TensorCalculus said:
Yes - I see what you mean!
This is actually really, really cool (you have convinced me to learn differential geometry just by showing me this haha). Thank you for showing it to me :D
I don't really understand what this means though - do you mind explaining it to me? I've tried to do some research to clarify on the internet, but I just ended up getting bogged down in endless articles that I don't understand, AI explanations that don't really answer the question and more and more difficult maths :(
Great, I'm glad you feel its worth learning. But I'd hold your horses about learning this stuff seriously at this stage of your education. This is advanced stuff and the exterior covariant derivative is really advanced stuff! Typically, you will learn vector analysis in undergraduate courses and differential geometry in graduate courses and the exterior covariant derivative is most likely taught at a second course at this level. Certainly I wasn't introduced to it in my masters course on theoretical physics. Nor does the textbook I recommended, Lee's Smooth Manifolds explains this. I first learnt about it in Baez & Muniain's Gauge Fields, Knots and Gravity which I have already recommended. It takes the physics approach to differential geometry and so avoids all the very careful constructions in math whilst still being relatively rigorous. The chapter you should look at is chapter II - 3, Curvature and the Yang-Mills equation, pg.250. But I urge you to study the text systematically, or at least go through it trying to understand the concepts that they introduce to get a sense of the terrain. It pays to go through the text step by step. But I also think at this stage of your education you should be getting the fundamentals of physics down - that is really, really important - and not be distracted by advanced and really advanced stuff. I promise you they will still be waiting for you later on in your career. It's a problem that I've had in my own education, so I feel your frustrations. I explained what I did in my previous post as motivation that this stuff is worth learning and not as an indication that you should be learning this stuff at this stage of your education.
Nevertheless, I've written an exposition below which I hope will give you a sense of the terrain without getting bogged down in detail. I hope it helps without further confusing you and leading you astray!
First, to really get grips with the exterior covariant derivative you need to be comfortable with manifolds, tangent bundles and vector bundles as well as their spaces of sections - basically tangent fields and vector fields, as well as the wedge product (a generalisation of the cross product which works for all dimensions, the cross product only works in 3d) and tensor product and of course differential forms and the exterior derivative, oh as well as understanding that tangent fields can also be interpreted as certain differential operators. This is a long list of mathematical technology, so I've written an exposition that gives you a birds-eye view of some of the neccessary concepts without getting into the detail that's needed to really grasp these ideas and make use of them.
The first thing to learn is manifolds. This is basically a formalisation of one aspect of the notion of covariance. In fact Einstein said:
"So there is nothing for it but to regard all imaginable systems of coordinates, on principle, as equally suitable for description of nature"
Manifolds are spaces that are equipped with charts, that is coordinate systems. They are also equipped with transition functions that map between these different charts. This sounds complicated, but it's not so bad as we learn to manipulate manifolds. For example, if you had two manifolds ##M## & ##N## then you can multiply them ##M \times N## and you can take their disjoint sum ##M \sqcup N##. For example if we took the real line is a manifold as well as the circle. If we multiply them we get the infinite cylinder and if we add them we just get the line 'next' to the circle. So we have a calculus of manifolds. It's also important to distinguish between manifolds by themselves and manifolds that are embedded in Euclidean space. For example, if you imagine a sphere - by the way, mathematicians call the surface of a ball, a sphere and the interior of a ball, is just called a ball, then you will probably imagine a sphere in space. This makes it easy to see what the tangent planes on a sphere are. This is a sphere embedded in space. When a mathematician refers to a sphere, it is a sphere all by itself, sort of hanging in the void. This makes it difficult to understand what the tangent planes are as there's nowhere we can take the tangent plane. There's only the void surrounding the sphere. However, there is a way of introducing the tangent planes. This relies on maths but a quick way of seeing this, is simply to invoke a theorem that all manifolds are embeddable in a Euclidean space of high enough dimension and then just take the tangent planes as usual. The manifold with all the tangent spaces attached is called the tangent bundle over that manifold.
Now if ##M## is a manifold - for example a sphere - then ##TM## is the usual notation for the tangent bundle over ##M##. The map ##T## is called the tangent functor and this has nice properties. For example, if ##M## & ##N## are manifolds then ##T(M \times N) = TM \times TN## and obviously ##T(M \sqcup N) = TM \sqcup TN##. This is already important. For example, the tangent space to the infinite cylinder can now be easily calculated. It is ##T(S^1 \times \mathbb{R}) = T(S^1) \times T(\mathbb{R})##.
