Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain
|
|
- Χαρικλώ Μπότσαρης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 PHY 396 K. Solutions for homework set #5. Problem 1a: Starting with eq. 3 proved in class and applying the Leibniz rule, we obtain [ γ κ γ λ, S µν] γ κ [ γ λ, S µν] + [ γ κ, S µν] γ λ γ κ ig λµ γ ν ig λν γ µ + ig κµ γ ν ig κν γ µ γ λ ig λµ γ κ γ ν ig κν γ µ γ λ ig λν γ κ γ µ + ig κµ γ ν γ λ ig λµ γ κ γ ν g κν ig κν γ µ γ λ g λµ ig λν γ κ γ µ g κµ + ig κµ γ ν γ λ g λν S.1 2g λµ S κν 2g κν S µλ 2g λν S κµ + 2g κµ S νλ, where the last line follows from γ κ γ ν g κν 2iS κν proved in class. And since g κλ are c-numbers which commute with everything, [ S κλ, S µν] 2 i [ γ κ γ λ, S µν] ig λµ S κν ig κν S µλ ig λν S κµ + ig κµ S νλ. 4 Q.E.D. Problem 1b: By the multiple commutator formula AKA Hadamard Lemma, for any matrix F e F γ µ e +F γ µ + [ γ µ, F ] [[ γ µ, F ], F ] [[[ γ µ, F ], F ], F ] +. S.2 Let F 2 i Θ αβs αβ so that M D e F and MD 1 e F ; then MD 1 γµ M D follows via eq. S.2 from the multiple commutators of F with the γ µ. It turns out that all such commutators are linear combinations of the γ ν. Indeed, the single commutator here is [ γ µ, F ] i 2 Θ αβ [ γ µ, S αβ] 2 1Θ αβ g µα γ β g µβ γ α Θ αβ g µα γ β Θ µ β γβ, S.3 while the multiple commutators follow by iterating this formula: [[ γ µ, F ], F ] Θ µ λ Θλ νγ ν, [[[ γ µ, F ], F ], F ] Θ µ λ Θλ ρθ ρ νγ ν,.... S.4 1
2 Combining all these commutators as in eq. S.2, we obtain M 1 D γµ M D e F γ µ e +F γ µ + [ γ µ, F ] [[ γ µ, F ], F ] [[[ γ µ, F ], F ], F ] + γ µ + Θ µ νγ ν Θµ λ Θλ νγ ν Θµ λ Θλ ρθ ρ νγ ν + δ ν µ + Θ µ ν + 2 1Θµ λ Θλ ν + 6 1Θµ λ Θλ ρθ ρ ν + γ ν S.5 L µ νγ ν. Q.E.D. Problem 1c: This is an exercise in Leibniz rules for commutators and anticommutators, [A, BC] [A, B]C + B[A, C] {A, B}C B{A, C}, {A, BC} [A, B]C + B{A, C} {A, B}C B[A, C]. S.6 Applying these formulae to Dirac matrices we immediately obtain [γ ρ, γ µ γ ν ] {γ ρ, γ µ } γ ν {γ ρ, γ ν } γ µ 2g ρµ γ ν 2g ρν γ µ S.7 and hence { γ ρ, γ λ γ µ γ ν} { γ ρ, γ λ} γ µ γ ν γ λ [γ ρ, γ µ γ ν ] 2g ρλ γ µ γ ν γ λ 2g ρµ γ ν 2g ρν γ µ 2g ρλ γ µ γ ν 2g ρµ γ λ γ ν + 2g ρν γ λ γ µ, S.8 [γ ρ, γ κ γ λ γ µ γ ν] {γ ρ, γ κ } γ λ γ µ γ ν γ κ { γ ρ, γ λ γ µ γ ν} 2g ρκ γ λ γ µ γ ν γ κ 2g ρλ γ µ γ ν 2g ρµ γ λ γ ν + 2g ρν γ λ γ µ 2g ρκ γ λ γ µ γ ν 2g ρλ γ κ γ µ γ ν + 2g ρµ γ κ γ λ γ ν 2g ρν γ κ γ λ γ µ, S.9 etc., etc. 2
