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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

MATH 332: Vector and Tensor Analysis Vector Algebra

Symmetry δjk = δkj

Scalar Product

Unity Matrix

A · B = |B||A| cos θ Commutativity A·B=B·A Magnitude



 1 0 0 I = (δik ) =  0 1 0  0 0 1 Transformation of Rectangular Coordinates

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|A| = A · A x0i = αi0 k xk + x(0) i Vector Product A × B = n | A | | B | sin θ Anti-commutativity

Summation from 1 to 3 is assumed over all repeated indices Vector Form

A × B = −B × A A×A=0 Scalar Triple Product (A × B) · C Cyclic symmetry (A × B) · C = (B × C) · A = (C × A) · B

 x01 x0 = (x0i ) =  x02  , x03 

 x1 x = (xi ) =  x2  x3 

Transformation matrix 

 α10 1 α10 2 α10 3 α = (αi0 k ) =  α20 1 α20 2 α20 3  α30 1 α30 2 α30 3

Matrix Form of the Transformation x0 = αx + x0

Vector Triple Product A × (B × C) = B(A · C) − C(A · B)

Orthonormal Basis i k ,

(k = 1, 2, 3)

ik · ij = δkj Kronecker Symbol δik =

δki

=

δik

 =

1 if i = k 0 if i 6= k

Orientation of Orthonormal Basis: right-handed if ( i 1 × i 2 ) · i 3 = +1 left-handed if ( i 1 × i 2 ) · i 3 = −1

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

Transformation of Orthonormal Basis i 0j = αj 0 k i k ,

Levi-Civita (Alternating) Symbol  +1 if (i, j, k) = (1, 2, 3),     (2, 3, 1), (3, 1, 2)   −1 if (i, j, k) = (2, 1, 3), εijk =   (3, 2, 1), (1, 3, 2)     0 otherwise

i k = αj 0 k i 0j

αj 0 k = cos( i 0j , i k ) Orthogonality condition αi0 k αj 0 k = δij ,

αk0 i αk0 j = δij

εijk = εijk

Matrix Form of Orthogonality Condition Anti-symmetry

ααT = I T means transposition of a matrix (replacement of rows by columns) Proper transformation (no change of orientation) det(αj 0 k ) = 1

εijk = −εjik = −εikj = −εkji Cyclic symmetry εijk = εjki = εkij Orthonormal basis

Improper transformation (changes orientation) i j × i k = εjkl i l det(αj 0 k ) = −1 ( i j × i k ) · i l = εjkl Cartesian Vectors

Vector Product in Cartesian Components Ak = A · i k

A = Ak i k ,

(A × B)i = εijk Aj Bk

Scalar Product in Cartesian Components Scalar Triple Product A · B = Ai Bi (A × B) · C = εijk Ai Bj Ck

|A|2 = Ai Ai Vector Product

Tensor Notation



 i1 i2 i3 A × B = det  A1 A2 A3  B1 B2 B3

δii = δii = 3

Scalar Triple Product 



A1 A2 A3  (A × B) · C = det B1 B2 B3  C1 C2 C3

δij Aj = Ai δ ij Ai Bj = Ai Bi = A · B

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

A0 = αA εijj = εjij = εjji = 0 εijk δij = εijk δik = εijk δjk = 0

Matrix Form of a 2-tensor   A11 A12 A13 A = (Aik ) =  A21 A22 A23  A31 A32 A33

εijk Aj Ak = εijk Ai Ak = εijk Ai Aj = 0 A0 = αAαT εijk εmnl = δim δjn δkl + δjm δkn δil + δkm δin δjl −

δim δkn δjl

−

δjm δin δkl

−

δkm δjn δil

εijk εmnk = δim δjn − δjm δin εijk εmjk = 2δim εijk εijk = 6

Stress Tensor pik Stress pi = pik nk ni unit normal Moment of Inertia Tensor Iik =

N X

h i (j) (j) (j) (j) mj δik xl xl − xi xk

j=1

Angular Momentum

Cartesian Tensors (tensors in rectangular coordinates)

