Typeset in LATEX 2ε July 11, 2018 A Mathematical Introduction to Magnetohydrodynamics
Omar Maj
Max Planck Institute for Plasma Physics, D-85748 Garching, Germany. e-mail: [email protected]
1 2 Contents
Preamble 5
1 Basic elements of fluid dynamics7 1.1 Kinematics of fluids...... 7 1.2 Lagrangian trajectories and flow of a vector field...... 8 1.3 Deformation tensor and vorticity...... 19 1.4 Advective derivative and Reynolds transport theorem...... 22 1.5 Dynamics of fluids...... 25 1.6 Relation to kinetic theory and closure...... 29 1.7 Incompressible flows...... 44 1.8 Equations of state, isentropic flows and vorticity...... 45 1.9 Effects of Euler-type nonlinearities...... 46
2 Basic elements of classical electrodynamics 51 2.1 Maxwell’s equations...... 51 2.2 Lorentz force and motion of an electrically charged particle...... 64 2.3 Basic mathematical results for electrodynamics...... 69
3 From multi-fluid models to magnetohydrodynamics 83 3.1 A model for multiple electrically charged fluids...... 83 3.2 Quasi-neutral limit...... 88 3.3 From multi-fluid to a single-fluid model...... 96 3.4 The Ohm’s law for an electron-ion plasma...... 100 3.5 The equations of magnetohydrodynamics...... 104
4 Conservation laws in magnetohydrodynamics 109 4.1 Global conservation laws in resistive MHD...... 109 4.2 Global conservation laws in ideal MHD...... 113 4.3 Frozen-in law...... 116 4.4 Flux conservation...... 120 4.5 Topology of the magnetic field...... 124 4.6 Analogy with the vorticity of isentropic flows...... 132
5 Basic processes in magnetohydrodynamics 135 5.1 Linear MHD waves...... 135 5.2 Nonlinear shear Alfv´enwaves...... 143 5.3 Magnetic field diffusion...... 145 5.4 Magnetic reconnection: basic ideas and examples...... 150
6 Variational formulation 153 6.1 Basic elements of calculus of variations...... 153 6.2 Existence of a variational formulation...... 163 6.3 Variational principle for Maxwell’s equations...... 165 6.4 Motion of a changed particle in an electromagnetic field...... 167 6.5 First-order Lagrangian theories...... 168 6.6 Jet bundles and Noether’s theorem...... 170 6.7 Geodesics and Euler’s equations of fluid dynamics...... 179 6.8 Lagrangian formulation of ideal MHD...... 186
7 Hamiltonian formulation 191 7.1 Introduction to Hamiltonian systems...... 191 7.2 Hamiltonian structure of ideal MHD...... 191 7.3 Metriplectic systems and dissipation...... 191
A Proofs of the results on kinetic theory and closure 193 A.1 Proof of proposition 1.13...... 195 A.2 Proof of proposition 1.14...... 197 A.3 Proof of proposition 1.15...... 199
B Energy conservation in extended MHD models 203
3 C Magnetic vector potential in MHD 207
D Lie derivatives and passively advected quantities 209
References 222
4 Preamble
Magnetohydrodynamics is the theory of electrically conducting, neutral fluids. The basic equations of magnetohydrodynamics (MHD) have been proposed by Hannes Alfv´en[1,2], who realized the importance of both the electric currents carried by a plasma and the magnetic field they generate. Alfv´encombined the equations of fluid dynamics with Faraday’s and Amp`ere’slaws of electrodynam- ics, thus obtaining a novel mathematical theory, which helped us to understand space plasmas in Earth and planetary magnetospheres, as well as the physics of the Sun, solar wind, and stellar atmospheres. In fusion research, MHD is crucial to the understanding of plasma equilibria and their stability. Liquid metals and electrolytes, like salt water, can also be modeled by MHD equations. Besides the important physical applications, MHD equations exhibit a re- markably beautiful mathematical structure, with connections to geometry and topology that allows us to understand some of the dynamics of magnetic fields in plasmas in terms of topological ideas [3,4,5,6,7,8]. As a dynamical system MHD is an example of infinite-dimensional Hamiltonian system [9, 10]. The scope of this lecture is, in this regard, extremely limited. The goal is to introduce MHD equations in a reasonably self-contained way and to discuss some of their most important features. The style of this lectures is quite similar to the mathematical introduction to fluid dynamics by Chorin and Marsden [11] as the title of this note suggests. Particularly, we shall attempt to introduce the physics modeling in a mathematically precise albeit not always rigorous way. The physics literature on the subject is vast. As a reference for further read- ing, the book by Biskamp [12] provides a clear and comprehensive exposition, while the lectures by Schnack [13] offer a more gradual learning curve. For MHD of the solar atmosphere one can refer to Priest [14] as well as to Aschwanden’s book on the solar corona [15]. For an introduction to magnetohydrodynamics with emphasis on equilibria and stability of fusion plasmas one can refer to the books by Freidberg [16] and Zohm [17], while Goedbloed, Poedts and Keppens address applications to both astrophysical and fusion plasmas [18, 19]. A nice introduction to MHD with a broader perspective which includes applications to metals can be found in Davidson’s book [20]. On the mathematical side, MHD has received considerable attention from ap- plied mathematicians. Its rich mathematical structure has become a paradigm for the application of geometry and topology [8, 21] as well as for structure preserving discretization [22, 23, 24, 25]. As a system of partial differential equations, well-posedness of the Cauchy problem for MHD equations subject to appropriate boundary conditions have been studied first by Duvaut and Lions [26, in French], Sermange and Temam [27], Secchi [28] and more recently, by Chen and coworkers [29] and by Fefferman and coworkers [30, and references therein].
