms defined in equation (6). Before proceeding to calculate this integral, we shall introduce some common notation: the letter C (as the third letter of the alphabet) is used for this integral because it contains three propagators D, D and D 3. The C µν integral has a tensor Lorentzstructure. Likewise we introduce the vector C µ and the scalar C integrals as follows C µ = C = d d l l µ (9) (π) d D D D 3 d d l. (π) d D D D 3 Similarly we can introduce the B integrals. They contain only two propagators and are defined as: B µ (, ) = d d l l µ (π) d D D B µ (, 3) = B (, ) = d d l (π) d D D d d l l µ (π) d D D 3 B µ (, 3) = B (, 3) = d d l (π) d D D 3 d d l l µ (π) d D D 3 B (, 3) = () d d l (π) d D D 3 We will show that the tensor C µν can be written in terms of the scalar functions C and B. The scalar functions C and B can be then be calculated analytically using the Feynman trick (see later), and thus we obtain the expression for C µν. Note that for this case the dimension d does not appear in the answer. 5
6 In order to show that the tensor C µν can be written in terms of the scalar functions C and B we first write the Lorentzstructure of the tensor C µν explicitly in most general terms, using the available vectors k, k and the metric g µν, noting that C µν is symmetric under interchange of the indices µ and ν: C µν = k,µ k,ν C + k,µ k,ν C + {k, k } µν C 3 + g µν C 4 () with the symmetry operator {k, k } µν k,µ k,ν + k,ν k,µ. Contracting the tensor C µν with the momenta k gives C µν k ν s = k,µ C s + k,µ C 3 + k,µ C 4 () C µν k ν s = k,µ C s + k,µ C 3 + k,µ C 4 It is customary to define the projection operators P µ k and P µ k. When operated on the vectors k µ i, they are required to behave as A representation of these operators is P µ k i k j,µ = δ ij. (3) P µ k = s kµ (4) P µ k = s kµ Using these projection operators we can map the individual coefficients for the expansion of C µν as defined in equation (). It is convenient to define the sets of scalar functions R i as follows R 3 P µ k C µν k ν = s C 3 + C 4 (5) R 4 P µ k C µν k ν = s C R 5 P µ k C µν k ν = s C R 6 P µ k C µν k ν = s C 3 + C 4 after which we will express these functions R i in terms of reduced integrals. We now start to rewrite the expression of C µν () in terms of the B integrals (which have two denominators only). We do this by contracting C µν with the vecors k and k. First note that k l = (D D k ) (6) k l = (D 3 D k ) where we leave the term k and k in for later use. It means that when contracted with k µ i, i =,, one of the propagators of C µν cancels in the denominator and turns into a B function (note that here we can put the gluon on massshell, i.e. k = ): C µν k ν = (B µ(, 3) B µ (, 3)) (7) C µν k ν = (B µ(, ) B µ (, 3)) 6
7 Here we have the expression for the integral C µν k ν with three denominators reduced to integrals B µ with two denominators. We can perform the same expansion of the function B µ (i, j) in terms of its Lorentzstructure as we did for C µν in equation (). We will first do this for the B µ (, ) and B µ (, 3) functions, as they are the easiest. The only terms available for them are B µ (, ) = k,µ B (, ) (8) B µ (, 3) = k,µ B (, 3). The function B (i, j) can be written in terms of the scalar integrals of equation (), by applying the contraction of equation (4) again, such that k µ B µ (, ) = k B (, ) (9) = [ ( d d l ) ] d d l k (π) d D D (π) d D D the first integral in this expression is zero, which can be seen by applying a suitable shift in momentum l. We are left, after dividing by k on both sides, with B (, ) = B (, ) and similarly (3) B (, 3) = B (, 3) Now we turn to the expansion of B µ (, 3), which is a little more involved. writing our the integral gives d d l l µ B µ (, 3) = (π) d [(l k ) m ] [(l + k ) m ] Explicitly and we apply a shift to the momentum l, l = l + k, and integrate over l, such that we get two terms B µ (, 3) = B µ (, 3) + k,µb (, 3) (3) The first term