Li = Iik ωk

Scalar: 0-tensor ϕ Vector: 1-tensor Ai 2-tensor Aik n-tensor Ai1 ...in

ωk angular velocity Deformation Tensor   ∂uk ∂ul ∂ul 1 ∂ui + + uik = 2 ∂xk ∂xi ∂xi ∂xk

Transformation Laws

ui displacement vector

ϕ0 = ϕ A0i = αi0 k Ak A0ij = αi0 k αj 0 l Akl A0i1 ...in

= αi01 j1 · · · αi0n jn Aj1 ...jn

Matrix Form of the Transformation Laws Vector   A1 A = (Ai ) =  A2  A3

Rate of Deformation Tensor   1 ∂vi ∂vk vik = + 2 ∂xk ∂xi vi velocity vector field Isotropic Tensors (built from δik only; no preferred directions) Isotropic 2-tensor Aik = pδik

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

Isotropic 4-tensor

Antisymmetrization 1 T[ij] = (Tik − Tki ) 2

Aiklm = pδik δlm + ρδil δkm + λδim δkl

Decomposition of 2-Tensor

Tensor Algebra

Tik = T(ik) + T[ij]

Tensor Product Cik = Ai Bk ,

Cijkl = Aij Bkl

Contraction Djkl = Ciijkl Trace

Duality (Equivalence of antisymmetric 2-tensor to an axial vector) 1 A˜i = εijk Ajk , 2

Aij = εijk A˜k

A = Aii A˜1 = A23 ,

Inner Product Ci = Aik Bk ,

A˜2 = A31 ,

A˜3 = A12 ,

Dij = Cijkl Bkl Principal Axes

Symmetric Tensors Sij = Sji   S11 S12 S13 (Sik ) =  S12 S22 S23  S13 S23 S33 Anti-symmetric Tensors

Eigenvalues λ(r) (characteristic values) and Eigenvectors n (r) (characteristic or principal directions) (r) (r) Tik nk = λ(r) ni | n (r) | = 1 Characteristic Equation det(Tik − λδik ) = 0

Aij = −Aji 

 0 A12 A13 0 A23  (Aik ) =  −A12 −A13 −A23 0

 T11 − λ T12 T13 T22 − λ T23  = 0 det  T21 T31 T32 T33 − λ

Contraction of antisymmetric tensor

λ3 − I1 λ2 + I2 λ − I3 = 0

Aii = 0 Symmetrization 1 T(ij) = (Tik + Tki ) 2



Invariants of a 2-Tensor: Ik I1 = Tii = T11 + T22 + T33     T22 T23 T11 T12 I2 = det + det T32 T33 T21 T22

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas



 T11 T13 + det T31 T33   T11 T12 T13 I3 = det(Tik ) = det  T21 T22 T23  T31 T32 T33

Curvilinear Coordinates q i = q i ( r ),

(i = 1, 2, 3)

Cartesian Coordinates xi = xi (q)

The eigenvalues λr , (r = 1, 2, 3), of a symmetric Radius vector 2-tensor are real r(q) = xk (q) i k = x1 (q) i 1 + x2 (q) i 2 + x3 (q) i 3 A symmetric 2-tensor has three orthogonal principal axes n(r) , (r = 1, 2, 3) Basis (tangent vectors to coordinate curves) n (r) · n (p) = δrp ei = In the principal axes a symmetric 2-tensor Tik has Transformation of basis diagonal matrix e 0j = αk j 0 e k ,   λ1 0 0 0 (Tik ) =  0 λ2 0  αi k αk j 0 = δji , 0 0 λ3

∂r ∂q i

0

e k = αj k e 0j 0

αi k0 αk j = δji

Orientation: Decomposition in terms of orthonormal eigenvecright-handed if ( e 1 × e 2 ) · e 3 > 0 tors n (r) and eigenvalues λ(r) and left-handed if ( e 1 × e 2 ) · e 3 < 0 Tik =

3 X

(r) (r)

λ(r) ni nk

r=1

Traceless Tensors (Deviators) 1 T ik = Tik − δik T 3 where T = Tjj (trace) T = T ii = 0 Decomposition