5 6 1 Basic elements of fluid dynamics
The basic understanding of fluid dynamics is an essential prerequisite to the study of MHD. We shall start by recalling the basic elements thereof, following Chorin and Marsden [11], cf. also Marsden and Hughes [31]. First, we define the physical quantities that describe the dynamical state of a fluid (kinematics) and continue with the equations of motion (dynamics).
1.1 Kinematics of fluids. Fluid dynamics is built on the basis of the con- tinuum hypothesis: A fluid is a distribution of matter occupying a certain region of the continuous three-dimensional space. The considered region of space is a domain (i.e., an open, non-empty, and connected subset) Ω ⊆ R3. We neglect the fact that any fluid is ultimately made of atoms and molecules as we are interested in studying its collective motion on a much larger spatial scale. With the continuum hypothesis, one needs to quantify how matter is dis- tributed in Ω at any time t in a certain time interval I ⊆ R. Thus, the first phys- ical quantity of interest is the mass density, which is a positive time-dependent scalar field, ρ : I × Ω → R+ with R+ being the set of strictly positive real numbers, such that ρ(t, x) > 0 (1.1) gives the mass per unit of volume at time t ∈ I at the spatial location x ∈ Ω. By definition, the amount of mass contained in an arbitrary volume W ⊆ Ω (to be referred to as a control volume) is given by Z (mass in W at time t) = ρ(t, x)dx, W which implies that the mass density must be at least locally integrable. Physically we think of an infinitesimal volume of fluid centered around a point x ∈ Ω. The volume of this infinitesimal region is mathematically repre- sented by the Lebesgue measure dx in Ω and the mass is represented by the measure ρ(t, x)dx. Such infinitesimal portions of fluid are referred to as fluid elements. In plasma physics the mass density is often replaced by an equivalent positive scalar field referred to as the particle number density or simply number density which is defined in terms of the mass density by
n(t, x) = ρ(t, x)/m.
