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DIFFERENTIAL FORMS ON R3 : REWRITING MAXWELL’S EQUATIONS FORREST GORDON Abstract. As they are classically written, Maxwell’s Equations ~ and the magnetic field B ~ as vector describe the electric field E fields on R3 which are determined by a time-dependent smooth function ρ called the charge density and a time-dependent vector field J~ called the current density. These equations can be rewritten more elegantly in the language of differential geometry, which we introduce herefor the special case R3 . We will describe the exterior algebra Ω R3 of differential forms on R3 and the exterior derivative on this space. Along the way, we will encounter new analogues of a few familiar tools from elementary vector calculus, and for fun we will conclude by viewing Maxwell’s Equations written in terms of differential forms. 1. Introduction We begin with a quick review of vector calculus on R3 . For neatness, we will denote the coordinate partial derivative operators by ∂x , ∂y , ∂ , etc. We will use ∇ to denote the del operator, and ∂z , instead of ∂x which is the formal vector whose components are these three partial derivatives: ∇ ≡ h∂x , ∂y , ∂z i. This operator will allow us to describe the way smooth functions and vector fields are changing at each point in R3 . Let f : R3 → R be a smooth function. The gradient of f is the vector field on R3 determined by the operation of del on f : grad f = ∇f = h∂x f, ∂y f, ∂z f i. The value of grad f at a point p ∈ R3 is a vector (∇f )|p whose three components are the rates at which the value of f is increasing in the three coordinate directions. This vector points in the direction of the greatest rate of increase of f , with magnitude equal to this rate. Now let F : R3 → R3 be a vector field, written in coordinate form as F = hf (x, y, z), g(x, y, z), h(x, y, z)i, where f , g, and h are differentiable functions. The curl of F is the vector field on R3 that we obtain when we take the formal cross product of the del operator and F : Date: Spring, 2010. 1 2 FORREST GORDON curl F = ∇ × F = (∂y h − ∂z g), (∂z f − ∂x h), (∂x g − ∂y f ) . Perhaps the physical meaning of the curl is obscured by this strange arrangement of partial derivatives, but the curl operator does just what its name suggests: it tells us how tightly and in which direction the vector field F is “curling” at each point p ∈ R3 . Specifically, the vector (∇ × F ) |p points along the axis of this local rotation, and the magnitude of this vector is proportional to the local rate of rotation around this axis (with sign determined by the Right Hand Rule). The divergence of the vector field F is the real-valued function that we obtain when we take the formal dot product of del and F : div F = ∇ · F = ∂x f + ∂y g + ∂z h. Thus the divergence of F at a point p ∈ R3 is a measure of how F “diverges” from p. If (∇ · F )(p) > 0, then p is a source of F ; if (∇ · F )(p) < 0, then p is a sink of F . With grad, curl, and div fresh in mind, we behold Maxwell’s Equations, shown here with all physical constants set to 1: Maxwell’s Equations (1.1) (1.2) (1.3) (1.4) ~ =0 ∇·B ~ ~ + dB = 0 ∇×E dt ~ =ρ ∇·E ~ ~ − dE = J~ ∇×B dt Equation (1.1), Gauss’ Law for Magnetism, states that the diver~ is zero at every point in space. In other gence of the magnetic field B words, the magnetic field has no sources or sinks; there are no magnetic monopoles. Of course, there are electric monopoles, such as protons and electrons, and the distribution of these monopoles in space is described by the charge density function ρ. Equation (1.3), Gauss’ ~ is equal to ρ Law, states that the divergence of the electric field E 3 everywhere on R . Equation (1.2), Faraday’s Law of Induction, states that an electric field is induced whenever the magnetic field changes. On the other hand Equation (1.4), Ampere’s Law with Maxwell’s Correction, states that a magnetic field is induced whenever the electric field changes and whenever electric charge moves around in space. The flow of electric charge through space is described by the current density field J~ in Equation (1.4). DIFFERENTIAL FORMS ON R3 : REWRITING MAXWELL’S EQUATIONS 3 