First of all, as you know if you read all of my last post, the absolute value of a form is a pseudoform only in degree n. But also, the absolute value of a purely imaginary number is a real number. Does this mean that you dislike real numbers and will only use purely imaginary numbers? As geometric algebraists know, n-forms are like purely imaginary numbers; less well known is that analogously, pseudo-n-forms are like real numbers. It is not that I
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Most references only deal with oriented manifolds and thus ignore all of this pseudostuff. But Patrick mentioned a reference in his post (a reply to my last post that's been broken off from the thread), by Theodore Frankel.  I have no idea how good or bad it is. I like Frankel's book very much, but his discussion on pseudoforms was not very satisfying to me. Well, I've gotten around to looking at Frankel's book now. Overall, it seems to rely quite a bit on components and _base_s, although this effect seems to weaken as the book progresses, so perhaps I just haven't read enough of it yet. This even shows up when the volume pseudo-n-form is written vol = o(x) dx^1 ^ ... ^ dx^n, as quoted earlier on this board. I misinterpreted this notation when it was quoted earlier, but here the o(x) is a function of *the*entire*coordinate*chart* x, taking the value 1 or -1 depending on whether x agrees with the orientation. So this _expression_ doesn't even make sense without choosing an orientation, and vol, in Frankel's notation, *equals* either dx^1 ^ ... ^ dx^n or -dx^1 ^ ... ^ dx^n, depending on the relative orientation of the coordinate chart x. This goes along with Frankel's opening sentences on pages 85 and 86: The differential forms and vectors considered so far have not involved the notion of orientation of *space*. However, roughly half of the forms, vectors, and scalars that occur in physics are in fact pseudo-_object_s that make sense only when an orientation is prescribed. The magnetic field pseudovector B is perhaps the most famous example, and we shall discuss this later. This seems to me to get the point almost exactly backwards. Pseudofields *do* make sense in the absence of a prescribed orientation (that is on unoriented manifolds, whether orientable or not). Rather, what requires a prescribed orientation is the _expression_ of a pseudofield *in*local*coordinates*. This is not to say that I'm entirely unhappy with Frankel, however. Far from it!  Section 3.5, on Maxwell's equations, is excellent. This discussion is from a viewpoint employing pseudoforms. He clearly explains *why* the electric field is a 1form, the magnetic field is a 2form, charge is given by a pseudo3form, and current by a pseudo2form.  Then he introduces a metric on space, which allows identification of p-forms and pseudo(n-p)forms, and shows how this is needed only for Maxwell's inhomogeneous equations, not for the homogeneous equations or the Lorentz force law. Of course, this development of Maxwell's equations works just as well on unoriented manifolds, even unorientable ones, and Frankel never explicitly states otherwise; that's just the impression that one might get from previous sections. What I found really nice about this section, however, was the very short subsection 3.5d. Forms and Pseudoforms. Here Frankel writes a *form* measures an *intensity* whereas a *pseudoform* measures a *quantity* . I'd never seen this idea before, but it is true! It even makes sense in Maxwell's equations in dielectric media. Let's look at each quantity in turn: The charge form sigma measures *quantity* of charge; it's a pseudo3form. You integrate it over an unoriented region to get the total charge. The SI unit of the charge pseudo3form is the coulomb. If this confuses you, note that sigma = rho vol, where rho is the usual charge density (a scalar field) and vol is the volume element (a pseudo3form), which is in Cartesian coordinates dx dy dz, or dx ^ dy ^ dz using a right handed orientation. But sigma makes sense, as a density of charge, even if vol doesn't. The current form j measures *flux* of charge, or equivalently *quantity* of current; it's a pseudo2form. You integrate it over a transversely oriented surface to get the total current travelling through the surface in the indicated direction. The SI unit of the current density pseudo2form is the coulomb per second, or ampere. Note that j = J . surf, where J is the usual current density (a vector field) and surf is the surface integral element (a pseudovector valued 2form), which is in Cartesian coordinates (dy ^ dz - dz ^ dy, dz ^ dx - dx ^ dz, dx ^ dy - dy ^ dx) using a right handed orientation.  