has now the equivalent structure of B µ (, ) and B µ (, 3) with momenta k = k + k and becomes: ( B µ(, 3) = (k,µ + k,µ ) ) B (, 3) (33) such that the answer for B µ (, 3) becomes: (3) B µ (, 3) = (k,µ k,µ )B (, 3) (34) At this stage we are ready and have expressed the vector integrals B µ (i, j) in terms of scalar integrals B (i, j). We can use this result in equation (7), in order to write the contraction of C µν k ν in terms of scalar integrals B (i, j), as follows: C µν k ν = 4 k,µb (, 3) + 4 (k,µ k,ν )B (, 3) (35) C µν k ν = 4 k,µb (, ) 4 (k,µ k,ν )B (, 3). Note that in this case we keep track of the term k, which we have set to zero earlier in equation (6). We need this small offshellness to avoid dividing by zero in the determination of the coefficient B, where k cancels in the result. For the C functions this problem does not appear. 7
8 We now use the projection operators (3) to obtain the functions R i, as defined in equations (5), in terms of the scalar integrals B : R 3 = 4 B (, 3) (36) R 4 = 4 B (, 3) + 4 B (, ) R 5 = 4 B (, 3) + 4 B (, 3) R 6 = 4 B (, 3). We are now ready to determine the Lorentz factors of the expansion of C µν as in equation (), using the expression of R i as given in (5): C = s R 5 (37) C = s R 4 C 4 = ( ) B (, 3) + m C R 3 R 6 d C 3 = s (R 6 C 4 ). The expression for C 4 and C 3 are a little less trivial to obtain, since they cannot be obtained from equation (5) by direct inversion. Instead, consider the projection operator P µν defined as: P µν = ( g µν P µ k d k ν P µ ) k k ν. (38) When acted on C µν it indeed selects the coefficient C 4 P µν C µν = d [sc 3 dc 4 s C 3 C 4 s ] C 3 C 4 = C 4, (39) and when acted on the integral expression of C µν gives the result as given above in equation (37), after noting that d g µν l l m d d l C µν = + (π) d D D D 3 (π) d m = B (, 3) + m C. D D D 3 (4) We are now ready to calculate the scalar integrals B and C. We will first calculate B, then compute the cross section in terms of the scalar integral C and keep the calculation of this integral for desert. 6 Calculation of B So far, the result of the loop integral of the matrix element of equation (5), after the PassarinoVeltman reduction, contains the scalar integrals B and C in d = 4 ε dimensions 3. The dependence on the dimension d originates from the expression of C 4, as given in equation (37). ɛ µ. 3 Do not confuse the regulator ε of dimensional regularisation with the polarization vector of the gluons, 8
9 We expand the dependence on the dimension d in terms of ε as /(d ) (/)( + ε + ε +...). The complete integral of the trace, obtained from form, becomes: d d l M = Tr [( l+ k (π) d + m)γ µ ( l k + m)] = D D D 3 (ɛ ɛ ) [ ] 8m(ε + ε )B (, 3) 4mMHC + 6m 3 ( + ε + ε )C [ ] 6m +(k ɛ )(k ɛ ) ( ε ε )B MH (, 3) + 8mC + 3m3 ( ε ε )C MH This result can be written as M = (ɛ ɛ )a + (k ɛ )(k ɛ )b with (4) a = 8m(ε + ε )B (, 3) 4mM HC + 6m 3 ( + ε + ε )C b = a M H Note that the scalar function B only appears in this expression multiplied with the regulator ε or higher orders. As the regulator ε will be set to zero at the end of the calculation, we are only interested in eventual poles in ε of B. Any constant terms or terms proportional to ε in B will vanish. For the scalar integral C we need to keep track of the constant term as well, as it will not vanish for ε. Note that we do not expect poles in B proportional to /ε or higher inverse order, as this would make the cross section unrenormalisable. The same is true for pole terms /ε or higher inverse order for the scalar intergral C. In order to keep the correct dimensions we introduce µ the dimensional regularization (mass) scale µ, and write for B B = µ ε d d l (π) d (l m )((l + k + k ) m ) We shall do this integral using Feynman s trick, i.e. by rewriting a fraction as an integral as follows: AB = dx [xa + ( x)b]. (43) This technique can be extended to higher powers of the fraction: (4) A α A αn n = Γ(α + + α n ) Γ(α ) Γ(α n ) dx dx n δ( x x n ) x α x αn n [x A x n A n ] (α + +α n). (44) In addition we need the integral over d dimensions d d l (π) d [ l M + iɛ ] s = ( ) s Γ ) ( s d Γ(s) i(4π) ε 6π [ M iɛ ] d s (45) Now we have all the tools to calculate the scalar function B. We will use the Feynman trick to cast the scalar function B in the form suitable for equation (44). Then the relevant pole /ε is identified, which we can use in equation (4). 9