3 Finally, applying the Leibniz rule to the left side of the bracket, we obtain [ γ σ γ ρ, γ λ γ µ γ ν] γ σ { γ ρ, γ λ γ µ γ ν} { γ σ, γ λ γ µ γ ν} γ ρ γ σ 2g ρλ γ µ γ ν 2g ρµ γ λ γ ν + 2g ρν γ λ γ µ 2g σλ γ µ γ ν 2g σµ γ λ γ ν + 2g σν γ λ γ µ γ ρ S.1 and hence [ S ρσ, γ λ γ µ γ ν] ig σλ γ ρ γ µ γ ν + ig σµ γ λ γ ρ γ ν + ig σν γ λ γ µ γ ρ ig ρλ γ σ γ µ γ ν ig ρµ γ λ γ σ γ ν ig ρν γ λ γ µ γ σ. S.11 Problem 1d: γ α γ α 1 2 {γα, γ β }g αβ g αβ g αβ 4; γ α γ ν γ α 2g αν γ α γ ν γ α γ α 2γ ν γ ν 4 2γ ν ; γ α γ µ γ ν γ α 2g αµ γ ν γ α γ µ γ α γ ν γ α 2γ ν γ µ γ µ 2γ ν 2{γ ν, γ µ } 4g µν ; γ α γ λ γ µ γ ν γ α 2g αλ γ µ γ ν γ α γ λ γ α γ µ γ ν γ α 2γ µ γ ν γ λ γ λ 4g µν 2 γ µ γ ν 2g µν γ λ 2γ ν γ µ γ λ. S.12 Problem 2a: γ µ γ ν ±γ ν γ µ where the sign is + for µ ν and otherwise. Hence for any product Γ of the γ matrices, γ µ Γ 1 n Γγ µ, where n is the number of γ ν µ factors of Γ. For Γ γ 5 iγ γ 1 γ 2 γ 3, n 3 for any µ, 1, 2, 3, hence γ µ γ 5 γ 5 γ µ. As to the spin matrices, γ 5 γ µ γ ν γ µ γ 5 γ ν +γ µ γ ν γ 5 and therefore γ 5 S µν +S µν γ 5. 3
4 Problem 2b: First, γ 5 iγ γ 1 γ 2 γ 3 iγ 3 γ 2 γ 1 γ +iγ 3 γ 2 γ 1 γ +iγ 3 γ 2 γ 1 γ 1 3 i γ γ 3 γ 2 γ i γ γ 1 γ 3 γ i γ γ 1 γ 2 γ 3 S.13 +iγ γ 1 γ 2 γ 3 +γ 5. Second, γ 5 2 γ 5 γ 5 iγ γ 1 γ 2 γ 3 iγ 3 γ 2 γ 1 γ γ γ 1 γ 2 γ 3 γ 3 γ 2 γ 1 γ +γ γ 1 γ 2 γ 2 γ 1 γ γ γ 1 γ 1 γ +γ γ +1. S.14 Problem 2c: Since the four Dirac matrices γ, γ 1, γ 2, γ 3 all anticommute with each other, ɛ κλµν γ κ γ λ γ µ γ ν γ [ γ 1 γ 2 γ 3] 24γ γ 1 γ 2 γ 3 24iγ 5. S.15 To prove the other identity, we note that a totally antisymmetric product γ [κ γ λ γ µ γ ν] vanishes unless the Lorentz indices κ, λ, µ, ν are all distinct which makes them, 1, 2, 3 in some order. For such indices, the anticommutativity of Dirac matrices implies γ κ γ λ γ µ γ ν ɛ κλµν γ γ 1 γ 2 γ 3 and hence γ [κ γ λ γ µ γ ν] 24 ɛ κλµν γ γ 1 γ 2 γ 3 24 iɛ κλµν γ 5. S.16 Problem 2d: 6iɛ κλµν γ κ γ γ κ γ [κ γ λ γ µ γ ν] 4 1 γ κ γ κ γ [λ γ µ γ ν] γ [λ γ κ γ µ γ ν] + γ [λ γ µ γ κ γ ν] γ [λ γ µ γ ν] γ κ 4γ [λ γ µ γ ν] + 2γ [λ γ µ γ ν] + 4g [λµ γ ν] + 2γ [ν γ µ γ λ] γ[λ γ µ γ ν] γ [λ γ µ γ ν]. S.17 4
5 Problem 2e: Proof by inspection: In the Weyl basis and 2 2 block form, the 16 matrices are 1 1 +σ i 1 4 4, γ, γ i 1 1 σ i, 1 σ k 2 γ[i γ j] iɛ ijk 1 σ i σ k, 2 γ [ γ i] +σ i, S.18 1 σ i 1 γ 5 γ, γ 5 γ 1 +1 σ i, γ 5, +1 and their linear independence is self-evident. Since there are only 16 independent 4 4 matrices altogether, any such matrix Γ is a linear combination of the matrices S.18. Q.E.D. Algebraic Proof: Without making any assumption about the matrix form of the γ µ operators, let us consider the Clifford algebra γ µ γ ν + γ ν γ µ 2g µν. Because of these anticommutation relations, one may re-order any product of the γ s as ±γ γ γ 1 γ 1 γ 2 γ 2 γ 3 γ 3 and then further simplify it to ±γ or 1 γ 1 or 1 γ 2 or 1 γ 3 or 1. The net result is up to a sign or ±i factor one of the 16 operators 1, γ µ, 1 2 γ[µ γ ν], 1 6 γ[λ γ µ γ ν] iɛ λµνρ γ 5 γ ρ cf. d or i 24 γ[κ γ λ γ µ γ ν] ɛ κλµν γ 5 cf. c. Consequently, any operator Γ algebraically