These equations encapsulate our understanding of electromagnetism. Still, if we take a moment to look at all four of them at once, we may find our mathematical sensibilities a little offended. For one thing, Equations (1.1) and (1.2) look almost exactly like Equations (1.3) and ~ and E ~ switching roles. The only differences are the minus (1.4), with B sign in Equation (1.4) and the non-zero right hand sides in the second pair of equations. We might infer that there is some redundancy or indirectness in this presentation of Maxwell’s Equations, and since we have heard on the street that electricity and magnetism are different ~ and E ~ faces of the same force, we might wonder why separate fields B are needed at all, instead of some single unified field. Additionally, Equations (1.2) and (1.4) involve the cross product. This operation requires an arbitrary choice of handedness on R3 and is only defined for pairs of vectors in R3 , and so for instance we cannot comfortably work with these equations on Minkowski spacetime, which is a special version of R4 . There is some property of the cross product which is essential to these equations, though; we might wonder if we can isolate this property “behind” the handedness convention. One last complaint we might raise - and this is a more subtle thing - is that these equations are written in a way which depends on global coordinates on R3 . This is undesirable because, in addition to more technical reasons, “the universe does not come equipped with any standard set of coordinates”. Thus we would like to rewrite these equations in some elegant way which does not depend on a global coordinate system and involves only one unified field. But there is no obvious way for us to do this; our best lead right now is that we would like to get rid of the cross product in these equations. With this goal in mind, we set out to develop some new machinery. 2. The Exterior Algebra over a Vector Space Let us consider the characteristics of the cross product as it operates on the vector space R3 before we attempt to replace it with a more general product defined on arbitrary vector spaces, which we will denote by ∧ and refer to as the wedge product. For starters, the cross product is bilinear and distributive over vector addition on R3 , and R3 is closed under the cross pruduct. We will want our wedge product to share these three properties. The real distinguishing characteristic of the cross product, though, is that it is antisymmetric. That is, ~v × w ~ = −w ~ × ~v 4 FORREST GORDON for all vectors ~v and w ~ in R3 . This is ordinarily presented as a consequence of the Right Hand Rule: if ~v and w ~ are linearly independent 3 vectors in R , then they span a plane in R3 , and the cross products ~v × w ~ and w ~ × ~v are both normal to this plane but have opposite magnitudes, according to the Right Hand Rule. But antisymmetry would also follow from a Left Hand Rule convention on R3 , and since we are only interested in what is happening behind the handedness convention, we simply make our wedge product antisymmetric without any concern for generalizing the Right Hand Rule. This means, though, that if we would like to take a wedge product of two vectors on some vector space V , we are left without a natural way to place this product in V . Our solution to this apparent dilemma is simple, and perhaps surprising: we do not require wedge products to reside in V at all! Instead, we treat them simply as formal products, and rather than give them geometric meaning inside of V (as cross products have in R3 ), we deal with them outside of V via the properties of the wedge product which we have chosen above, along with one last property (which is not a property of the cross product): we decide that our wedge product will be associative. In vector calculus we often work with expressions like ~u + (~v × w), ~ where the addition of ~u and ~v × w ~ makes sense because both of these summands are elements of V . We will now be working with expressions like ~u + (~v ∧ w), ~ even though ~u is an element of V and ~v ∧ w ~ is not. We will also encounter expressions like ~u ∧ (~v ∧ w). ~ Manipulating wedge products “alongside” the vectors in V in this way will generate a superspace of V called the exterior algebra over V , denoted ΛV . We pause to note that including associativity may seem like a strange and unwarranted step: why make the wedge product associative when the cross product - our starting point for creating the wedge product - is not associative itself? In fact, it is easily shown that the cross product could not have been associative and still satisfied the Right Hand Rule. Since we have abandoned the Right Hand Rule, we have no reason not to assume associativity. The major benefit of an associative wedge product is that we will be able to deal with longer wedge products, such as the 3-fold wedge product ~u ∧~v ∧ w, ~ without worrying about the order in which wedge products are evaluated, since (~u ∧ ~v ) ∧ w ~ = ~u ∧ (~v ∧ w). ~ The bilinearity, distributivity, antisymmetry, and associativity of the wedge product impose a nice, natural structure on the exterior algebra ΛV , which we now describe. Let {~v1 , . . . , ~vn } be a basis for V . The bilinearity of ∧ has the important consequence that every k-fold wedge product of vectors in V can be written as a linear combination of k-fold wedge products of basis DIFFERENTIAL FORMS ON R3 : REWRITING MAXWELL’S EQUATIONS 5 vectors. Furthermore, because ∧ is associative and antisymmetric, every k-fold wedge product of basis vectors may be rearranged so that its k basis vectors are written out in non-descending order (with respect to the indices of the basis vectors), and this only involves multiplication by some power of −1. For example, the 4-fold wedge product ~v1 ∧ ~v4 ∧ ~v2 ∧ ~v7 is equivalent to the non-descending wedge product −~v1 ∧ ~v2 ∧ ~v4 ∧ ~v7 . Thus every k-fold wedge product of vectors in V is a linear combination of non-descending k-fold wedge products of basis vectors: X ασ ~vσ(1) ∧ · · · ∧ ~vσ(k) , (2.1) w ~1 ∧ · · · ∧ w ~k = σ∈L where • L is the set of all non-descending lists of k (not necessarily distinct) basis vectors, • ~vσ(i) is the ith entry in the list σ, and • each ασ is a real number. The bilinearity of ∧ also gives us that w ~ ∧0 = 0∧w ~ = 0 for each w ~ in V , and this is easily extended to the statement that for all w ~ i in V , (2.2) w ~1 ∧ · · · ∧ 0 ∧ · · · ∧ w ~ k = 0. The antisymmetry of ∧ gives us that for all w ~ in V , (2.3) w ~ ∧w ~ = 0. Therefore, in (2.1), for each list σ which is not strictly ascending, we have by (2.3) that the wedge product ~vσ(1) ∧ · · · ∧ ~vσ(k) contains a zero, and then by (2.2) that the whole product is zero. Thus the set of all strictly ascending k-fold wedge products of basis vectors ~vi is a basis for the vector space (over R) spanned by k-fold wedge products of vectors k in V . This space is denoted Λ V . If we write n = dim V , then it is n clear that there are exactly k elements in this basis, and so we have: n k dim Λ V = . k Now an element of Λ0 V - that is, a linear combination over R of 0-fold wedge products - is not really a wedge product of vectors at all; it is just some coeffecient in R. Likewise, an element of Λ1 V is really just an element of V . Thus we have natural isomorphisms Λ0 V ∼ = R and Λ1 V ∼ V , which agree with the above equation. In general, an element = of ΛV can be broken down uniquely into (n + 1) components, with one 6 FORREST GORDON component beloning to each Λk V as k varies from 0 to n. That is, the exterior algebra over V is the direct sum of these vector spaces: n M ΛV = Λk V . k=0 Notice that the exterior algebra over V is a finite dimensional vector space: n X dim ΛV = dim Λk V = 2n . k=0 Notice also that ΛV is closed under the wedge product for every vector space V , just as R3 is closed under the cross product. 