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# C.1 Gradient, Divergence, Curl, Laplacian

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674

Appendix C Identities and Formulas

2. divergence (F = Fr er + Fθ eθ + Fz k):

div F =

1 ∂

1 ∂

(rFr ) +

Fθ +

Fz .

r ∂r

r ∂θ

∂z

3. Laplacian:

Δf =

1 ∂

r ∂r

r

∂f

∂r

+

1 ∂2f

∂2f

∂2f

1 ∂f

∂2f

1 ∂2f

+ 2 =

+

+ 2 2 + 2.

2

2

2

r ∂θ

∂z

∂r

r ∂r

r ∂θ

∂z

4. curl :

curl F =

e re e

1 r θ z

∂r ∂θ ∂z .

r F rF F

r

θ z

Spherical coordinates

x = r cos θ sin ψ, y = r sin θ sin ψ, z = r cos ψ

(r > 0, 0 ≤ θ ≤ 2π, 0 ≤ ψ ≤ π)

er = cos θ sin ψi + sin θ sin ψj + cos ψk

eθ = − sin θi + cos θj

eψ = cos θ cos ψi + sin θ cos ψj − sin ψk.

∇f =

1 ∂f

1 ∂f

∂f

er +

eθ +

eψ .

∂r

r sin ψ ∂θ

r ∂ψ

2. divergence (F = Fr er + Fθ eθ + Fψ eψ ):

div F =

1 ∂

2

1

Fr + Fr +

Fθ +

Fψ + cot ψFψ .

∂r

r

r sin ψ ∂θ

∂ψ

spherical part

3. Laplacian:

Δf =

2 ∂f

1

∂2f

+

+

∂r 2

r ∂r r 2

∂2f

1

∂ 2f

∂f

+

+ cot ψ

(sin ψ)2 ∂θ2

∂ψ2

∂ψ

spherical part (Laplace-Beltrami operator)

4. curl :

rot F =

er reψ r sin ψeθ

1

.

∂θ

r ∂ψ

r 2 sin ψ

Fr rFψ r sin ψFz

.

C.2 Formulas

675

C.2 Formulas

Gauss’ formulas

In Rn , n ≥ 2, let:

1.

2.

3.

4.

5.

6.

7.

8.

Ω be a bounded smooth domain and and ν the outward unit normal on ∂Ω.

u, v be vector ﬁelds of class C 1 Ω .

ϕ, ψ be real functions of class C 1 Ω .

dσ be the area element on ∂Ω.

div u dx = ∂Ω u · ν dσ

∇ϕ dx = ∂Ω ϕν dσ.

Ω

Δϕ dx = ∂Ω ∇ϕ · ν dσ =

Ω

(Divergence Theorem).

Ω

∂Ω

∂ν ϕ dσ.

ψ div F dx = ∂Ω ψF · ν dσ − Ω ∇ψ · F dx

ψΔϕ dx = ∂Ω ψ∂ν ϕ dσ − Ω ∇ϕ · ∇ψ dx

Ω

(ψΔϕ − ϕΔψ) dx = ∂Ω (ψ∂ν ϕ − ϕ∂ν ψ) dσ

Ω

curl u dx = − ∂Ω u × ν dσ.

Ω

Ω

Ω

u· curl v dx =

Ω

v· curl u dx −

∂Ω

(Integration by parts).

(Green’s identity I).

(Green’s identity II).

(u × v) · ν dσ.

Identities

1. div curl u = 0.

2. curl ∇ϕ = 0.

3. div (ϕu) = ϕ div u + ∇ϕ · u.

4. curl (ϕu) = ϕ curl u + ∇ϕ × u.

5. curl (u × v) = (v · ∇) u − (u · ∇) v + (div v) u − (div u) v.

6. div (u × v) = curlu · v−curlv · u.

7. ∇ (u · v) = u× curl v + v× curl u + (u · ∇) v + (v · ∇) u.

2

8. (u · ∇) u = curlu × u + 12 ∇ |u| .

9. curl curl u = ∇(div u) − Δu.

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[2] Friedman, A.: Partial Diﬀerential Equations of parabolic Type, Prentice-Hall,

Englewood Cliﬀs, 1964.

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[4] Grisvard, P.: Elliptic Problems in nonsmooth domains, Pitman, Boston, 1985.