constructed of the γ µ s is a linear combination of these 16 operators. Incidentally, this proof explains why the Dirac matrices are 4 4 in d 4 spacetime dimensions: the 16 linearly-independent products of Dirac matrices require matrix size to be Technically, we may also use matrices of size 4n 4n, but then we would have γ µ γ µ n n, and ditto for all their products. Physically, this means combining the Dirac spinor index with some other index i 1,..., n which has nothing to do with Lorentz symmetry. Nobody wants such an index confusion, so physicists always stick to 4 4 Dirac matrices in 4 spacetime dimensions. Problems 3f and 3g: As in part a of this problem, γ µ Γ 1 n Γγ µ where n is the number of γ ν µ factors of Γ. For Γ γ γ 1 γ d 1 moduli the overall sign or ±i factor, each γ µ appear once, with 5
6 the remaining d 1 factors being γ ν µ. Thus, n d 1 and γ µ Γ 1 d 1 Γγ µ : for even spacetime dimensions d, Γ anticommutes with all d γ µ matrices, but for odd d, Γ commutes with all the γ µ instead of anticommuting. Now consider the dimension of the Clifford algebra made out of Dirac matrix products and their linear combinations. As in part e, any product can be re-arranged as ±γ or 1 γ 1 or 1 γ d 1 or 1, and the number of distinct products of this type is clearly 2 d. Alternatively, we count a single unit matrix, d 1 d of γ µ matrices, d 2 of independent 1 2 γ [µ γ ν] products, d 3 of 1 6 γ [λ γ µ γ ν] products, etc., etc., all the way up to d d 1 d of γ [β γ ω] ɛ αβ ω γ α Γ, and d d 1 of γ [α γ ω] ɛ αβ ω Γ; the net number of all such matrices is d d k k 2 d. For even dimensions d, all the 2 d matrix products are distinct, and we need matrices of size 2 d/2 2 d/2 to accommodate them. But for odd d, the Γ matrix commutes with the entire Clifford algebra since it commutes with all the γ µ, so to avoid redundancy we should set Γ 1 or Γ 1. Consequently, any product of k distinct Dirac matrices becomes identical up to a sign or ±i with the product of the other d k matrices, for example γ γ k 1 i?? γ k γ d 1. This halves the net dimension of the Clifford algebra from 2 d to 2 d 1 and calls for matrix size 2 d 1/2 2 d 1/2. Thus, the Dirac matrices in 2 or 3 space time dimensions are 2 2, in 4 or 5 dimensions they are 4 4, in 6 or 7 dimensions 8 8, in 8 or 9 dimensions 16 16, in 1 or 11 dimensions 32 32, etc., etc. Problem 3: Eq. S.18 gives explicit form of various products of Dirac γ matrices in the Weyl convention. In particular, the second like of S.18 spells out 1 2 γ[µ γ ν] 1 2 [γµ, γ ν ] 2iS µν. In terms of the S µν matrices themselves, S ij ɛijk 2 σ k σ k, S i S i 1 2 iσ i +iσ i. S.19 Note that all these matrices are block-diagonal, hence all the ML exp i 2 Θ µνs µν ma- 6