10 First apply the Feynman trick to obtain d B = µ ε d l dx (π) d [( x)(l m ) + x(l + k + k ) m ]. (46) Denote k = k + k and rewrite the expression as d B = µ ε d l dx (π) d [l + xl k + xs m ]. (47) We shift the variable l by defining l = l xk such that d B = µ ε d l dx (π) d [l + xs x s m ] (48) (we have dropped the prime from l) and we have cast the integral into a form in which it can be calculated using expression (45), with M = xs x s m. The result for B becomes ε Γ(ε) i(4π) ε B = µ dx [ x s xs + m ] ε (49) Γ() 6π In this expression we need to identify the pole /ε. It occurs at only one place, which is in Γ(ε), as it can be written as: Γ(ε) = ε Γ( + ε) ; Γ( + ε) e εγ E ; γ E.577 (5) For the integral over x we are therefore only interested in the constant term (as any other terms will vanish in the final result of the matrix element). Note that a ε = ε ln a + O(ε ). (5) Hence the whole integral over x can be approximated as. Using this we obtain ε i(4π)ε B = µ = i 6π 6π e εγ E ( ε ε γ E + ln 4π ) µ ε. (5) We are ready to substitute this result 4 in the expression of the matrix element, equation (4), and take the limit ε. The result for the matrix element can be written in equation (4) with the coefficients: a = 8m 6π b = a M H with τ defined in equation (6). ( M H ( τ)6π C ) 4 The combination of terms /ε γ E + ln 4π occurs often in the dimensional renormalisation. In many situations in higher order gauge theories the real and virtual contributions both contain these poles, that have to be subtracted to reach the final finite result. It is common use to not only subtract the pole /ε (minimal subtraction, or MS scheme), but to subtract the γ E and ln 4π terms as well (the MS scheme). (53)
11 7 Calculation of the cross section Our next step is to determine the cross section in terms of the scalar integral C, without calculating this integral explicitly yet. Any cross section can be written in general terms as ˆσ = dp S ŝ spins,color M (54) For our process, the oneparticle phase space is just d 4 q dp S = (π) (π)δ +(q M 4 H )(π)4 δ 4 (k + k q) = πδ(ŝ MH ) (55) Now note that averaging over the spins in the initial state gives a factor ( ), averaging over color gives a factor (. 8) Also we use that (g s ) = 6π αs yt and that ( = m t = ) v GF m t. The color factor trace equals Tr[t at b ] = δ ab and hence the color contribution is 4 δ abδ ab =. The cross section becomes ˆσ = α sπ 3 G F 6M H m t MH spins M MH δ(ŝ M H ) (56) where we redefined the matrix element M with all constants, coupling constants etc, left out. M = a(ɛ ɛ ) a(ɛ MH k )(ɛ k ) (57) with ɛ i = ɛ µ (λ i, k i ). We have to sum over the spin states λ, λ and one normally uses the completeness relation for this. Note however that in this case we only sum over the polarization states, the gluon being massless onshell. One way of implementing this is by introducing the auxillary vector n µ in the completeness relation, with n = : It follows that so that λ i ɛ µ (λ i, k i )ɛ ν (λ i, k i ) = g µν + n µk i,ν + n ν k i,µ n k i P i,µν (58) P µν i k i,ν = P µν i n ν = λ,λ M = a P µν P,µν + b (P µν k,µ k,ν )(P ρσ k,ρ k,σ ) + m HR(a b) = a ( + 4 4) = a (59) where we used the notation for b as in equation (53). The factor a is defined in equation (53) ˆσ = α s GF τ 6π M H ( τ)6π C MH δ(ŝ M H ) (6) We now have the complete expression for the cross section as given at the start in equation (), in terms of the integral C. The calculation of this integral is our remaining task.