[5] Guenter, R.B. and Lee, J.W.: Partial Diﬀerential Equations of Mathematical

Physics and Integral Equations, Dover Publications, Inc., New York, 1998.

[6] Helms, O.: Introduction to Potential Theory, Krieger Publishing Company,

New York, 1975.

[7] John, F.: Partial Diﬀerential Equations, 4th ed., Springer-Verlag, New York,

1982.

[8] Kellog, O.: Foundations of Potential Theory, Dover, New York, 1954.

[9] Galdi, G.: Introduction to the Mathematical Theory of Navier-Stokes Equations, vols. I and II, Springer-Verlag, New York, 1994.

[10] Lieberman, G.M.: Second Order Parabolic Partial Diﬀerential Equations,

World Scientiﬁc, Singapore, 1996.

[11] Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and

Applications, ols. 1, 2, Springer-Verlag, New York, 1972.

[12] McOwen, R.: Partial Diﬀerential Equations: Methods and Applications, Prentice-Hall, New Jersey, 1996.

[13] Olver, P.J.: Introduction to Partial Diﬀerential Equations, Springer International Publishing Switzerland, 2014.

[14] Protter, M. and Weinberger, H.: Maximum Principles in Diﬀerential Equations, Prentice-Hall, Englewood Cliﬀs, 1984.

[15] Renardy, M. and Rogers, R.C.: An Introduction to Partial Diﬀerential Equations, Springer-Verlag, New York, 1993.

[16] Rauch, J.: Partial Diﬀerential Equations, Springer-Verlag, Heidelberg, 1992.

© Springer International Publishing Switzerland 2016

S. Salsa, Partial Diﬀerential Equations in Action. From Modelling to Theory, 3rd Ed.,

UNITEXT – La Matematica per il 3+2 99, DOI 10.1007/978-3-319-31238-5

678

References

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[18] Smoller, J.: Shock Waves and Reaction-Diﬀusion Equations, Springer-Verlag,

New York, 1983.

[19] Strauss, W.: Partial Diﬀerential Equation: An Introduction, Wiley, 1992.

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Mathematical Modelling

[21] Acheson, A.J.: Elementary Fluid Dynamics, Clarendon Press, Oxford, 1990.

[22] Billingham, J. and King, A.C.: Wave Motion, Cambridge University Press,

2000.

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the Natural Sciences, SIAM Classics in Applied Mathematics, 4th ed., 1995.

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Heidelberg, 2001.

[27] Rhee, H., Aris, R., and Amundson, N.: First Order Partial Diﬀerential Equations, vola. 1 and 2, Dover, New York, 1986.

[28] Scherzer, O., Grasmair, M., Grosshauer, H.; Haltmeier, M., and Lenzen, F.:

Variational Methods in Imaging, Applied Mathematical Sciences 167, Springer,

New York, 2008.

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[30] Whitham, G.B.: Linear and Nonlinear Waves, Wiley-Interscience, 1974.

ODEs, Analysis and Functional Analysis

[32] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Diﬀerential Equations, Springer, New York, 2010.

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[35] Maz’ya, V.G.: Sobolev Spaces, Springer-Verlag, Berlin Heidelberg, 1985.

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[37] Schwartz, L.: Théorie des Distributions, Hermann, Paris, 1966.

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[40] Ziemer, W.: Weakly Diﬀerentiable Functions, Springer-Verlag, Berlin Heidelberg, 1989.

[41] Zygmund, R. and Wheeden, R.: Measure and Integral, Marcel Dekker, 1977.

Numerical Analysis

[42] Dautray, R. and Lions, J.L.: Mathematical Analysis and Numerical Methods

for Science and Technology, vols. 4 and 6, Springer-Verlag, Berlin Heidelberg,

1985.

[43] Quarteroni, A.: Numerical Models for Diﬀerential Problems, MS&A, SpringerVerlag Italia, Milan, 2014.

[44] Quarteroni, A. and Valli, A.: Numerical Approximation of Partial Diﬀerential

Equations, Springer-Verlag, Berlin Heidelberg, 1994.

[45] Godlewski, E. and Raviart, P.A.: Numerical Approximation of Hyperbolic

Systems of Conservation Laws, Springer-Verlag, New York, 1996.

Stochastic Processes and Finance

[46] Baxter, M. and Rennie, A.: Financial Calculus: An Introduction to Derivative

Pricing, Cambridge University Press, 1996.