7 trices are also block-diagonal as in eq. 6. Specifically, let Θ ij ɛ ijk a k, Θ i Θ i b i S.2 for some real 3-vectors a and b, then Θ µν S µν a + ib σ 1 2 a ib σ S.21 and hence M D exp i 2 Θ µνs µν exp 12 ia + b σ exp 1 2 ia b σ. S.22 Moreover, comparing the diagonal blocks here to eqs. 8 and 9 from the last homework, we immediately see that M D ML M R where M L and M R denoted M and M in the last homework represent the same Lorentz symmetry L expθ in 2 and 2 irreducible multiplets of the SL2, C Spin3, 1. 6 In particular, a pure rotation through angle ϕ around the z axis 1 L µ cos ϕ sin ϕ ν sin ϕ cos ϕ exp +ϕ ϕ 1 S.23 corresponds to a,, ϕ, b,,, hence M L M R exp i 2 ϕσz S.24 More generally, for a pure rotation around any axis n we have a ϕn, b, hence M L M R exp i ϕ 2 n σ. 7 7
8 For a pure Lorentz boost in the z direction γ +βγ 1 +r L µ 1 ν 1 exp +βγ γ +r S.25 where r is the rapidity of the boost; it is related to the β v/c and γ 1/ 1 β 2 parameters as β tanhr and γ coshr. For successive boosts in the same direction, their rapidities add up. Anyhow, the boost S.25 corresponds to a,, while b,, r, where the minus sign comes from Θ i +Θi Θi b i. Similarly, a Lorentz boost in a general direction n has a while b rn, hence M L exp r 2 n σ while M R exp + r 2 n σ. S.26 In terms of β and γ parameters of the boost, M L 2 exp rn σ coshr sinhr n σ S.27 and likewise hence γ βγ n σ M R 2 exp +rn σ coshr + sinhr n σ γ + βγ n σ, S.28 M L γ 1 βn σ, M R γ 1 + βn σ. 8 Equivalently, for a boost L that turns a particle at rest to a moving particle with energymomentum p µ E, p, m ML L E p σ, m MR L E + p σ. S.29 We will need this formula in problem 4 below parts b and c. 8
9 Problem 4a: For p, p +m and p m mγ 1. Hence, up, s satisfy γ 1u, or in the Weyl basis u u ζ S.3 ζ where ζ is an arbitrary two-component spinor. It s normalization follows from u u 2ζ ζ, so if we want u u 2E p 2m for p we need ζ ζ m. Equivalently, we want ζ mξ and hence u as in eq. 1 for a conventionally normalized spinor ξ, ξ ξ. Note that there are two independent choices of ξ, normalized to ξ sξ s δ s,s so that u, su, s 2mδ s,s 2E p δ s,s. They correspond to two spin states of a p electron. In terms of the spin vector, S 2 1ξ sσξ s. Problem 4b: The Dirac equation is Lorentz-covariant, so we may obtain solutions for all p µ +E p, p by simply Lorentz-boosting the solutions 1 from the rest frame where p µ +m,. Thus, up, s M D L up, s ML M R m ξs m ξs m ML ξ s m MR ξ s S.31 where M D, M L, and M R are respectively Dirac-spinor, LH Weyl-spinor, and RH Weyl-spinor representations or the Lorentz boost m, E, p. Eq. S.29 above gives the explicit form of the M L and M R matrices for this boost. Plugging them into eq. S.31 immediately gives us up, s E p σ ξ s. 11 E + p σ ξ s Q.E.D. Problem 4c: The negative-frequency solutions e +ipx vp, s have Dirac spinors v satisfying p + mv. 9