Here the fluid is regarded as a collection of particles that all have the same mass m; thus, the number of particles contained in a control volume W is given by 1 Z Z (number of particles in W at time t) = ρ(t, x)dx = n(t, x)dx. m W W According to this definition, the number of particles does not need to be an integer, due to the continuum hypothesis. Next we need to describe the motion of the fluid. We introduce a velocity field defined as a time-dependent vector field, u : I × Ω → R3 such that,
u(t, x) ∈ R3 (1.2)
7 gives the velocity of the fluid element at the point x ∈ Ω and time t ∈ I. The vector u(t, x) is referred to as the fluid velocity. The mass times the velocity of the fluid element, namely, ρ(t, x)u(t, x)dx, gives the linear momentum of the fluid element, hence Z (momentum in W at time t) = ρ(t, x)u(t, x)dx. W In addition to the quantities ρ(t, x) and u(t, x), that can be regarded as the counterpart in fluid dynamics of mass and velocity of a particle in mechan- ics, we need to specify another scalar field for the internal energy of the fluid element. Differently from a point-mass particle, a fluid element is a thermody- namical system that can undergo expansions and compressions, thus absorbing and releasing energy. The thermodynamic status of a fluid element is specified by the internal energy density U : I ×Ω → R+. If each fluid element is regarded as an ideal gas composed by n(t, x)dx particles, we can equivalently express the internal energy density in terms of a new variable. Specifically the laws of thermodynamics for a perfect gas allow us to write the internal energy density in the form 3 U(t, x) = n(t, x)k T (t, x), (1.3) 2 B where kB is the Boltzmann constant and T : I × Ω → R+ is a strictly positive scalar field, such that T (t, x) > 0 represents the local temperature of the fluid element at time t ∈ I and position x ∈ Ω. Therefore, the total energy carried by a fluid element is the sum of the kinetic energy associated to its motion plus the internal energy associated to its thermodynamics, namely, Z 1 2 3 (energy in W at time t) = ρ(t, x)u(t, x) + n(t, x)kBT (t, x) dx. W 2 2 When the law of a perfect gas does not apply we can still define T by equa- tion (1.3) which is now viewed as a purely mathematical change of variable U 7→ T . In general, we shall regard T as a measure of the internal energy, without necessarily implying thermodynamical equilibrium. Summarizing, we shall describe the dynamical state of a fluid by the triple of functions ρ, u, T , where • the mass density ρ is a positive scalar field, • the fluid velocity u is a vector field, and • the temperature T is a positive scalar field. The equations of fluid dynamics are a system of partial differential equations governing the time evolution of (ρ, u, T ).
1.2 Lagrangian trajectories and flow of a vector field. Under appro- priate hypotheses, we can associate to any velocity field u : I ×Ω → R3 a family of maps Ft :Ω → Ω, parametrized by time t in a possibly smaller interval
Iε ⊆ I. Such a one-parameter family of maps {Ft}t∈Iε is referred to as the
8 flow of the vector field. It gives an equivalent description of the motion of the fluid, i.e., the vector field u and its flow Ft contain the same information on the fluid motion. In this section we shall define the flow and prove some of its basic properties. Although it is often overlooked in the physics literature, the flow is a key concept in the mathematical theory of fluid dynamics and thus of MHD. Let us start from a given velocity field u : I × Ω → R3. The associated flow Ft is constructed from the solution of the Cauchy problem dx(t) = u t, x(t), x(t ) = x . (1.4) dt 0 0 Physically, the solution t 7→ x(t) represents the trajectory of a fluid element as it moves with the fluid velocity from the initial position x0 ∈ Ω at time t = t0 ∈ I. Such curves are referred to as Lagrangian trajectories. Basic results from the theory of ordinary differential equations (ODE) guar- antee the existence and uniqueness of the solution of the Cauchy problem (1.4) at least for a short time. A compact account of results on ODEs can be found, for instance, in Marsden et al. [32] as part of the theory of vector fields. The first chapter of both H¨ormander’s[33] and Tao’s [34] lectures on nonlinear par- tial differential equations gives a very nice and compact overview of the theory. Specifically we have the following standard result that we recall without proof. Let us fix constants τ, r > 0 such that the interval Iτ = [t0 − τ, t0 + τ] and 3 the ball Br(x0) = {x ∈ R | |x−x0| ≤ r} are contained in I and Ω, respectively, and let u be continuous with V = sup |u(t, x)| for (t, x) in Iτ × Br(x0). Theorem 1.1 (Local existence and uniqueness for ODEs). If u is continuous and satisfies the Lipschitz condition
|u(t, x) − u(t, y)| ≤ L|x − y|, with constant L ≥ 0 on Iτ × Br(x0), then for any positive ε ≤ min{τ, ρ/V } with V = sup{|u(t, x)| | (t, x) ∈ Iτ × Br(x0)}, there exists a solution x ∈ 1 C [t0 − ε, t0 + ε],Br(x0) of the Cauchy problem (1.4) and any other solution 1 x˜ ∈ C ([t0−ε,˜ t0+˜ε]) must satisfy x(t) =x ˜(t) on the intersection of the domains. We can see that the upper limit of the domain of definition is determined by the minimum time r/V needed to traverse the ball Br(x0). The interval [t0 − ε, t0 + ε] is referred to as the lifespan of the solution. This has a physical significance: the maximum lifespan of the solution is determined by how fast the trajectory can travel up to the boundary of the considered ball. In general the maximum lifespan depends on the initial condition x0. For instance, if the initial condition is very close to the boundary of Ω, r and thus ε can be rather small. We can at most refine a bit this result and make the lifespan of the solution uniform for all initial conditions in a small neighborhood of x0. This can be established by applying the basic existence result to a smaller ball centered on x0: For all initial conditions y0 ∈ Br/2(x0), theorem 1.1 with x0 and r replaced by y0 and r/2, respectively, gives a solution of the Cauchy problem with initial condition x(t0) = y0; then such a solution is contained in Br(x0) and the lifespan is ≤ min{τ, r/(2V )} for all y0 ∈ Br/2(x0). Hence,
Corollary 1.2. Let u, t0, x0, r, τ, and V be as in theorem 1.1. Then there exists a neighborhood U ⊂ Br(x0) and 0 < ε ≤ min{τ, r/(2V )} such that for every 1 y0 ∈ U the Cauchy problem (1.4) has a solution x ∈ C ([t0 − ε, t0 + ε],Br).