But j makes sense, as a density of current, even if surf and the dot product don't. The electric field strength form E measures *strength* of electric force; it's a 1form. If a small body carrying a unit charge travels through space, then it will trace a curve and orient it by its direction of travel; you integrate E over the curve to get the electric work done on the body. The SI unit of the electric field strength 1form is the joule per coulomb. Note that E = E . line, where E is the usual electric field (a vector field) and line is the line integral element (a vector valued 1form), which is in Cartesian coordinates (dx, dy, dz). But E makes sense, as a measure of field strength, even if the dot product doesn't. The magnetic field strength form B measures *strength* of magnetic force; it's a 2form. If a thin conductor carrying a unit current travels through space, then it will trace a surface and orient it by first the direction of the current and then the direction of travel; you integrate B over the surface to get the work done on the conductor. The SI unit of the magnetic field strength 2form is the joule per ampere. Note that B = B . surf, where B is the usual magnetic field (a pseudovector field).  But B makes sense, as a measure of field strength, even if surf and the dot product don't. The electric displacement form D measures *quantity* of free charge; it's a pseudo2form. You integrate it over the boundary of an unoriented region (which is a transversely oriented closed surface, with outside and inside clearly distinguished) to get the total free charge in that region. The SI unit of the electric displacement pseudo2form is the coulomb. Note that D = D . surf, where D is the usual electric displacement (a vector field).  But D makes sense, as a density of charge, even if surf and the dot product don't. The magnetic displacement form H measures *quantity* of free current; it's a pseudo1form. You integrate it over the boundary of a transversely oriented surface (which is a transversely oriented closed curve, with a sense of positive circulation *around* it) to get the total free current passing through that surface together with the time derivative of the integral of D on that surface (which guarantees the conservation of free charge). The SI unit of the magnetic displacement pseudo1form is the ampere. Note that H = H . line, where H is the usual magnetic displacement (a vector field).  But H makes sense, as a density of current, even if the dot product doesn't. Now you can write down the Maxwell equations, dD = sigma, dB = 0, dE + B' = 0, and dH = j + D'. But the real fun comes in when you introduce geometry, which is needed to put D and H in terms of E and B. In vacuum, D = e0 *E and H = *B/m0 for fundamental constants e0 and m0. In nice media, D = epsilon *E and H = *B/mu for medium dependent constants epsilon and mu. More general media can be analysed in terms of the polarisation forms P := D - e0 *E and M := *B/m0 - H. So the value of using pseudoforms carefully, rather than including Hodge duals willy nilly, is that we see exactly when geometry is needed

My gymnastics make this understandable as an example of integration of a pseudo1form on a pseudo1chain. Or do you have another way of understanding it? I admit, I didn't understand it at first. Now, I think you can just use regular forms. Not by your method above.  Let me give you a specific example, so that you know that I'm not lying to you. Let f be the constant function 1, and let c be the unit circle, oriented counterclockwise. (For a possible parametrisation of c, take c(t) := e^(2 i pi t) for 0 <= t <= 1.) Then int_c f dz = 0, but int_c f |dz| = 2 pi. (Go ahead and calculate it using the lowbrow definitions above!) OK, now let me do what I promised and explain complex integrals using forms. The first thing to realise is that there is no reason why forms can't be complex valued; this even works on real manifolds. You just do the real parts and imaginary parts separately. Then a form like dz should make perfect sense on the manifold C. z is the identity function from C to C; its real and imaginary parts are called x and y . Then dz is its exterior derivative, a complex valued 1form. You can write things like dz = dx + i dy. If f is a complex valued function on C (*not* assumed to be complex differentiable, only smooth as a function of 2 real variables), then f dz becomes a perfectly good 1form, just complex valued. Meanwhile, C is, among other things, a 2D real manifold. So if c: [a,b] - C is a smooth curve, then it's an unoriented chain in a real manifold. Place an orientation on [a,b], and it's an oriented chain. Unless otherwise specified, it's traditional to give [a,b] the