Reciprocal Basis e j · e k = δkj ej =

ek × el ( e 1 × e 2) · e 3

where (j, k, l) = (1, 2, 3), (2, 3, 1), (3, 1, 2) Contravariant Components A = Ai e i ,

Ai = A · e i

Covariant components 1 Tik = T ik + δik Tjj 3

A = Ai e i ,

Ai = A · e i

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

ds2 = gik dq i dq k

Transformation of components 0

A0i = αi k Ak ,

A0i = αk i0 Ak

Volume Element √ dV = G dq 1 dq 2 dq 3 ,

G = det(gik )

Metric Tensor

gik = e i · e k =

∂r ∂r · , ∂q i ∂q k

g ik = e i · e k

Contravariant components Aik Covariant components Aik Mixed components Ai k , Ai k Relations

Symmetry g ik = g ki

gik = gki ,

Tensors in Curvilinear Coordinate System

Aik = gin An k = gkn Ai n = gin gkm Anm

gik g kn = δin

Aik = g in An k = g kn Ai n = g in g km Anm

Determinant of the Metric Tensor det(g ik ) =

G = det(gik ) ,

1 G

Ai k = g in Ank = gkn Ain etc

ik

The matrix g is the inverse matrix of the matrix Orthogonal Coordinate System gik given by g ik = (−1)ik

∂r ∂r · = 0, ∂q i ∂q k

det M ik G

where M ik is a 2 × 2 matrix obtained from the 3 × 3 matrix gij by removing the i-th row and the k-th column

Basis e i · e k = e i · e k = 0 if i 6= k | e i | = hi ,

Relations between components Ai = gik Ak ,

Ai = g ik Ak

if i 6= k

| e i| =

1 hi

Metric gik = 0,

if i 6= k

gii = h2i (no summation!) Displacement dr =

∂r i dq ∂q i

Line Element (Arc Length) ds2 = d r · d r =

∂r ∂r i k · dq dq ∂q i ∂q k

g ik = 0, if i 6= k 1 g ii = 2 (no summation!) hi Metric Coefficients ∂r hi = i ∂q

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

s hi =

∂x1 ∂q i

2

 +

∂x2 ∂q i

2

 +

∂x3 ∂q i

2

Relation between covariant and contravariant components Ai =

h2i Ai

x1 = ρ cos ϕ,

x2 = ρ sin ϕ,

x3 = z

Radius Vector r = ρ cos ϕ i 1 + ρ sin ϕ i 2 + z i 3 Orthonormal basis

(no summation!)

e ρ = cos ϕ i 1 + sin ϕ i 2 e ϕ = − sin ϕ i 1 + cos ϕ i 2

Orthonormal Basis

ez = i3

∂r ∂q i e i = e i = ∂r ∂q i

Line Element ds2 = (dρ)2 + ρ2 (dϕ)2 + (dz)2 Volume Element

Line Element 2

ds =

h21

dV = ρ dρ dϕ dz 1 2

(dq ) +

h22

2 2

(dq ) +

h23

3 2

(dq )

Volume Element

Metric Coefficients hρ = 1,

hϕ = ρ,

hz = 1

dV = h1 h2 h3 dq 1 dq 2 dq 3 Spherical Coordinates Cartesian Coordinates Line Element ds2 = (dx1 )2 + (dx2 )2 + (dx3 )2 Volume Element dV = dx1 dx2 dx3

r ≥ 0,

0 ≤ ϕ < 2π p q x21 + x22 2 2 2 r = x1 + x2 + x3 , tan θ = , x3 x2 tan ϕ = x1 x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ

Metric Coefficients h1 = h2 = h3 = 1

0 ≤ θ ≤ π,

Radius Vector r = r sin θ cos ϕ i 1 + r sin θ sin ϕ i 2 + r cos θ i 3

Cylindrical Coordinates ρ ≥ 0, 0 ≤ ϕ < 2π, −∞ < z < ∞ q x2 ρ = x21 + x22 , tan ϕ = , z = x3 x1