12 8 Calculation of C The explicit form of C is d d l C = (π) d (l m )(l k l m )(l + k l m ) We will use the Feynman trick twice to write the integral in the form (use equation (44) for the second line): d d l C = dx (π) d [l xl k m ] (6) [l + k l m ] d d l = dx dy Γ(3) (π) d Γ()Γ() y [ l x( y)k l + yk l m ] 3 again shift the integration parameter l to l = l + K, with K = x( y)k yk (such that K = x( = y)ys) to obtain: d d l Γ(3) C = dx dy (π) d Γ()Γ() y [ l (m + K ) ] 3 (63) Here C is cast in form of equation (45), and we can perform the integral. Since the power in the denominator is 3, we will find the term Γ( + ε) in the answer, which contains no pole in ε. The answer is therefore finite in the limit ε and we can take this limit here straightaway. The result becomes: C = ( ) 6π i y dx dy (64) (m xsy( y)) This integral is tedious but can be done with conventional techniques. First integrate over x: C = i dy 6π ( y)( s) ln ( m sy( y)x ) = i 6π = i 6π s dy ln ( ξy( y)) ( y)( s) (6) dy ln ( ξy( y)) (65) y with ξ = s. In the third line we changed integration variable y y. Note that the m logarithm in the argument becomes imaginary for ξ > 4. The calculation of the integral in (65) now follows along the lines of Chapter 7 of the book by Smith and De Wit. Integration by parts gives I = ξ = = dy ln( ξy( y)) y ( y) ln y dy y( y)ξ ( y) ln y dy (y y + )(y y ) dy( y) (ln y) β ( y y + y y ) (66)
13 where in the second but last line we used the principal value integral ( ) x y iɛ = P x y ± iπδ(x y) (67) with the poles of the denominator given by y ± = ± 4/ξ ± β ; β = We now have to determine the integral for the various cases. 4m s (68) 8. Imaginary part for ξ > 4 Lets start with the imaginary part of I, which only appears for ξ > 4, when the argument of the logarithm becomes negative, ln( x) = ln x + iπ. The result becomes iii = iπ β dyδ(y y + )( y) ln y + iπ β = iπ β [( y ) ln y + ( y + ) ln y + ] = iπ β [β ln y β ln y + ] dyδ(y y )( y) ln y = iπ ln y + y = iπ ln + τ τ (69) where τ = 4m /s. 8. Real part For the real part of the integral I, with ξ > 4, we need the trick to first determine the derivative to ξ: di dξ = y dy (7) y( y)ξ 3
14 We change the integration variable to u, defined as y = (+u) and hence y = ( u). The expression for the derivative becomes 5 di dξ = = βξ P = βξ ln u du 4 ξ + ξu ( du ( ) + β β ( u + β u β )) (7) We now need to integrate over ξ. We change again variables as dξ di dξ dβ dξ di dβ dξ (73) where dξ/dβ = βξ /. For ξ > 4 the integral becomes: ( ) I = dβ βξ + β βξ ln β ( ) ( ) + β d + β = dβ ln β dβ ln β = ( ) + β ln + Cnst. (74) β Whereas for ξ < 4 the integral becomes ( ) ( )] (75) dξ arcsin ξ/4 = dx( 4) [arcsin arcsin x = ξ/4 ξ(4 ξ) x The integration constant is chosen such that the integral is continuous at ξ = 4, i.e. Cnst = π. All results can be summarized for the scalar integral C as C = i 6π s [ ln ( ) + β iπ] β = τ, τ < β arcsin ( τ ) τ > This result can be substituted in Eq. (56), and we re done. 5 The principal value integral is defined as integral with infinitesimal distance to the pole [ P du β δ = lim du ] u β δ u β + du β+δ u β [ ( ) ( )] δ β = lim ln + ln δ β δ ( ) β = ln + β (7) 4
15 Appendix In this appendix the form algebraic manipulations, used in several places in the text, are presented. The input program to arrive at equation (7) is as follows: FORM by J.Vermaseren,version 3.(Sep 9 3) Run at: Wed Sep 4 3::44 3 * * Calculation of gg>h via top quark loop * Result using the algebraic manipulation program FORM * * Declarations Symbol m, d; Vector k k l; Index mu=d nu=d; * Local M = ( ( g_(,l)+g_(,k)+m)*g_(,mu)*( g_(,l)+m)*g_(,nu)*( g_(,l)g_(,k)+m) + (g_(,l)+g_(,k)+m)*g_(,nu)*(g_(,l)+m)*g_(,mu)*(g_(,l)g_(,k)+m) ); tracen,;.sort Time =. sec Generated terms = 4 M Terms in output = 8 Bytes used = 4 bracket m; print;.end Time =. sec Generated terms = 8 M Terms in output = 8 Bytes used = 7 M = + m * ( 8*k(mu)*k(nu)  8*k(nu)*k(mu)  6*k(nu)*l(mu) + 6*k(mu) *l(nu) + 3*l(mu)*l(nu)  8*d_(mu,nu)*k.k  8*d_(mu,nu)*l.l ) + m^3 * ( 8*d_(mu,nu) ); References [] J. A. M. Vermaseren, (), mathph/5. 5