9 We shall however work under the assumption that the lifespan of Lagrangian trajectories is uniform on the whole domain Ω, i.e., we assume that there is an ε > 0 depending only on t0, such that for every initial condition x0 ∈ Ω there is a Lagrangian trajectory x : Iε → Ω with Iε = [t0 − ε, t0 + ε]. For a generic ordinary differential equation, this is a very strong assumption. For our problem, however, this is not so strong because, in practice, it just means that the domain Ω and the boundary conditions for the vector field u have been chosen properly, in the sense that “Ω contains the fluid”. We shall fix the initial time to be t0 = 0 and let Iε = [−ε, ε] the interval of existence of the Lagrangian trajectories. At this point we are ready to define the flow of the velocity field.
Definition 1.1 (Flow). For every t ∈ Iε the map Ft :Ω → Ω is defined by
x0 7→ Ft(x0) = x(t), where x(t) is the Lagrangian trajectory passing through x0 at the time t = 0. In addition this defines a map F : Iε × Ω → Ω given by F (t, x) = Ft(x). The uniqueness of the solution of the Cauchy problem for Lagrangian tra- jectories is essential in the definition of Ft. In fact, for Ft to be unambiguously defined we need that Ft(x) 6= Ft(y) implies x 6= y; it is not admissible that the same point is mapped into two different points. The reader can check that definition 1.1 is well posed in this sense because of the uniqueness of Lagrangian trajectories. From a physical point of view the flow describes the displacement of the fluid as time advances, i.e., given a control volume W ⊆ Ω, then Ft(W ) ⊆ Ω is the volume occupied by the fluid initially in W after it has evolved for a time t. In summary, we can consider a one-parameter family of maps Ft which can be used in two different ways, namely,
• t 7→ Ft(x) is the Lagrangian trajectory passing through x at time t = 0;
• x 7→ Ft(x) is the displacement of the point x after a time t. As a consequence of its definition, the flow satisfies, cf. equation (1.4), d F (x ) = u t, F (x ), dt t 0 t 0 (1.5) F0(x0) = x0, where the initial point x0 ∈ Ω is regarded as a parameter, so that we write a total derivative instead of a partial derivative and consider this an ordinary differential equation rather than a partial differential equation. For the flow Ft, we always imply initial conditions at t0 = 0. For general initial time t0 = s, we define the map Ft,s for t, s ∈ Iε by the Cauchy problem d F (x ) = u t, F (x ), dt t,s 0 t,s 0 Fs,s(x0) = x0, with t0 = s; hence Ft = Ft,0. For an autonomous vector field u = u(x) the maps Ft,s can be written in terms of Ft. Specifically, we observe that the curve
10 cs(t) = Ft−s(x0) solves the Cauchy problem dc (t) s = u c (t), c (s) = x , dt s s 0 and by uniqueness of the solution of Cauchy problems, we deduce that for autonomous fields Ft,s(x0) = cs(t) = Ft−s(x0). We shall now establish a few key properties of the flow Ft that essentially descend from equation (1.5).
Proposition 1.3. For every t, s ∈ Iε such that t + s ∈ Iε, we have in general Ft+s = Ft+s,s ◦ Fs = Ft+s,t ◦ Ft, while for autonomous vector fields we have the semi-group property Ft+s = Ft ◦ Fs = Fs ◦ Ft.