It also turns out that if we have maps ##f: M \rightarrow N## and another map ##g: N \rightarrow P##, then ##T(g\circ f) = Tg \circ Tf##. To understand this intuitively, it best to see the situation geometrically. When we map one manifold to another, then we also map the tangent planes on the first manifold to the second manifold. Then if we have a composition of maps then the tangent maps also compose. It's best to see this as an illustration but I don't know if Physics Forums supports images. Its also worth mentioning that this rule is basically the chain rule we learn to love in calculus, generalised to multivariable calculus and then to manifolds. The way to show this is to work locally, that is by fixing charts and then working out the various expressions.
Now the covariant derivative ##D## is a replacement for the directional derivative in differential geometry. The usual notion of a directional derivative doesn't work as it relies on the embedding of the manifold in Euclidean space. However, that theorem of embedding the manifold in a Euclidean space of high enough dimension comes in useful again. It turns out that the covariant derivative is the orthogonal projection of the directional derivative onto the tangent space. I say tangent space rather than tangent plane because whilst a 2d manifold has a 2d tangent space - aka a tangent plane - a n-dimensional manifold has a n-dimensional tangent space. Usually, when the covariant derivative is introduced it is done intrinsically but I find this geometric way of defining it more intuitive.
We need the covariant derivative because the exterior covariant derivative combines the exterior derivative and the covariant derivative. I haven't introduced the exterior derivative but we can at least see how the exterior covariant derivative is defined on Baez & Muniain, pg.250
We are going to go for an inductive definition. This is because we have differential 0-forms, 1-forms, 2-forms all the way upto ##n##-forms where ##n## is the dimesion of the manifolds. A differential k-form is said to have order ##k##. The space of k-forms on a manifold ##M## is denoted by ##\Omega^k(M)##. It will turn out that the exterior derivative of a k-form will result in a form of one degree higher, aka a k+1-form. In symbols:
##d: \Omega^k(M) \rightarrow \Omega^{k+1}(M)##
We really ought to have a superscript ##k## on the ##d## but this is just usually left implicit to avoid notational clutter.
Now, its important to realise 0-forms on a manifold are precisely real valued functions on the manifold. Then the exterior derivative ##d## of a differential 0- form - aka a function ##f## - is defined by:
##df(v) := v(f)##
Here ##v## is a vector field, really a tangent field. Now I've already said above that tangent fields can be reinterpreted as a certain differential operator. So ##v(f)## is saying ##v## is deriving ##f##.
Now the exterior covariant derivative is defined by introducing an auxilary covariant derivative ##D## and this must live on some vector bundle ##E##. We haven't introduced vector bundles before. They are a generalisation of tangent bundles. And just as tangent bundles have a covariant derivative, so do vector bundles. Now instead of having k-forms we have instead E-valued k-forms. The space of all E-valued k-forms is written as ##\Omega^k(M, E)##. Then the exterior covariant derivative maps E-valued k-forms to E-valued k+1-forms. Or in symbols:
##d^D : \Omega^k(M, E) \rightarrow \Omega^{k+1}(M, E)##
Here again the ##d^D## ought to have a superscript ##k##, but its left implicit. Now to run the inductive definition of ##d^D## we need to find out what E-valued 0-forms are. These are just vector fields of E. We denote them by latin letters, so in the following formula for ##d^D##, by the letter s.
##(d^Ds)(v) = D_vs##
Thus we see in the zeroth step of the inductive definition we see we have replaced the derivative ##v## acting on ##f## by ##D_v## on ##s##. Here ##f## is a 0-form, aka a function, and ##s## is an E-valued 0-form, aka a vector field in ##E##.
We then inductively define ##d## for higher differential forms via a variation of the Liebniz rule for the derivative of products (Baez & Muniain, pg.63) :
##d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta##
Here ##k## is the order of the differential form ##\alpha##. The symbol ##\wedge## is the wedge or exterior product - it's the generalisation of the cross product to any dimension because the usual cross product just works in 3d only. The corresponding inductive definition for the exterior covariant derivative is on Baez & Muniain, pg.250. It is also a variation on the Liebniz product rule:
##d^D(s \otimes \alpha) = d^Ds \wedge \alpha + s \otimes d\alpha##
Here ##s## is an E-valued 0-form and ##\alpha## is an E-valued k-form. Here the symbol ##\otimes## is the tensor product.
It's been a long post but I hope this gives you some sense on how the exterior covariant derivative comes about. To be honest, I don't expect you to grasp all this as I've skipped over a lot of detail - for example, what is the wedge product. And don't feel frustrated if it escapes you, as I've already said the exterior covariant derivative is really, really advanced stuff. But I do hope what I've written gives you a feeling for the shape of the terrain you would have to cover to rigourously learn this stuff. If you have any questions on what I've written then feel free to ask.