[47] Øksendal, B.K.: Stochastic Diﬀerential Equations: An Introduction with Applications, 4th ed., Springer-Verlag, Berlin Heidelberg, 1995.

[48] Wilmott, P., Howison, S., and Dewinne, J.: The Mathematics of Financial

Derivatives. A Student Introduction, Cambridge University Press, 1996.

Index

A

Absorbing barriers 111

Adjoint of a bilinear form 402

Alternative

– for the Dirichlet problem 526

– for the Neumann problem 529

Angular frequency 260

Arbitrage 92

B

Barenblatt solutions 103

Barrier 143

Bernoulli’s equation 330

Bessel function 73, 297

Bilinear form 382

Bond number 333

Boundary conditions 21

– Dirichlet 21, 33

– mixed 22, 33

– Neumann 22, 33

– Robin 22, 33

Breaking time 201

Brownian motion 55

Brownian path 55

Burgers, viscous 216

C

Canonical form 293, 295

Canonical isometry 379

Characteristic 181, 230, 620, 621

– parallelogram 277

– strip 246

– system 245

Chebyshev polynomials 370

Closure 8

Comparison 38

Compatibility conditions 403, 405

Condition

– compatibility 118

– E 220

– entropy 638

– Rankine-Hugoniot 631

Conjugate exponent 11, 358

Conormal derivative 527

Contact discontinuity 626, 632, 640

Continuous isomorphism 385

Convection 61

Convergence

– least squares 660

– uniform 11

– weak 394

Convolution 430, 449

Cost functional 571

Critical mass 75

Critical survival value 66

Curve

– rarefaction 633

– shock 640

D

d’Alembert formula 276

d-harmonic function 121

Darcy’s law 102

Diﬀusion 18

Diﬀusion coeﬃcient 54

Dirac comb 436

Dirac measure 44

Direct product 452

Direct sum 364

Dirichlet eigenfunctions 517

© Springer International Publishing Switzerland 2016

S. Salsa, Partial Diﬀerential Equations in Action. From Modelling to Theory, 3rd Ed.,

UNITEXT – La Matematica per il 3+2 99, DOI 10.1007/978-3-319-31238-5

682

Index

Dirichlet Principle 546

Dispersion 287

– relation 261, 287, 335

Dissipation

– external/internal 286

Distribution 434

– composition 445

– division 448

Distributional derivative 438

Domain 8

– C^1, C^k 12

– Lipschitz 14

– of dependence 278, 315

– smooth 12

Drift 60, 89

Duhamel method 285

E

Eigenfunction 370

– of a bilinear form 411

Eigenspace 407, 409, 411

Eigenvalue 370, 409

– of a bilinear form 411

Eigenvector 409

Elastic restoring force 111

Elliptic equation 505

Entropy condition 209, 210

Equal area rule 202

Equation

– backward 94

– backward heat 39

– Bessel 73

– Bessel’s 372

– biharmonic 555

– Black-Scholes 3, 93

– Bukley-Leverett 243

– Burgers 4

– Chebyshev 370

– diﬀusion 2, 17

– Eiconal 5

– eikonal 249

– elastostatics 557

– elliptic 289

– Fisher 4

– fully nonlinear 2

– Hermite’s 371

– hyperbolic 289

– Klein-Gordon 287

– Laplace 3

– Legendre’s 371

– linear elasticity 5

– linear, nonlinear 2

– Maxwell 5

– minimal surface 4

– Navier 557

– Navier Stokes 5

– Navier-Stokes 153, 561

– Navier-Stokes, stationary 566

– parabolic 289, 581

– parametric Bessel’s (of order p) 372

– partial diﬀerential 2

– Poisson 3, 115

– porous media 103

– Porous medium 4