3. Differential Forms on R3 We are starting to get somewhere now. In the wedge product and the exterior algebra it generates over any given vector space, we have found our sought-after replacement for the cross product. We could try to use the exterior albegra ΛR3 to rewrite Maxwell’s Equations. It turns out - for reasons which we will not discuss in this short survey - that this is not what we want to do. Instead, we will work with the exterior algebra over the vector space of 1-forms on R3 . These objects are defined on every smooth manifold, but for concreteness we will describe them only in the special case of R3 . Let C ∞ (R3 ) denote the set of all smooth (that is, infinitely differentiable) functions on R3 , and let Vect(R3 ) denote the set of all smooth vector fields on R3 . Notice that if f is an element of C ∞ (R3 ) and V is an element of Vect(R3 ), then we can multiply V by f “pointwise” on R3 to obtain another vector field, f V . That is, at each point x in R3 , (f V )|x = f (x) · V |x . We can also “operate” on f with V to obtain a new smooth function, written V f , whose value at every point x in R3 is equal to the directional derivative of f along the vector V |x based at x: (V f )(x) = DV |x (f ). ∞ 3 Since C (R ) and Vect(R3 ) are both vector spaces, it makes sense to think about maps between them. A 1-form on R3 is a map ω : Vect(R3 ) → C ∞ (R3 ) which is linear over C ∞ (R3 ). In other words, for all V, W ∈ Vect(R3 ) and f ∈ C ∞ (R3 ), ω(f V ) = f ω(V ) and DIFFERENTIAL FORMS ON R3 : REWRITING MAXWELL’S EQUATIONS 7 ω(V + W ) = ω(V ) + ω(W ). Perhaps this definition alone does not give us much of a sense for what a 1-form really is, but the following important class of examples of 1forms ought to give us a good hint. Let f be any smooth function on R3 , and define the differential of f to be the map from Vect(R3 ) to C ∞ (R3 ) given by df : V 7→ V f . The linearity of this map over C ∞ (R3 ) is easily checked (try it), and so we find that df is a 1-form on R3 . Now define a 1-form dx = d(f x ), where f x denotes the first coordinate function on R3 . Define dy and dz similarly. It can be shown that the set of 1-forms on R3 , denoted Ω1 (R3 ), is a vector space over C ∞ (R3 ), with basis {dx, dy, dz}. That is, every 1-form ω on R3 can be written uniquely as some linear combination ω = f dx + gdy + hdz, where f , g, and h are smooth functions on R3 . This is the vector space over which we will build and use an external algebra to revisit Maxwell’s Equations. We define the space of differential forms on R3 to be L k 3 Ω(R3 ) = Ω (R ), the exterior algebra generated over Ω1 (R3 ) by the wedge product we introduced earlier, but this time with coefficients in C ∞ (R3 ) instead of just R. In this case then, we have Ω0 (R3 ) ∼ = C ∞ (R3 ). Notice that the differential operator defined above determines a map d : Ω0 (R3 ) → Ω1 (R3 ). We now generalize this to define the exterior derivative as the map d : Ωk (R3 ) → Ωk+1 (R3 ), determined uniquely for each integer k, which satisfies the following: (1) d : Ω0 (R3 ) → Ω1 (R3 ) is the differential (2) d is linear over R (3) d(ω ∧µ) = dω ∧µ + (−1)k ω ∧dµ for ω ∈ Ωk (R3 ) and µ ∈ Ω(R3 ) (4) d(dω) = 0 for all ω ∈ Ω(R3 ). These may seem like some strange conditions. Let us get our hands dirty by explicitly calculating exterior derivatives of general k-forms for k = 0, 1, and 2. We will find a nice surprise when we do this. First let us consider the case d : Ω0 (R3 ) → Ω1 (R3 ). Let ω be an element of Ω0 (R3 ). Then of course ω is just a smooth function on R3 , and dω will be a 1-form on R3 , so for all V ∈ Vect (R3 ), we have dω (V ) = V ω ∈ C ∞ (R3 ). 8 FORREST GORDON Now for all p ∈ R3 , d (ω(p + tV |p )) (dω(V ))(p) = dt d = ( ωhpx + tVx (p), py + tVy (p), pz + tVz (p)i ) dt d d = (px + tVx (p))∂x ω(p) + (py + tVy (p))∂y ω(p)+ dt dt d (pz + tVz (p))∂z ω(p) dt = Vx (p)(∂x ω(p)) + Vy (p)(∂y ω(p)) + Vz (p)(∂z ω(p)) = ∇ω|p · V |p . Thus globally we have dω(V ) = ∇ω · V . Since V is arbitrary in this equality, we may write simply dω ≡ ∇ω, (3.1) and so we see that taking the exterior derivative of a 0-form on R3 is “the same as” taking the gradient of a smooth function on R3 ! Let us move on to the case d : Ω1 (R3 ) → Ω2 (R3 ). Let ω be an element of Ω1 (R3 ). Since {dx, dy, dz} is a basis for Ω1 (R3 ), we may write ω = ωx dx + ωy dy + ωz dz, ∞ where ωx , ωy , ωz ∈ C (R3 ). By the linearity of d over R, we have dω = d(ωx dx + ωy dy + ωz dz) = d(ωx dx) + d(ωy dy) + d(ωz dz). Now by the definitions of the wedge product and the exterior derivative and by (3.1), we can manipulate the first term in the right-hand side of this last equation: d(ωx