index notation

δ

hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 S In Example 4.5 the gradient vector for a function is calculated. ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadM j ∈ 22 0000043701 00000 n v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R =

= a c [email protected]@,  Product WGPbGaamOAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadMgacaWG c

nk a S ∂ k = Product  x

x k These lessons and worksheets look at the rules required at GCSE for index notation. ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr c The worksheets have solutions attached. .  Let 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa = vanishes under the symmetry condition τ⁎a = 0. n beaakiabes7aKnaaBaaaleaacaWGTbGaam4AaaqabaaakeaacqGHii δ B δ The 2 in y 2 is called the power or index, and y is the base. Revue Econ.

x Let x be a (three dimensional) vector and let S be a second order tensor.

The balance equation for the total angular momentum J is obtained if the balance of the internal angular momentum (7.24) is added to (7.23). ≡   λ=

Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa aSbaaSqaaiaaikdacaaIXaaabeaakiabg2da9iabes7aKnaaBaaale 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf k The symbol (:) indicates that the product is contracted twice, as evidenced in the index notation by the presence of the repeated indices k and I. [email protected]@[email protected]@+= 6. 1 hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr



aaqabaaabaGaam4Aaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLd 4AaiaadMgaaeqaaOGaeyypa0JaeyOeI0IaeyicI48aaSbaaSqaaiaa 2det(S)

= x where superscript T denotes transpose of a vector or a matrix. qabaGaamivaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa km

A similar significance has to be assigned to the conservation of the angular momentum as characteristic of the invariance of any field against rotational motions. daWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaam4qamaaBaaaleaaca

aaleaacaWGPbGaeyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoakiab feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn

qaaiaaigdaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacaWGHbWa If x ∈ Rnthen we denote, as usual, the coordinates of x by x1, …, xn. e PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH

0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaa ∈ ) jm

[email protected]@, We [email protected]@,  Transpose  a

ij This is shown in Fig. ANS. 3 eyypa0JaaCyqamaaCaaaleqabaGaamivaaaakiaahkeacaaMc8UaaG 12〈110〉 dislocations split into two partial dislocations, bounding an area of stacking fault (SF).

0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa 0000021861 00000 n θ( v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R kn λ= S gaaeqaaOGaamyyamaaBaaaleaacaWGQbaabeaakiabg2da9iaadgga Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa δ strain relation, σ feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn j

i tensor.   Let [email protected]@[email protected]@+= C ν b→p1,p2,p3 on the {111} plane. and notice that we can replace the remaining ii by kk, σ a ∈ ∈

hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr



= aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca δ = The respective rate equation for Ω may be expressed in the generally valid form (5.81). where [email protected]@[email protected]@+= [email protected]@ ij v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R n 322 A caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadgeadaWgaa jm

b⋅d e

aiaadUgacaWGUbaabeaakiabgkHiTiabes7aKnaaBaaaleaacaWGQb

c The rate equation for J must correspond with the mathematical form (5.81) of every conservation law: 0 (

− ) P GHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9maalaaa 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam qaaiaadQgacaWGPbaabaGaeyOeI0IaaGymaaaakiabg2da9maalaaa

δ B Likewise, atoms in C are at the corresponding centres of the triangles in B or, equivalently, in the centres of the alternate triangles in A. BaaaleaacaWGQbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaamyAai hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr for linear elasticity may be expressed as, σ 2 k n hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 1 ik [email protected]@[email protected]@+=

[email protected]@, 2.4. ) ,          This simple observation lies behind the well-known leading-trailing partial rule usually formulated using the Thompson tetrahedron notation, see Fig.

caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadUgaaeqaaa ggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4UdWMaeyypa0 A

on Cartesian components of vectors and tensors may be expressed very We will also discuss the divergence observed in Fig. 1+ν Gaay5waiaaw2faamaaCaaaleqabaGaamivaaaakmaadmaabaGaamOq il caGaaeqabaGadeaadaaakqaabeqaaiabeU7aSjabg2da9iaadggada a %%EOF v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R da9iaadgeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamyDamaaBaaa kj Q qpdaWadaqaaiaadgeaaiaawUfacaGLDbaadaWadaqaaiaadkeaaiaa v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R ijk GHHjIUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH7oaB i .  Calculate JbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGPaVlabgIGiopaaBa      ≡      ) S A [email protected]@[email protected]@+=

v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R 1. 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf Anticipating. j=1 ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU ∈ ik 333 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf x of two tensors  =1 baGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadUgacaWGPb where the axial vector κ denotes the local production term; the angular flux density JIΩ is a second-order tensor. [email protected]@[email protected]@+= ki

qpcaaMc8UaaGPaVlaaykW7cqGHiiIZdaWgaaWcbaGaamyAaiaadQga

c IZdaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakiabgIGiopaaBaaa

b→p1 and Since the gradient of a function is used while discussing methods of optimum design, we define and discuss its geometrical significance. Here p is the base, and 4 is the index (or power) of p. Expressing numbers with a base and an index is called index notation.

If Y ∈ g and if 1 ij

Grigore, in Handbook of Global Analysis, 2008. δ 33 m

kiabg2da9iaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaOGaamiEam δ 1−2ν

hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 ey4kaSIaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccaWG4bWaaS aMc8UaaGPaVlaaykW7cqaH7oaBcqGH9aqpcaWGHbWaaSbaaSqaaiaa This necessary and sufficient condition is satisfied if the result of all the forces acting upon any volume, as well as of the moments of these forces, is zero for all events. (c) A perfect dislocation loop with Burgers vector P + 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEam , δ b

[email protected]@[email protected]@+= aOqaaiabeU7aSjabg2da9iaahgeacqGHflY1cqGHflY1caaMc8UaaC 0000004330 00000 n i=1 baGaamyAaaqabaGccqGHRaWkdaWcaaqaaiaadcfadaWgaaWcbaGaam k and special symbols for index notation,                                                                 Â,                                                              Â, Substitute hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l [email protected]@[email protected]@+= [ B ] =

gaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab [email protected]@[email protected]@+= aaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVl [email protected]@,  Summation hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9

e ) index notation,  Addition.   jn

= ij 0000018205 00000 n PaVlaadMgacaGGSaGaamOAaiaacYcacaWGRbGaeyypa0JaaGPaVlaa E c×d [email protected]@, Substitute k A hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 As a consequence, every volume element of a fluid that can carry out either a translational or a rotational nonuniform motion will remain in the state of virtual and actual material stability. Hence, the production vector k may be expressed in index notation: To satisfy (7.25), the quantity k must be identified with an axial vector that is proportional to the antisymmetric part JIa of the momentum flux density JI. 1 i 12 〈110〉 dislocations determines their mobility and, through that, the overall plasticity behaviour of FCC metals and alloys. 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf

∈

k    ⇔ = [email protected]@[email protected]@+= a QbGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyi1HS 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf 3 x

kj GccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGa j v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R

startxref

baaSqaaiaaiodaaeqaaaGcbaGaam4yamaaBaaaleaacaaIZaaabeaa

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