positive orientation and orient c thus. So, the standard way to interpret a reversal of orientation is to define the orientation reversal of c on [-b,-a], as alluded to earlier. However, since you and I know that [a,b] does have another orientation, we can be more clever than that and reverse the orientation of c without changing its domain of definition. Anyway, to integrate int_c f dz is now trivial. f dz is a complex valued 1form on the 2fold C, and c is an oriented 1chain in the 2fold C, so the integral can be found by the usual methods. And the formula used as a defintion by most books, int_a^b f(c(t)) c'(t) dt, is easily derived, since (f.c) dc is obviously the pullback of f dz, if you remember that z is just the identity function on C. (The . in (f.c) indicates composition of functions.) Now, int_c f |dz| is conceptually more difficult, since there is no f |dz| defined on the complex plane. It can only be defined once the curve c has been specified. So, first pull back the 0form f to a 0form on [a,b] and pull back the 1form dz to a 1form on [a,b]. Now since [a,b] is a 1fold, you can take the absolute value of the pullback of dz to get a pseudo1form on [a,b]. Now you can multiply the 0form and the pseudo1form to get another pseudo1form on [a,b]. And this you can integrate, regardless of the orientation of [a,b]. Let's calculate the usual textbook formula in detail. As a warmup, I'll calculate int_c f dz = int_a^b f(c(t)) c'(t) dt first. So, int_c f dz should be int_[a,b] c*(f dz), where * indicates pullback. The pullback of a 1form w under a manifold morphism c is given by <c*(w), v = <w, c*(v), where v is any vector tangent to the domain of c, say at the point p. Also, the pushforward c*(v) of v is a vector tangent to the codomain of c at the point c(p) defined by c*(v)[g] = v[c.g], where g is any function defined near c(p). So, <c*(f dz), v = <f dz, c*(v) = f(c(p)) <dz, c*(v) = f(c(p)) c*(v)[z] = (f.c)(p) v[c.z] = (f.c)(p) v[c] = (f.c)(p) <dc, v = <(f.c) dc, v. Therefore, c*(f dz) = (f.c) dc, and int_c f dz = int_[a,b] (f.c) dc. Now if you remember that [a,b] is by default oriented positively, then you recognise that this is what is normally written as int_a^b f(c(t)) c'(t) dt , using dc = c' dt. OK, now let's work on int_c f |dz| = int_a^b f(c(t)) |c'(t)| dt. Since f |dz| makes no sense on C, f and dz must first be pulled back separately. The
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I think that the de_script_ion via multivariable calculus is lacking something and I can't really put my finger on it (yet). I won't assume too much about integration in multivariable calculus then. Since integration on manifolds is usually reduced to this, then we'd better make sure that we make sense of it too! Now, R^n has a lot of structure, right down to the level of a standard basis. While one normally says that an n-manifold is locally diffeomorphic to R^n, it's actually sufficient to have a local diffeomorphism to an nD vector space. So I'll try to define multiple integration over a vector space, not R^n. This is really just a good way to make sure that the definition doesn't use any extraneous structure (like the preferred basis). So let V be an nD vector space. Naively, we'd try to integrate a function f over a reasonably nice subset R of V. For my reasonably nice subset , I'll take a simplex, since you like to triangulate your manifolds, but actually a much wider class of shapes could be used. I would like to keep to shapes where the Riemann integral is well behaved, so that we can use that theory if we feel like it, and of course simplices fit that bill just fine. So R is a region (simplex) in V and f is a smooth function defined on R. (Smoothness really isn't necessary, but I'll keep things nice.) Can we define int_R f?  No, absolutely not! After all, int_R 1 would be the volume of the region R, and there is no intrinsic notion of size on a mere vector space. Had we worked with R^n, we might have tricked ourselves on this score, using the standard basis of R^n, but we really shouldn't use that. Still, if only V had a basis on it, then the integral could be defined. So what we *can* define is int_{R,E} f, where E is some (ordered) basis. How is this defined?  Simply lay down the coordinate system of E, write the integral over R as an iterated integral (in the right order)

are combined in the Faraday 2form F = E ^ dt + B. This looks mighty strange, it is almost physically painful to look at it. I don't want to dispute the utility of forms in E&M, but this can't be right. Whatever E and B are, they at a very minimum have to live in the same algebra. So, properly speaking, one should write F = Ek dxk^dt - 1/2 eijk Bi dxj^dxk