Mastering Differential Geometry with the Covariant Derivative

Added: 2026-05-12

[00:00:00]In order to understand the complex mathematics of differential geometry, one must first master the concept of the coariant derivative.

[00:00:10]This powerful tool allows the results of ordinary differential calculus to be extended to the domain of curved surfaces leading directly to precise formulations for concepts like geodeics connections and remmanian curvature.

[00:00:29]ultimately unleashing powerful new ways to analyze abstract geometries including that of general relativities for dimensional spacetime.

[00:00:41]This is dialect with the covariant derivative.

[00:00:58]The covariant derivative is another one of those topics in differential geometry that sounds intimidating and sophisticated, but when all is said and done, winds up being pretty straightforward.

[00:01:09]What is a coariant derivative? Well, really, it's nothing more than the ordinary derivative you would take with a vector. That is start with a vector on a surface, have it undergo a small displacement that subsequently alters it and then take the change of that vector with respect to the change in some parameterization of the displacement.

[00:01:34]And finally, in the infantismal limit, you arrive at your derivative.

[00:01:40]Nothing too unfamiliar hopefully.

[00:01:45]So what's the difference then between an ordinary vector derivative and a covariant one?

[00:01:54]Well, that all just boils down to context.

[00:02:00]That is let's imagine a very basic scenario where we take a vector on a surface or manifold and displace it by a small amount such that the vector remains unchanged during this displacement.

[00:02:12]Its derivative in this scenario is obviously zero which we can restate by saying that its uklidian x and y components are equivalent at either location.

[00:02:26]Next, however, imagine that we decide to assign to our manifold a coordinate system which changes smoothly from point to point.

[00:02:38]For instance, say that at the initial position where our vector is located, that coordinate system looks like this.

[00:02:46]While at the final position, it looks like this.

[00:02:51]Now, in this new context, the components of our vector at either location are now no longer equivalent.

[00:03:00]Indeed, from within the perspective of this variable coordinate system, it's the vector itself that appears to change.

[00:03:11]We therefore now have two derivatives in play.

[00:03:16]A coordinative derivative taken from within the changing reference system and the actual derivative or the one that refers to the geometry of the manifold.

[00:03:30]It is this latter derivative that is termed the covariant one.

[00:03:37]We can thus see that a coariant derivative is nothing more than an ordinary derivative but one that refers to a context which may involve the use of changing coordinate systems or frames of reference such that the coordinative picture must be held distinct from the manifold picture.

[00:03:59]Now the purpose of using such changing reference frames in the first place is to be able to work from within what's called curvy linear coordinates.

[00:04:09]That is in a traditional uklidian setup one assigns x and y coordinates based on the distance out from those respective axes.

[00:04:20]But with a curvy linear system, you instead assign coordinates in essentially any arbitrary, albeit smooth manner.

[00:04:31]Now, if you haven't seen our prior video, an introduction to curvy linear coordinates, we'd highly recommend watching that first.

[00:04:39]But the short gist is that these coordinate frames precisely characterize how a system of curvy linear coordinates is behaving about any infantismal region.

[00:04:49]Essentially, you can think of them as linear approximations of the curvy linear grid about that point.

[00:05:00]Now, each frame spans or parameterizes a space tangent to the manifold there.

[00:05:07]And the total collection of these tangent spaces is called a tangent bundle.

[00:05:14]Such a bundle allows us to work with vectors and other mathematical objects upon the manifold from within the curvy linear coordinate system itself.

[00:05:26]For a two-dimensional manifold like this one, any tangent frame is characterized by a pair of vectors called basis vectors which tangentially point along the directions of the coordinate curves at that respective location.

[00:05:41]If we designate our curvy linear coordinates with the variables u and v, then this basis vector which we'll label eu indicates the positive direction of the u coordinate curve at that point.

[00:05:56]While this basis vector e v indicates the positive direction of the vcoin curve at that point.

[00:06:05]Additionally, the length of these vectors represents one unit of their respective coordinate distance.

[00:06:12]So this length indicates how much manifold distance one unit of U coordinate spans about that infantessimal region.

[00:06:20]While this length indicates how much manifold distance one unit of V coordinate spans there.

[00:06:29]Now thanks to these coordinate basis frames any vector V on our manifold can be expressed as a linear combination of the curvy linear basis about that point.

[00:06:40]That is as some u coordinate component vu * eu plus some v coordinate component vv * e v.