Orthonormal Basis e r = sin θ cos ϕ i 1 + sin θ sin ϕ i 2 + cos θ i 3 e θ = cos θ cos ϕ i 1 + cos θ sin ϕ i 2 − sin θ i 3

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

e ϕ = − sin ϕ i 1 + cos ϕ i 2

Speed dr ds |v| = = dt dt

Line Element ds2 = (dr)2 + r2 (dθ)2 + r2 sin2 θ(dϕ)2

Acceleration

Volume Element

a=

dV = r2 sin θ dr dθ dϕ

Arc Length (Line Element)

Metric Coefficients hr = 1,

dv d2 r = 2 dt dt

ds = |v|dt hθ = r,

hϕ = r sin θ

Fields in Cartesian Coordinates

Vector And Tensor Analysis

Partial derivatives ∂i =

Functions of Single variable

∂ ∂xi

Nabla (Del) Operator Product Rules dϕ dA d (ϕ A ) = A +ϕ dt dt dt d dB dA (B · A) = ·A+B· dt dt dt dB dA d (B × A) = ×A+B× dt dt dt Trajectory r(t) = xk (t) ik ,

a≤t≤b

Velocity dr v= dt Unit Tangent Vector dr dr u = = dt ds dr dt

∇ = ik ∂k = i1 ∂1 + i2 ∂2 + i3 ∂3 Gradient grad f = ∇f = ik ∂k f ( grad f )i = ∂i f Directional Derivative df dr = · gradf = u · gradf ds ds Flow Lines

dr =βF dt dx1 dx2 dx3 = = F1 F2 F3

Divergence div F = ∇ · F = ∂i Fi

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

Curl i1 i2 i3 curl F = ∂1 ∂2 ∂3 F1 F2 F3 (curl F)i = εijk ∂j Fk curl F = ∇ × F = il εljk ∂j Fk Laplacian ∆ = div grad = ∇2 = ∇ · ∇

∇×r=0 q r = | r | = x21 + x22 + x23 |∇r| = 1

df r dr r r dF ∇ · F (r) = · r dr r dF ∇ × F (r) = × r dr ∇f (r) =

∆ = ∂i ∂i = ∂12 + ∂22 + ∂32

Vector identities:

r , r

∇r =

(F · ∇)r = F

∇ × (∇ × F ) = −∆ F + ∇(∇ · F )

Fields in Orthogonal Coordinate System (in orthonormal basis e i )

curl curl F = −∆ F + grad div F

Vector Components

∇ × ∇ϕ = 0,

Fi = F · e i

curl grad = 0 ∇ · (∇ × F) = 0, div curl = 0 ∇(f g) = (∇f )g + f (∇g)

grad f = e 1

1 ∂ 1 ∂ 1 ∂ f + e2 f + e3 f 1 2 h1 ∂q h2 ∂q h3 ∂q 3

∇(f F) = (∇f ) · F + f ∇F ∇ × (f F) = (∇f ) × F + f (∇ × F) ∇( F × G ) = G · (∇ × F ) − F · (∇ × G ) df ∇ϕ dϕ dF ∇ · F (ϕ) = ∇ϕ · dϕ dF ∇ × F (ϕ) = ∇ϕ × dϕ ∇f (ϕ) =

∇r = 3

1 div F = h1 h2 h3

(

∂ (h2 h3 F1 ) ∂q 1 ) ∂ ∂ + 2 (h3 h1 F2 ) + 3 (h1 h2 F3 ) ∂q ∂q

  1 ∂ ∂ curl F = e 1 (h3 F3 ) − [ 3 (h2 F2 ) h2 h3 ∂q 2 ∂q   1 ∂ ∂ +e2 (h1 F1 ) − [ 1 (h3 F3 ) h3 h1 ∂q 3 ∂q

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

  1 ∂ ∂ +e3 (h2 F2 ) − [ 2 (h1 F1 ) h1 h2 ∂q 1 ∂q ∆f = (

  ∂ h2 h3 ∂ 1 f ∆f = h1 h2 h3 ∂q 1 h1 ∂q 1    ) h2 h3 ∂ h2 h3 ∂ ∂ ∂ + 1 f + 1 f ∂q h1 ∂q 1 ∂q h1 ∂q 1