0 Proof. For every x0 ∈ Ω, let us consider the Lagrangian trajectory x(t ) corre- 0 sponding to the initial condition x(0) = x0 at t = 0. By definition, Ft+s(x0) = x(t + s). The curve t0 7→ x(t0) also solves the problem
dx(t0) = u t0, x(t0), x(s) = F (x ), dt0 s 0
0 0 hence, x(t ) = Ft0,s Fs(x0) and, replacing s by t, x(t ) = Ft0,t Ft(x0) . Then Ft+s(x0) = x(t + s) = Ft+s,s ◦ Fs(x0) = Ft+s,t ◦ Ft(x0). For an autonomous vector field we have Ft0,s = Ft0−s, hence Ft+s = Ft ◦ Fs and Ft+s = Fs ◦ Ft. Proposition 1.3 has an immediate consequence.
Corollary 1.4. For every t ∈ Iε, Ft :Ω → Ω is invertible and the inverse map −1 −1 is given by Ft = F0,t. For autonomous fields Ft = F0,t = F−t.
Proof. We have F0(x) = x for all x ∈ Ω and the identity Ft+s = Ft+s,t ◦ Ft in proposition 1.3 with s = −t gives x = F0(x) = F0,t ◦ Ft(x) which shows that F0,t is the inverse of Ft. For autonomous fields we have Ft,s = Ft−s hence −1 Ft = F−t.
Another basic result from ODE theory implies that Ft :Ω → Ω is Lipschitz continuous in Ω for all t ∈ Iε. Proposition 1.5. If u is continuous and satisfies the Lipschitz condition of theorem 1.1 on Iε × Ω and Ft :Ω → Ω is defined on Ω for all t ∈ Iε, then
L|t| Ft(x) − Ft(y) ≤ |x − y|e , for all x, y ∈ Ω and t ∈ Iε. Here L is the Lipschitz constant of u. Proof. Let us first consider the half-interval t ≥ 0. For every x, y ∈ Ω fixed, let h(t) = Ft(x) − Ft(y) and, by the Lipschitz condition for u,
dh(t) = u t, Ft(x) − u t, Ft(y) ≤ L|h(t)|. dt We actually need to control the derivative of the norm, rather then the norm of the derivative. With this aim we can estimate
1 d 2 dh(t) dh(t) 2 h(t) = h(t) · ≤ h(t) · ≤ Lh(t) , 2 dt dt dt
11 and for t ≥ 0, d h d i h(t)2e−2Lt = h(t)2 − 2Lh(t)2 e−2Lt ≤ 0, dt dt hence h(t)2e−2Lt ≤ h(0)2 which is equivalent to the claim for t ≥ 0. As for the other half-interval t ≤ 0, let s = −t ≥ 0 and h(s) = F−s(x) − F−s(y) and we notice that dh(s) = −u − s, F (x) + u − s, F (y), ds −s −s and repeat the argument, integrating in the variable s.
We now know that Ft is a continuous transformation of Ω into itself for all t ∈ Iε. We also shall need to understand when Ft is differentiable and in those cases have a convenient way to compute its Jacobian matrix and determinant, namely, t DFt(x0) = ∇x0 Ft(x0) ,Jt(x0) = det DFt(x0) , where tA denotes the transpose of a tensor A. In this note, the gradient ∇v(t, x) of a generic vector field v(t, x) is defined according to standard dyadic vector calculus, which differs from the definition adopted by Chorin and Marsden [11]. The Jacobian matrix is then denoted by Dv and it is the transpose of the gradient, namely, ∂vj ∇v(t, x) ij = = Dv(t, x) ji. ∂xi We derive an evolution equation for the Jacobian matrix of the flow. 1 Proposition 1.6. If the velocity field u is of class C and Ft :Ω → Ω is defined 1 2 on Ω for all t ∈ Iε, then Ft ∈ C (Ω), the map t 7→ DFt(x) is in C and satisfies the Cauchy problem, d DF (x ) = Du t, F (x )DF (x ), dt t 0 t 0 t 0 (1.6) DF0(x0) = I, with I being the identity matrix. By induction, if u ∈ Ck for k ≥ 1 then k k+1 Ft ∈ C (Ω) and t 7→ Ft(x) is C . Proof. Cf. lemma 4.1.9 of Marsden et al. [32].
2 If we can say that the function F (t, x) = Ft(x) is of class C , equation (1.6) is a direct consequence of the chain rule. In fact, the time derivative and the gradient of the flow commute and ∂t∇x0 Ft(x0) = ∇x0 ∂tFt(x0) = ∇x0 u t, Ft(x0) = ∇x0 Ft(x0) · ∇u t, Ft(x0) = ∇Ft(x0) · ∇u t, Ft(x0) .