[00:06:52]This in turn sets us up to take vector derivatives from within the curvy linear coordinates themselves.

[00:07:00]To see how, let's first write out our vector in a simpler form like this. V= V K E K.

[00:07:08]The K here is a dummy index which stands in for either the U or the V variable and its appearance in the upper and lower indices indicates that a summation should occur over both the U and V terms.

[00:07:26]This notation, the Einstein notation, allows us to preserve the conceptual structure of the vector while avoiding the writing out of repetitive and lengthy summations.

[00:07:37]And it also conveniently generalizes to higher dimensions.

[00:07:44]Next, let's say that our vector, which starts at point A on the manifold, is displaced to point B.

[00:07:52]As we move from A to B, this vector undergo some change DV with respect to the manifold.

[00:08:01]But of course, our vector isn't the only thing changing. As it changes, our basis vectors change underneath it, possibly both in length and direction, depending on however the coordinate curves there are behaving, which again can be in essentially any arbitrary fashion.

[00:08:24]So how do we express the change of our vector DV from within such an arbitrary coordinate system then?

[00:08:34]Well, the key is to recognize that since these basis vectors change smoothly across the manifold, they constitute functions of the manifold itself.

[00:08:45]That is in the coordinative view the basis vectors always have 1 0 or 01 components but on the manifold they'll have uklitian x and y components.

[00:08:59]The magnitude of these uklitian components in turn can be expressed as functions of the curvy linear coordinates that is as functions of u and v.

[00:09:12]For instance, in the most familiar curvy linear coordinate system, polar coordinates, the ukidian x and y components of a radial basis vector are given by cosine theta and sin theta respectively and those of the theta basis vector by minus r sin theta and r cosine theta respectively.

[00:09:39]Thus, when we consider a vector V= VK EK being expressed in terms of some curvy linear basis, we're actually looking at a product function.

[00:09:50]Since both how the basis vector EK appears across the manifold and what component or proportion VK of that basis vector is currently in play are going to be functions of U and V that smoothly differ at every point across the manifold.

[00:10:10]This means that expressing the infantessimal change of our vector with respect to the manifold requires invoking the chain rule. That is DV equals DV K E K plus VK DE K.

[00:10:34]Now this first term on the right is the coordinative differential.

[00:10:40]It's how the change in the vector appears from within the curvy linear coordinates.

[00:10:51]The second term meanwhile is the correction term.

[00:10:55]It accounts for and offsets that apparent change.

[00:11:04]As a simplified example, let's consider the case where we have a vector that undergoes no change on the manifold when displaced and that we are working with basis vectors that are essentially uklidian with the exception that when displaced along this respective direction the u basis vector shrinks.

[00:11:26]Then in this case from the coordinative view the vector V appears to have grown by an amount equal to DVU.

[00:11:36]But the covariance implies that with respect to the manifold picture this change is being offset by a second term VUDu where deu is the change of the EU basis vector with respect to the manifold.

[00:11:56]We now have our expression for DV or the infantessimal change in our vector with respect to the manifold.

[00:12:06]To make this expression fully coariant, however, we need to express de not in terms of ukitian x and y vectors but rather in terms of the curvy linear basis itself.

[00:12:20]More precisely, this means we need to find how the basis vectors change across the manifold when displaced from point A to point B and then express that change in terms of the original basis at point A.

[00:12:34]This requires a few steps. First, going back to our expression dvk ek plus vkdee k. Let's note that because these are separate summations, we can swap the dummy index K in this second term for any other dummy index we might like.

[00:12:53]In this case, we'll use the dummy index J to rewrite VKD K as VJ DEJ.

[00:13:02]Our expression for DV means exactly the same thing as it did before, but as we'll soon see, reabeling the indices like this will come in handy by the end.

[00:13:14]Next, let's recognize that as we transport a given basis vector EJ from point A to point B.

[00:13:22]This involves both a displacement du in the u coordinate direction and a displacement dv in the vcoordinate direction.

[00:13:32]We can thus decompose the change deej of our basis vector into that component which occurred along the u coordinate displacement deej du du and that component which occurred along the vcoordinate displacement deej dv dv.

[00:13:53]We can thereby rewrite deej as dee ej du du plus dee ej dv dv.

[00:14:08]Like before, we're going to express these two terms as a single term with an implied summation. DEJ DXI DXI, where DXI stands in for either DU or DV depending on the I index.