1 1 ∂θ (sin θ∂θ f ) ∂r (r2 ∂r f ) + 2 2 r r sin θ +

r2

1 ∂2 f sin2 θ ϕ

Integrals Parametrization of a curve C

Cylindrical Coordinates:

r = r (t),

1 grad f = eρ ∂ρ f + eϕ ∂ϕ f + ez ∂z f ρ

a≤t≤b

Line Integrals

1 1 div F = ∂ρ (ρFρ ) + ∂ϕ Fϕ + ∂z Fz ρ ρ

Zb

Z F · dr =

F(x(t)) · a

C

eρ ρeϕ ez 1 curl F = ∂ρ ∂ϕ ∂z ρ Fρ ρFϕ Fz

dr dt dt

Circulation of vector field along a closed contour I F · dr C

1 1 ∆f = ∂ρ (ρ ∂ρ f ) + 2 ∂ϕ2 f + ∂z2 f ρ ρ

Parametrization of a surface S r = r(u, v),

Spherical Coordinates

a ≤ u ≤ b, 1 1 grad f = er ∂r f + eθ ∂θ f + eϕ ∂ϕ f r r sin θ

Unit Normal n=

div F =

1 1 ∂r (r2 Fr ) + ∂θ (sin θ Fθ ) 2 r r sin θ 1 + ∂ϕ Fϕ r sin θ

e reθ r sin θ eϕ 1 r ∂r ∂θ ∂ϕ curl F = 2 r sin θ Fr rFθ r sin θFϕ



c≤v≤d

∂u r × ∂v r |∂u r × ∂v r|

For a surface given by f( r ) = C the Unit Normal is n=

∇f |∇f |

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Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

Surface Element dS = n dS = ∂u r × ∂v r du dv

Circulation of a Gradient along a closed contour I grad ϕ · d r = 0 C

d S = |∂u r × ∂v r| du dv For a surface given by z = f (x, y)

Gauss (Divergence) Theorem ZZZ ZZ div F dV = F · dS

a ≤ x ≤ b, y1 (x) ≤ y ≤ y2 (x) the Surface Element is q dS = 1 + (∂x f )2 + (∂y f )2 dy dx

Surface Integral of a scalar field ϕ( r (u, v)) |∂u r × ∂v r| du dv

ϕ dS = c

S

a

Zb yZ2 (x) = ϕ(x, y, z(x, y)) a y1 (x)

∂D is a closed surface, which is the boundary of the solid region D Green’s Theorem ZZ I (∂1 F2 − ∂2 F1 )dx1 dx2 = (F1 dx1 + F2 dx2 ) ∂S

∂S is a closed plane curve, which is the boundary of the region S in the x1 x2 -plane Stokes’ Theorem ZZ I curl F · dS = F · dr S

q

1 + (∂x f )2 + (∂y f )2 dy dx

Flux of a Vector Field F through the surface S ZZ ZZ F · dS = F · n dS S

∂D

S

Zd Zb

ZZ

D

S

∂S

∂S is a closed space curve, which is the boundary of the surface S Flux of a curl through a closed surface S ZZ curl F · dS = 0 S

Line Integral of a Gradient

Tensor Fields

ZQ grad ϕ · d r = ϕ(Q) − ϕ(P ) P

Flux of a Tensor Field ZZ Tik nk dS, S

ZZ Tik ni dS S

Ivan Avramidi, MATH 332: Vector and Tensor Analysis, Formulas

Divergence of a Tensor (in Cartesian Coordinates) ∂i Tik , ∂k Tik Directional Derivative (in Cartesian Coordinates) dTik dxj = ∂j Tik ds ds Analog of curl of an antisymmetric 2-tensor 3εijk ∂i Ajk = ∂1 A23 + ∂2 A31 + ∂3 A12 ˜ = ∂1 A˜1 + ∂2 A˜2 + ∂3 A˜3 = div A

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