dx) = d(ωx ∧ dx) = dωx ∧ dx + (−1)0 ω ∧ d(dx) = dωx ∧ dx + 0 = ((∂x ωx )dx + (∂y ωx )dy + (∂z ωx )dz) ∧ dx = (∂x ωx )dx ∧ dx + (∂y ωx )dy ∧ dx + (∂z ωx )dz ∧ dx = 0 + (−∂y ωx )dx ∧ dy + (∂z ωx )dz ∧ dx. By similar arguments, we obtain d(ωy dy) = (−∂z ωy )dy ∧ dz + (∂x ωy )dx ∧ dy and DIFFERENTIAL FORMS ON R3 : REWRITING MAXWELL’S EQUATIONS 9 d(ωz dz) = (−∂x ωz )dz ∧ dx + (∂y ωz )dy ∧ dz, and so we add these three summands together to find that dω = d(ωx dx) + d(ωy dy) + d(ωz dz) = (∂x ωy − ∂y ωx )dx ∧ dy + (∂y ωz − ∂z ωy )dy ∧ dz + (∂z ωx − ∂x ωz )dz ∧ dx. Thus the exterior derivative of a 1-form on R3 is “the same as” the curl of a vector field on R3 ! Finally, let us consider the case d : Ω2 (R3 ) → Ω3 (R3 ). Let ω be an element of Ω2 (R3 ); we may write ω = ωxy dx ∧ dy + ωyz dy ∧ dz + ωzx dz ∧ dx, where again ωxy , ωyz , ωzx ∈ C ∞ (R3 ). Then dω = d(ωxy dx ∧ dy + ωyz dy ∧ dz + ωzx dz ∧ dx) = d(ωxy dx ∧ dy) + d(ωyz dy ∧ dz) + d(ωzx dz ∧ dx). Again, we use the definitions above and (3.1) to manipulate the first term in the right hand side: d(ωxy dx ∧ dy) = d(ωxy ∧ (dx ∧ dy)) = (dωxy ) ∧ (dx ∧ dy) + (−1)0 ωxy ∧ d(dx ∧ dy) = [(∂x ωxy )dx + (∂y ωxy )dy + (∂z ωxy )dz] ∧ dx ∧ dy = (∂z ωxy ) dz ∧ dx ∧ dy = ∂z ωxy dx ∧ dy ∧ dz. Similarly, we obtain d(ωyz ∧ dy ∧ dz) = (∂x ωyz ) ∧ dx ∧ dy ∧ dz and d(ωzx ∧ dz ∧ dx) = (∂y ωzx ) ∧ dx ∧ dy ∧ dz, and so by adding these three terms together we find that dω = (∂x ωyz + ∂y ωzx + ∂z ωxy ) dx ∧ dy ∧ dz. That is, the exterior derivative of a 2-form on R3 is “the same as” the divergence of a vector field on R3 ! So we have seen that the gradient of a function, the curl of a vector field, and the divergence of a vector field are all special cases of this exterior derivative with which we are dealing. With this new revelation in mind, we return to Maxwell’s Equations. 10 FORREST GORDON 4. Maxwell’s Equations Revisited Let us consider again the first pair of Maxwell’s Equations: ~ =0 ∇·B ~ ~ + dB = 0 ∇×E dt ~ and the curl of These involve the divergence of the magnetic field B ~ Thus, in this pair of equations, we see B ~ acting the electric field E. ~ acting like a 1-form. Instead of writing these as like a 2-form and E ~ ~ = hEx , Ey , Ez i, let us write the vector fields B = hBx , By , Bz i and E magnetic field as a 2-form: B = Bxy dx ∧ dy + Byz dy ∧ dz + Bzx dz ∧ dx ∈ Ω2 (R3 ) and the electric field as a 1-form: E = Ex dx + Ey dy + Ez dz ∈ Ω1 (R3 ). We are very close to unifying these two fields, but to do so we need move out into R4 . The extra dimension will of course represent time, and we will denote by dt the 1-form on R4 corresponding to this new coordinate. The wedge product of the 1-form E with this new 1-form will be a 2-form, which we can combine with the 2-form B via simple addition. Thus we define the unified electromagnetic field F as a 2-form on Minkowski spacetime R4 : F = B + E ∧ dt. Written in terms of this new 2-form, Equations (1.1) and (1.2) together become the simple conservation law dF = 0. The electromagnetic field is a 2-form on spacetime whose exterior derivative is zero. The verification of this equality involves splitting the exterior derivative d into so-called “space-like” and “time-like” parts, and we will not describe this process here. Equations (1.3) and (1.4) are trickier to rewrite. Unlike the first pair, these lack a property called general covariance; what this means is that re-writing them in terms of F involves the use of a Riemannian metric on Minkowski spacetime. Their re-written form also involves the Hodge star operator, which swaps k-forms with (n − k)-forms in a particularly nice way. This machinery is above the level of this survey, but as promised, we conclude by displaying Maxwell’s Equations rewritten in terms of differential forms: dF = 0 ?d ? F = J. DIFFERENTIAL FORMS ON R3 : REWRITING MAXWELL’S EQUATIONS 11 References [1] Baez, John and Javier P. Muniain, Gauge Fields, Knots and Gravity, Series on Knots and Everything, Volume 4. World Scientific, Singapore, 1994. [2] Lee, John M. Introduction to Smooth Manifolds Springer, New York, 2003. Mathematics Department, Louisiana State University, Baton Rouge, Louisiana E-mail address: [email protected]