[00:14:27]Now the term dej dxi expressing the derivative of the basis vector with respect to the manifold along the respective i coordinate direction can be expressed in terms of the curvy linear basis via use of the chrisel symbols that is deej dxi equals gamma k i j e k where big gamma is the christophal symbol corresponding to the K component of the derivative of the EJ basis vector with respect to the I coordinate direction. And again, there's an implied summation over the K.

[00:15:12]If you'd like a deeper dive into the concept of the chrisel symbols or how to calculate them, we have a number of videos in our differential geometry playlist which tackle those subjects.

[00:15:24]But essentially the christophal symbols act as little signposts across the manifold telling you how the coordinate curves are changing there.

[00:15:34]Now substituting this definition of those Christophal symbols into the equation for deej above yields deej equals gamma k i j dx e k.

[00:15:51]Substituting this in turn back into our main expression yields dv equals dv k e k plus vj gamma ki j dxi e k.

[00:16:08]Now the K here could have been any dummy index we'd have liked but obviously we chose it so that these indices match up which allows us to neatly factor out the EK and express our equation solely in terms of the curvy linear basis vectors EU and EV.

[00:16:31]Lastly, we can simply divide our expression for DV by the differential for any arbitrary parameterization of the displacement D lambda to arrive at the final expression for our covariant derivative.

[00:16:50]Let's try out our shiny new equation with an example.

[00:16:56]First, as our curvy linear system, we're going to adopt polar coordinates.

[00:17:02]Then we'll consider a vector V with components V R= 0 and V theta= theta.

[00:17:14]Next we'll displace this vector along a path of constant r coordinate.

[00:17:20]And lastly we'll parameterize that path via the amount of theta coordinates traversed.

[00:17:28]So basically we're describing a vector that if displaced along a circle remains tangent to that circle but grows steadily in magnitude.

[00:17:37]This doesn't necessarily represent anything meaningful. It's just a demonstrative example.

[00:17:44]Now in the parameter space our coordinative derivative is quite simple.

[00:17:49]dv d theta merely becomes the vector 01 or e theta.

[00:17:59]But performing the derivative only in the parameter space misses the co-variant component vj gamma k i j dxi d theta e k which we must now calculate.

[00:18:11]Now to do this systematically you'd have to run through all the possible permutations of I, J and K being equal to R and theta which would give you a total of eight terms to calculate.

[00:18:25]However, we can shortcut this a lot by noting that VR equals zero. So we only need to consider terms where J equals theta.

[00:18:37]Furthermore, since our displacement is only in the theta direction, dr r d theta equals zero and we need not consider terms where i equals r.

[00:18:48]Lastly, if we consult the list of christophal components for polar coordinates, we'll see there's only one nonzero component wherein both i and j equal theta, the component minus r.

[00:19:04]Substituting in the appropriate values then for our final remaining term and simplifying we arrive at the expression for our coarent derivative that is dv d theta equ= e theta - r theta er in the parameter space the extra components tells us that there is an additional change on the manifold of our vector due to the change of the coordinates along that displacement path.

[00:19:34]and that this change occurs in the r coordinate direction.

[00:19:41]In the manifold picture, we can see this additional component accounts for the centrial curvature of our theta coordinate curves.

[00:19:54]Now it's important to note that if we had instead chosen as our vector V either of the polar basis vectors E R or E theta and chosen as our transport path either of the coordinate directions R or theta and subsequently parameterized that path via its respective coordinate distance. Then by design the covariant derivative merely reduces to the respective christophal components there.

[00:20:26]Thus we can see that the covariant derivative allows us to conduct essential differential operations from within a system of curvy linear coordinates while still accounting for the curvature of those coordinates themselves.

[00:20:41]This functionality becomes extremely important when we move to a curved manifold because in such a case the curvature of the manifold forces the curvature of any coordinates which are assigned upon it.

[00:20:54]The covariant derivative then becomes not only a useful tool for performing differential operations but in fact gives us a means for intrinsically determining the curvature of a manifold itself.

[00:21:10]However, in order to use the coariant derivative on a curved surface, we first need a method for comparing or connecting any two vectors which lie in different tangent planes.

[00:21:24]Once equipped with that, we'll be just steps away from our goal of being able to precisely calculate, quantify, and decode the Romanian curvature tensor.

[00:21:37]This has been dialect. Thanks for watching.