16 [] H. M. Georgi, S. L. Glashow, M. E. Machacek, and D. V. Nanopoulos, Phys. Rev. Lett. 4, 69 (978). [3] M. Spira, A. Djouadi, D. Graudenz, and P. M. Zerwas, Nucl. Phys. B453, 7 (995), hepph/
1 Anomalies and the Standard Model
1 Anomalies and the Standard Model The GlashowWeinbergSalam model of the electroweak interactions has been very successful in explaining a wide range of experimental observations. The only major prediction
More informationIntroduction to SME and Scattering Theory. Don Colladay. New College of Florida Sarasota, FL, 34243, U.S.A.
June 2012 Introduction to SME and Scattering Theory Don Colladay New College of Florida Sarasota, FL, 34243, U.S.A. This lecture was given at the IUCSS summer school during June of 2012. It contains a
More informationarxiv:1008.4792v2 [hepph] 20 Jun 2013
A Note on the IR Finiteness of Fermion Loop Diagrams Ambresh Shivaji HarishChandra Research Initute, Chhatnag Road, Junsi, Allahabad09, India arxiv:008.479v hepph] 0 Jun 03 Abract We show that the mo
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationFeynman diagrams. 1 Aim of the game 2
Feynman diagrams Contents 1 Aim of the game 2 2 Rules 2 2.1 Vertices................................ 3 2.2 Antiparticles............................. 3 2.3 Distinct diagrams...........................
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationGravity and running coupling constants
Gravity and running coupling constants 1) Motivation and history 2) Brief review of running couplings 3) Gravity as an effective field theory 4) Running couplings in effective field theory 5) Summary 6)
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationPURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our
More informationarxiv:hepth/0404072v2 15 Apr 2004
hepth/0404072 Tree Amplitudes in Gauge Theory as Scalar MHV Diagrams George Georgiou and Valentin V. Khoze arxiv:hepth/0404072v2 15 Apr 2004 Centre for Particle Theory, Department of Physics and IPPP,
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationTime Ordered Perturbation Theory
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field.
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationTensors on a vector space
APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationarxiv:1006.5339v1 [hepph] 28 Jun 2010
FRPHENO2010021 arxiv:1006.5339v1 [hepph] 28 Jun 2010 NLO QCD corrections to 4 bquark production University of Illinois at UrbanaChampaign, Urbana IL, 61801 USA Email: ngreiner@illinois.edu Alberto
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationG(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s).
Transfer Functions The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input, i.e., Y (s)/u(s). Denoting this ratio by G(s), i.e.,
More informationSchouten identities for Feynman graph amplitudes; The Master Integrals for the twoloop massive sunrise graph
Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 880 0 33 377 www.elsevier.com/locate/nuclphysb Schouten identities for Feynman graph amplitudes; The Master Integrals for the twoloop
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationElectroweak effects in Higgs boson production
Electroweak effects in Higgs boson production Frank Petriello University of Wisconsin, Madison w/c. Anastasiou, R. Boughezal 0811.3458 w/ W. Y. Keung, WIP Outline Brief review of experiment, theory for
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationStructure of the Root Spaces for Simple Lie Algebras
Structure of the Root Spaces for Simple Lie Algebras I. Introduction A Cartan subalgebra, H, of a Lie algebra, G, is a subalgebra, H G, such that a. H is nilpotent, i.e., there is some n such that (H)
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSCATTERING CROSS SECTIONS AND LORENTZ VIOLATION DON COLLADAY
SCATTERING CROSS SECTIONS AND LORENTZ VIOLATION DON COLLADAY New College of Florida, 5700 Tamiami Trail, Sarasota, FL 34243, USA Email: colladay@sar.usf.edu To date, a significant effort has been made
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationPhysics Department, Southampton University Highfield, Southampton, S09 5NH, U.K.