10 For a particle at rest, p µ +m,, this equation becomes mγ + 1v, or in the Weyl basis ηs vp, s vp, s m η s S.32 for a pair of independent two-component spinors η s. As in part a, the m factor translates the normalization v p, svp, s 2E p δ s,s 2mδ s,s for p to η sη s δ s,s. For p we proceed similarly to part b, namely Lorentz-boost the rest-frame solution S.32 to the frame where p µ +e p, p: ML + m ηs V p, s M D L V p, s M R m η s + m ML ξ s m M R ξ s + E p σ η s 11 E + p σ η s. Q.E.D. Problem 4d: Physically, the holes in a Fermi sea have opposite free energies, momenta, and spins from the missing particle states. I shall explain this point in class. A positron is a hole in the Dirac sea of electrons, so it should have opposite p µ E, p from the missing electron states that s why the vp, s spinors accompany the e +ipx e +iet ipx plane waves instead of the e ipx E ietipx factors of the up, s spinors. Likewise, the positron should have opposite spin state from the missing electron, and that s why the η s should have opposite spin from the ξ s. More accurately, the η s should carry an opposite spin vector η ssη s from ξ ssξ s. The solution to this spin relation is η s σ 2 ξs where denotes complex conjugation. Indeed, using the Pauli matrix relation σ 2 σ σ 2 σ from the last homework, we see that η σ 2 ξ η ξ σ 2 η ση ξ σ 2 σσ 2 ξ ξ σ 2 σ σ 2 ξ ξ σξ ξ σξ 1
11 η Sη 1 2 η ση ξ Sξ. S.33 Also, for η s σ 2 ξ s we have + E p σ η s + E p σ vp, s σ 2 ξs E + p σ η s E + p σ σ 2 ξs +σ2 σ 2 E p σ σ 2 ξs σ 2 σ 2 E + p σ σ 2 ξ s using σ 2 σσ 2 σ σ 2 E p σ σ 2 +σ2 E + p σ ξ s σ 2 E p σ ξ s +σ2 + E p σ ξ s σ 2 + E + p σ ξ s γ 2 u p, s. E ± p σ S.34 Similarly, up, s γ 2 v p, s. Indeed, γ 2 v p, s γ 2 γ 2 u p, s γ 2 γ 2 up, s up, s since γ 2 γ 2 γ 2 γ Problem 4e: The 3D spinors ξ λ of definite helicity λ 1 2 satisfy p σξ p ξ. S.35 Plugging these ξ λ into the positive-energy Dirac spinors 11, we obtain E ± p ξ up, λ 1 2. S.36 E p ξ In the ultra-relativistic limit E p m the square roots here simplify to E + p 2E and E p by comparison with the other root. we have E p and hence 11
12 E + p 2E; by comparison, E p. Consequently, eq. S.36 simplifies to up, L 2E ξl, up, R 2E ξ R. S.37 In other words, positive-energy ultra-relativistic Dirac spinors of definite helicity are chiral dominated by the LH Weyl components for the left helicity or by the RH Weyl components for the right helicity. Now consider the negative-energy Dirac spinors 12. Because the η s spinors have exactly opposite spin from the ξ s, their helicities are also opposite and hence p ση ± p η S.38 