– quasilinear 2

– reduced wave 178

– Schrodinger 4

– semilinear 2

– stationary Fisher 553

– stochastic diﬀerential 89

– Sturm-Liouville 370

– transport 2

– Tricomi 289

– uniformly parabolic 582

– vibrating plate 3

– wave 3

Equicontinuity 392

Equipartition of energy 342

Escape probability 137

Essential support 429

Essential supremum 358

Euler equation 388

European options 88

Expectation 58, 70

Expiry date 88

Extension operator 477

Exterior Dirichlet problem 163

Exterior domain 164

Exterior Robin problem 165, 177

F

Fick’s law 61

Final payoﬀ 94

First exit time 135

First integral 238, 240

First variation 388

Flux function 179

Focussing eﬀect 345

Forward cone 313

Fourier coeﬃcients 367

Fourier law 20

Fourier series 28

Fourier transform 454, 473

Fourier-Bessel series 74, 373

Frequency 260

Froude number 333

Index

Function

– Bessel’s of ﬁrst kind and order p 373

– characteristic 10

– compactly supported 10

– complementary error 220

– continuous 10

– d-harmonic 119

– essentially bounded 358

– Green’s 157

– Hölder continuous 357

– harmonic 18, 115

– Heaviside 44

– piecewise continuous 205

– summable 357

– test 48, 429

– weigth 370

Functional 377

Fundamental solution 43, 48, 148, 282,

311

G

Gas dynamics 617

Gaussian law 56, 68

Genuinely nonlinear 634

Global Cauchy problem 23, 34, 76

– nonhomogeneous 80

Gram-Schmidt process 369

Greatest lower bound 9

Group velocity 261

H

Harmonic lifting 141

Harmonic measure 138

Harnack’s inequality 131

Heisenberg Uncertainty Principle

– for the ﬁrst eigenvalue 421

Helmholtz decomposition formula 151

Hermite polynomials 371

Hilbert triplet 401

Hooke’s law 556

Hopf’s maximum principle 126

Hopf-Cole transformation 218

Hugoniot line 626

I

Identity

– Green’s (ﬁrst and second) 15

– strong Parseval’s 460

– weak Parseval’s 458

Inequality

– Hölder 358

Inﬁmum 9

Inﬂow/outﬂow boundary 239

683

Inﬂow/outﬂow characteristics 185

Inner product space 359

Integral surface 230

integration by parts 15

Interior shere condition 126

Invasion problem 113

Inward heat ﬂux 33

Isometry

– isometric 360

Ito’s formula 90

K

Kernel 374

Kinematic condition 331

Kinetic energy 266

L

Lagrange multiplier 565

Lattice 66, 118

Least squares 28

Least upper bound 9

Lebesgue spine 145

Legendre polynomials 371

Light cone 249

Linearly degenerate 642

Liouville Theorem 132

little o 11

Local chart 12

Local wave speed 190

Localization 477

Logarithmic potential 150

Logistic growth 105

Lognormal density 91

M

Mach number 308

Markov properties 57, 69

Mass conservation 60

Material derivative 154

Maximum principle 83, 120

– weak 36, 532, 601

Mean value property 123

Method 23

– Duhamel 81

– electrostatic images 157

– Galerkin’s 388

– of characteristics 189

– of descent 316

– of Faedo-Galerkin 591, 605

– of stationary phase 263

– reﬂection 477

– separation of variables 23, 26, 269, 304,

407, 520

684

Index

– time reversal 325

– vanishing viscosity 214

Metric space 353

Minimax property (of the eigenvalues)