By transposing this identity and considering the initial condition ∇F0 = I, which follows from F0(x0) = x0 in equation (1.5), we obtain equation (1.6). Without assuming further regularity, the proof require some more work. The full argument can be found in Marsden et al. [32], with the only difference that here we are assuming that Ft is defined uniformly on the whole domain Ω and not just locally. The evolution equation for the determinant follows from proposition 1.6.
12 Proposition 1.7. Under the same hypotheses of proposition 1.6, we have
d J (x ) = ∇ · u t, X(t, x )J (x ), dt t 0 0 t 0 (1.7) J0(x0) = 1.
This result is a special case of the Liouville’s formula, which can be proven from the properties of the determinant.
Lemma 1.8 (Liouville’s formula). Let A, ψ be functions from an interval I ⊆ R with values in the space Rn×n of n × n matrices, such that ψ ∈ C1 and dψ/dt = A(t)ψ(t). Then d det ψ(t)/dt = tr A(t) det ψ(t), where tr(A) is the trace of the matrix A.
Proof. See, for instance, proposition 1.2.4 in H¨ormander’s lectures [33].
Here however we give a proof which relies on the following basic identity from vector calculus. This establishes a relationships between the volume of a parallelepiped spanned by three vectors, expressed by the scalar triple product of the vectors, and the determinant of the matrix defined by the vectors.
Lemma 1.9. Let A be a 3-by-3 matrix and we write it as a column of row t vectors, i.e., A = (A1,A2,A3) with Ai = (aij)j. Then
det(A) = A1 · (A2 × A3) = A2 · (A3 × A1) = A3 · (A1 × A2).
Proof. By definition, the determinant is X det(A) = sign(σ)a1σ(1)a2σ(2)a3σ(3),
σ∈S3 where the sum runs over the set S3 of permutations of three elements (1, 2, 3). In terms of the completely anti-symmetric (Levi-Civita) symbol 1 (i, j, k) is an even permutation of (1, 2, 3), ijk = 0 (i, j, k) is not a permutation of (1, 2, 3), −1 (i, j, k) is an odd permutation of (1, 2, 3), we write X X det(A) = sign(σ)a1σ(1)a2σ(2)a3σ(3) = ijka1ia2ja3k,
σ∈S3 ijk and the right-hand side is just the scalar triple product A1 · (A2 × A3). The other two identities follow on noting that the triple product is invariant under cyclic permutations.
It is worth noting that the identity of lemma 1.9 holds in any dimension, but then the cross product has to be replaced by an appropriate anti-symmetric multi-linear operation. The proof of proposition 1.7 now follows by direct calculation.
13 k k+1 Proof of proposition 1.7. By proposition 1.6, Ft is C in space and C in time. By lemma 1.9, det ∇x0 Ft = ∇x0 X1 · ∇x0 X2 × ∇x0 X3 , where Ft(x0) = X1(t, x0),X2(t, x0),X3(t, x0) and Xi(t, x0) are the Cartesian coordinates of the position vector of the Lagrangian trajectory. We note that ∇Xi are the rows of the Jacobian matrix DFt, hence, from equation (1.6),
∂t∇x0 Xi = ∇x0 X · ∇ui, where u = (u1, u2, u3) is the fluid velocity in Cartesian components, and ∇ui is evaluated at t, Ft(x0) . Then, we compute ∂t det ∇x0 Ft = (∂t∇x0 X1) · ∇x0 X2 × ∇x0 X3 + ∇x0 X1 · (∂t∇x0 X2) × ∇x0 X3 + ∇x0 X1 · ∇x0 X2 × (∂t∇x0 X3) = (∇x0 X · ∇u1) · ∇x0 X2 × ∇x0 X3 + ∇x0 X1 · (∇x0 X · ∇u2) × ∇x0 X3 + ∇x0 X1 · ∇x0 X2 × (∇x0 X · ∇u3) 3 X h i∂u1 = ∇ X · ∇ X × ∇ X x0 i x0 2 x0 3 ∂x i=1 i 3 X h i∂u2 + ∇ X · ∇ X × ∇ X x0 1 x0 i x0 3 ∂x i=1 i 3 X h i∂u3 + ∇ X · ∇ X × ∇ X . x0 1 x0 2 x0 i ∂x i=1 i
Since the scalar triple product A · [B × C] vanishes if any pair of its factors are equal, in the first sum the only contribution comes from i = 1, in the second sum from i = 2, and in third sum from i = 3. We therefore have