note opposite sign from eq. S.35. Therefore, the negative-energy Dirac spinors v of definite helicity are and in the ultra-relativistic limit they become vp, L 2E + E p η vp, λ 1 2, S.39 E ± p η η L, vp, R + 2E ηr. S.4 Again, the ultra-relativistic spinors are chiral, but this time the chirality is opposite from the helicity the left-helicity spinor has dominant RH Weyl components while the right-helicity spinor has dominant LH Weyl components Problem 4f: First, let s check the normalization conditions for the Dirac spinors 11 and 12 for general moments: u p, sup, s ξ s E p σ 2 + E + p σ 2 ξ s ξ s2eξ s 2Eδ s,s, v p, svp, s η s + E p σ 2 + E + p σ 2 η s η s+2eη s 2Eδ s,s. S.41 Note that in the second equation η sη s ξ s σ 2 σ 2 ξ s ξ sξ s δs,s. 12
13 Now consider the Lorentz invariant products ūu and vv. The ū and v are given by ūp, s u p, sγ E p σ ξ s 12 2 E + p σ ξ s ξ s E + p σ, ξ s E p σ, + E p σ vp, s v p, sγ η s 12 2 E + p σ η s S.42 η s E + p σ, +η s E p σ. Consequently, ūp, s up, s ξ s E + p σ E p σ ξ s + ξ s E p σ E + p σ ξ s S.43 2m ξ sξ s 2mδ s,s because E + p σ E p σ E p σ E + p σ E 2 p σ 2 E 2 p 2 m. S.44 Likewise, vp, s vp, s η s E + p σ E p σ η s η s E p σ E + p σ η s 2m η sη s 2mδ s,s. S.45 Problem 4g: In matrix notations column row matrix, we have up, s up, s E pσ ξs ξ E + pσ s E + pσ, ξ s E pσ ξs E pσ ξs ξ s E + pσ ξs ξ s E + pσ E + pσ E pσ ξ s ξ s E + pσ ξ s ξ s E pσ E pσ, S.46 13
14 vp, s vp, s + E pσ ηs E + pσ η s E pσ η s η s η s E + pσ, +η s E pσ E + pσ + E pσ η s η s E pσ S E + pσ η s η s E + pσ E + pσ η s η s E pσ Summing over two spin polarizations replaces ξ s ξ s with s ξ s ξ s η s η s with s η s η s Consequently, and likewise up, s up, s s [ E pσ s ξ s ξ s ] E + pσ E pσ [ E s ξ s ξ s] pσ ] E + pσ E + pσ [ E s ξ s ξ s] pσ [ E + pσ s ξ s ξ s E pσ E + pσ E pσ E pσ E + pσ E + pσ E + pσ E pσ m E pσ E + pσ m m E γ p γ p + m. vp, s vp, s s [ E pσ s η s η s ] E + pσ + E pσ [ E s η s η s] pσ + E + pσ [ ] E s η s η s + pσ E + pσ [ E s η s η s] pσ E pσ E + pσ + E pσ E pσ + E + pσ E + pσ E + pσ E pσ m E pσ E + pσ m m E γ p γ S.48 p m. S.49 Q.E.D. 14
Now, suppose the electron field Ψ(x) satisfies the covariant Dirac equation (i D m)ψ = 0.
PHY 396 K. Solutions for homework set #7. Problem 1a: γ α γ α 1 {γα, γ β }g αβ g αβ g αβ 4; S.1 γ α γ ν γ α γ α γ ν g να γ ν γ α γ α γ ν γ ν γ α γ α 4 γ ν ; S. γ α γ µ γ ν γ α γ α γ µ g µα γ µ γ α γ ν
Διαβάστε περισσότεραDirac Matrices and Lorentz Spinors
Dirac Matrices and Lorentz Spinors Background: In 3D, the spinor j = 1 representation of the Spin3) rotation group is constructed from the Pauli matrices σ x, σ y, and σ k, which obey both commutation
Διαβάστε περισσότερα= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.