– for the ﬁrst eigenvalue 416, 426

Molliﬁer 430

Monotone iteration scheme 551

Multidimensional symmetric random

walk 66

Multiplicity (of an eigenvalue) 409

N

Neumann eigenfunctions 519

Neumann function 163

Norm

– Integral of order p 357

– least squares 355

– maximum 355

– maximum of order k 356

Normal probability density 43

Normed space 353

Numerical sets 7

O

Omeomorphism 418

Open covering 477

Operator

– bounded,continuous 374

– compact 397

– discrete Laplace 119

– linear 373

– mean value 118

Optimal control 572

Optimal state 572

Orthonormal basis 367

P

Parabolic

– boundary 23, 34

Parabolic dilations 40

Parallelogram law 360

Partition of unity 478

Perron method 142

Phase speed 260

Poincaré’s inequality 466, 488

Point 7

– boundary 8

– interior 7

– limit 8

Point source solution

– two dimensional 345

Poisson formula 131

Potential 115

– double layer 166

– energy 267

– Newtonian 149

– retarded 318, 345

– single layer 170

Principal Dirichlet eigenvalue 518

Principle of virtual work 560

Probability

– measure 670

– space 670

Problem

– abstract parabolic 586

– abstract variational 382

– Characteristic Cauchy 343

– eigenvalue 27

– Goursat 343

– ill posed (heat equation) 343

– inverse 325

– Riemann 623

– well posed 7, 21

Projected characteristics 238

Projection

– on closed convex sets 424

Put-call parity 97

Q

Qantum mechanics harmonic oscillator 422

R

Random variable 54

Random walk 49

– with drift 58

Range 374

– of inﬂuence 278, 313

Rankine-Hugoniot condition 198, 205

Rarefaction/simple waves 194

Rayleigh quotient 414, 518

Reaction 63

Reﬂecting barriers 111

Regular point 144

Resolvent 407

– of a bilinear form 411

– of a bounded operator 408

Retarded potential 318

retrocone 301

Reynolds number 562

Riemann invariant 628

Riemann problem 212

Rodrigues’ formula 371

Index

S

Schwarz inequality 360

Schwarz reﬂection principle 175

Self-ﬁnancing portfolio 92, 100

Sequence

– Cauchy 354

– fundamental 354

Set

– bounded 8

– closed 8

– compact 8

– compactly contained 8

– connected 8

– convex 8

– dense 8

– open 8

– precompact 391

– sequentially closed 8

– sequentially compact 8, 391

Shock

– curve 198

– speed 198

– wave 198

Similarity, self-similar solutions 41

Sobolev exponent 491

Solution

– classical 507

– distributional 507

– integral 205

– self-similar 103

– strong 507

– unit source 46

– variational 507

– viscosity 507

– weak 205

Sommerfeld condition 178

Space

– separable 367

Space-like curve 250

Spectral decomposition

– of a matrix 407

– of an operator 411

Spectrum 407

– continous 409

– of a bilinear form 411

– of a bounded operator 408

– point 409

– residual 409

Spherical waves 261

Stability estimate 386

Standing wave 271

Stationary phase (method of) 340

Steepest descent 575

Stiﬀness matrix 389

Stochastic process 55, 68

Stokes System

– equazione biarmonica 562

Stopping time 57, 135

Strike price 88

Strip condition 247

Strong Huygens’ principle 313, 315

Sub/superharmonic function 141

Sub/supersolution 36

Superposition principle 17, 77, 268

Support 10

– of a distribution 437

Surface

– of the unit sphere (ωn ) 7

– tension 328, 330

Symbol o(h) 53

Symbol “big O” 63

System

– hyperbolic 616

– p-system 619

T

Tempered distribution 456

Tensor

– deformation 556

– stress 153, 556

Tensor product 452

Term by term

– diﬀerentiation 12

– integration 12

Theorem

– Ascoli-Arzelà 392

– Contraction mapping 417

– Dominated Convergence 668

– Fubini 669

– Lax-MIlgram 383

– Leray-Shauder 420

– Monotone Convergence 668

– projection 364

– Rellich 487

– Riesz’s representation 378

– Riesz-Fréchet-Kolmogoroﬀ 393

– Schauder 419

time-like curve 250

Topology 7, 354

– euclidean 8

– relative 9

Trace 479

– inequality 486

Traﬃc in a tunnel 253

685

686

Index

Transition function 69

Transition layer 216

Transition probability 57, 121

Transmission conditions 547

Travelling wave 182, 190, 215

trivial extension 477

Tychonov class 83

U

Uniform ellipticity 521

Unit impulse 45

Upper,lower limit 9

V

Value function 88

Variational formulation

– Dirichlet problem 510, 523

– Mixed problem 516, 530

– Neumann problem 513, 528

– Robin problem 515

Variational inequality

– on closed convex sets 424

Variational principle

– for the ﬁrst eigenvalue 414

– for the k-th eigenvalue 415

Volatility 89

W

Wave

– capillarity 337

– cylindrical 296

– gravity 336

– harmonic 259

– incoming/outgoing 298

– linear 328

– linear gravity 346

– monochromatic/harmonic 296

– number 260

– packet 262

– plane 261, 296

– rarefaction, p-system 646, 647

– shock 640

– shock, p-system 645, 646

– simple 633

– spherical 297

– standing 260

– travelling 259

Weak coerciveness 525

Weak formulation

– Cauchy-Dirichlet problem 584

– Cauchy-Robin/Neumann problem 594–

596

– Initial-Dirichlet problem (wave eq.) 604

Weakly coercive (bilinear form) 402, 596

Weierstrass test 11, 29

Y

Young modulus 342