\ \ IFT Instituto de Física Teórica Universidade Estadual Paulista July/92 IFTP.025/92 LEPTON MASSES IN AN SU(Z) L U(1) N GAUGE MODEL R. Foot a, O.F. Hernandez ", F. Pisano e, and V. Pleitez 0 Physics
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations
More informationPartial Fractions. p(x) q(x)
Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break
More informationSecond Rank Tensor Field Theory
Physics 411 Lecture 25 Second Rank Tensor Field Theory Lecture 25 Physics 411 Classical Mechanics II October 31st, 2007 We have done the warmup: E&M. The first rank tensor field theory we developed last
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationCircuits 1 M H Miller
Introduction to Graph Theory Introduction These notes are primarily a digression to provide general background remarks. The subject is an efficient procedure for the determination of voltages and currents
More informationGenerally Covariant Quantum Mechanics
Chapter 15 Generally Covariant Quantum Mechanics by Myron W. Evans, Alpha Foundation s Institutute for Advance Study (AIAS). (emyrone@oal.com, www.aias.us, www.atomicprecision.com) Dedicated to the Late
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationLimits. Graphical Limits Let be a function defined on the interval [6,11] whose graph is given as:
Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes
More informationMatrix Solution of Equations
Contents 8 Matrix Solution of Equations 8.1 Solution by Cramer s Rule 2 8.2 Solution by Inverse Matrix Method 13 8.3 Solution by Gauss Elimination 22 Learning outcomes In this Workbook you will learn to
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationPhysik des Higgs Bosons. Higgs decays V( ) Re( ) Im( ) Figures and calculations from A. Djouadi, Phys.Rept. 457 (2008) 1216
: Higgs decays V( ) Re( ) Im( ) Figures and calculations from A. Djouadi, Phys.Rept. 457 (2008) 1216 1 Reminder 10.6.2014 Higgs couplings: 2 Reminder 10.6.2014 Higgs BF as a function of mh Higgs total
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationSolving a System of Equations
11 Solving a System of Equations 111 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined
More informationLinear Algebra: Matrices
B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationLinear DC Motors. 15.1 Magnetic Flux. 15.1.1 Permanent Bar Magnets
Linear DC Motors The purpose of this supplement is to present the basic material needed to understand the operation of simple DC motors. This is intended to be used as the reference material for the linear
More information2 Session Two  Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision  Session 2: Complex Numbers and Vectors 1 2 Session Two  Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationCoefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationChapter 15 Collision Theory
Chapter 15 Collision Theory 151 Introduction 1 15 Reference Frames Relative and Velocities 1 151 Center of Mass Reference Frame 15 Relative Velocities 3 153 Characterizing Collisions 5 154 OneDimensional
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationFOREWORD. Executive Secretary
FOREWORD The Botswana Examinations Council is pleased to authorise the publication of the revised assessment procedures for the Junior Certificate Examination programme. According to the Revised National
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More informationCross section, Flux, Luminosity, Scattering Rates
Cross section, Flux, Luminosity, Scattering Rates Table of Contents Paul Avery (Andrey Korytov) Sep. 9, 013 1 Introduction... 1 Cross section, flux and scattering... 1 3 Scattering length λ and λ ρ...
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics K. Schulten Department of Physics and Beckman Institute University of Illinois at Urbana Champaign 405 N. Mathews Street, Urbana, IL 680 USA April 8, 000 Preface i Preface The
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More informationCDROM Appendix E: Matlab
CDROM Appendix E: Matlab Susan A. Fugett Matlab version 7 or 6.5 is a very powerful tool useful for many kinds of mathematical tasks. For the purposes of this text, however, Matlab 7 or 6.5 will be used
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationRecitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere
Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationTheoretical Particle Physics FYTN04: Oral Exam Questions, version ht15
Theoretical Particle Physics FYTN04: Oral Exam Questions, version ht15 Examples of The questions are roughly ordered by chapter but are often connected across the different chapters. Ordering is as in
More information