By A. Blechman
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Additional resources for Mathematics Primer for Physics Grad. Students [Vers. 2.0]
My old calculus professor used to call it the “Jumping-d theorem”, since the “d” jumps from the manifold to the form. In words, this theorem says that the integral of a form over the boundary of a sufficiently nice manifold is the same thing as the integral of the derivative of the form over the whole mainfold itself. You have used this theorem many times before. Let’s rewrite it in more familiar notation, for the case of R3 : k 0 1 2 dφ ∇f · dx (∇ × f ) · dS (∇ · f )d3 x X Path from a to b Surface (Ω) Volume (V) X dφ = ∂X φ b a ∇f · dx = f (b) − f (a) (∇ × f ) · dS = ∂Ω f · dx Ω (∇ · f )d3x = ∂V f · dS V Theorem Name FTOC Stokes Theorem Gauss Theorem Here, I am using vector notation (even though technically I am supposed to be working with forms) and for the case of R3 , I’ve taken advantage of the following notations: dx = (dx, dy, dz) dS = (dy ∧ dz, dz ∧ dx, dx ∧ dy) d3 x = dx ∧ dy ∧ dz As you can see, all of the theorems of vector calculus in three dimensions are reproduced as specific cases of this generalized Stokes Theorem.
You must be very careful which convention is being used - particle physicists often use the minus prescription I use here, while GR people tend to use the mostly plus prescription. But sometimes they switch! Of course, physics doesn’t change, but the formulas might pick up random minus signs, so be careful. 31 Chapter 4 Geometry II: Curved Space Consider the planet Earth as a sphere. We know that in Euclidean geometry, if two lines are perpendicular to another line, then they are necessarely parallel to each other (they never intersect).
In all the examples we have considered so far, we have been in three dimensions, whereas Minkowski space has four. But this is no problem, since nothing above depended on the number of dimensions at all, so that generalizes very nicely (now gµν is a 4 × 4 matrix). There is one more difference, however, that must be addressed. Minkowski space is a “hyperbolic space”: the loci of points equadistant from the origin form hyperbolas. This is obvious when you look at the metric in spacetime: ∆s2 = c2 ∆t2 − ∆x2 (where I am looking in only one spacial dimension for simplicity).
Mathematics Primer for Physics Grad. Students [Vers. 2.0] by A. Blechman