PHY 396 T: SUSY Solutions for problem set #1. Problem 2(a): First of all, [D α, D 2 D α D α ] = {D α, D α }D α D α {D α, D α } = {D α, D α }D α + D α {D α, D α } (S.1) = {{D α, D α }, D α }. Second, {D
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραHomework 4 Solutions Weyl or Chiral representation for γ-matrices. Phys624 Dirac Equation Homework 4
Homework 4 Solutions 4.1 - Weyl or Chiral representation for γ-matrices 4.1.1: Anti-commutation relations We can write out the γ µ matrices as where ( ) 0 σ γ µ µ = σ µ 0 σ µ = (1, σ), σ µ = (1 2, σ) The
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραDirac Matrices and Lorentz Spinors
Dirac Matrices and Lorentz Spinors Background: In 3D, the spinor j = 2 1 representation of the Spin3) rotation group is constructed from the Pauli matrices σ x, σ y, and σ z, which obey both commutation
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραDirac Trace Techniques
Dirac Trace Techniques Consider a QED amplitude involving one incoming electron with momentum p and spin s, one outgoing electron with momentum p and spin s, and some photons. There may be several Feynman
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότερα4 Dirac Equation. and α k, β are N N matrices. Using the matrix notation, we can write the equations as imc
4 Dirac Equation To solve the negative probability density problem of the Klein-Gordon equation, people were looking for an equation which is first order in / t. Such an equation is found by Dirac. It
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραOrbital angular momentum and the spherical harmonics
Orbital angular momentum and the spherical harmonics March 8, 03 Orbital angular momentum We compare our result on representations of rotations with our previous experience of angular momentum, defined
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραGeneral 2 2 PT -Symmetric Matrices and Jordan Blocks 1
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 Qing-hai Wang National University of Singapore Quantum Physics with Non-Hermitian Operators Max-Planck-Institut für Physik komplexer Systeme Dresden,
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραTutorial problem set 6,
GENERAL RELATIVITY Tutorial problem set 6, 01.11.2013. SOLUTIONS PROBLEM 1 Killing vectors. a Show that the commutator of two Killing vectors is a Killing vector. Show that a linear combination with constant
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότερα( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότερα1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραLecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
Διαβάστε περισσότεραPHY 396 K/L. Solutions for problem set #12. Problem 1: Note the correct muon decay amplitude. The complex conjugate of this amplitude
PHY 396 K/L. Solutions for problem set #. Problem : Note the correct muon decay amplitude M(µ e ν µ ν e = G F ū(νµ ( γ 5 γ α u(µ ū(e ( γ 5 γ α v( ν e. ( The complex conjugate of this amplitude M = G F
Διαβάστε περισσότεραDerivations of Useful Trigonometric Identities
Derivations of Useful Trigonometric Identities Pythagorean Identity This is a basic and very useful relationship which comes directly from the definition of the trigonometric ratios of sine and cosine
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραLecture 15 - Root System Axiomatics
Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραSOLVING CUBICS AND QUARTICS BY RADICALS
SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with
Διαβάστε περισσότεραHigher Derivative Gravity Theories
Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)
Διαβάστε περισσότεραPhys624 Quantization of Scalar Fields II Homework 3. Homework 3 Solutions. 3.1: U(1) symmetry for complex scalar
Homework 3 Solutions 3.1: U(1) symmetry for complex scalar 1 3.: Two complex scalars The Lagrangian for two complex scalar fields is given by, L µ φ 1 µ φ 1 m φ 1φ 1 + µ φ µ φ m φ φ (1) This can be written
Διαβάστε περισσότεραTMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότερα1 Lorentz transformation of the Maxwell equations
1 Lorentz transformation of the Maxwell equations 1.1 The transformations of the fields Now that we have written the Maxwell equations in covariant form, we know exactly how they transform under Lorentz
Διαβάστε περισσότεραSymmetric Stress-Energy Tensor
Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραDERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response
Διαβάστε περισσότεραNew bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραSymmetry. March 31, 2013
Symmetry March 3, 203 The Lie Derivative With or without the covariant derivative, which requires a connection on all of spacetime, there is another sort of derivation called the Lie derivative, which
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραg-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
Διαβάστε περισσότεραdim(u) = n 1 and {v j } j i
SOLUTIONS Math B4900 Homework 1 2/7/2018 Unless otherwise specified, U, V, and W denote vector spaces over a common field F ; ϕ and ψ denote linear transformations; A, B, and C denote bases; A, B, and
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότερα= λ 1 1 e. = λ 1 =12. has the properties e 1. e 3,V(Y
Stat 50 Homework Solutions Spring 005. (a λ λ λ 44 (b trace( λ + λ + λ 0 (c V (e x e e λ e e λ e (λ e by definition, the eigenvector e has the properties e λ e and e e. (d λ e e + λ e e + λ e e 8 6 4 4
Διαβάστε περισσότερα