Tensor Calculus 16: Geodesic Examples on Plane and Sphere

[dr.ambiguous]

Maths is amazing. It feels like magic, until you understand it. Then you really appreciate how beautiful it is.

[kimchi_taco]

My professor in physics department should have written warning ⚠️ finally understood Christopher symbols. Big kudos!!

[JgM-iejy]

Eagerly awaiting to get to the great circle for finding the shortest path on a sphere. And then it just trivially falls into place. Brilliant.

[sunsetclub]

Thank you for explaining this, I've been doing quantum gravity research as an undergrad for a while and have never grasped tensors at all until watching your videos.

[tomasflores]

Muchísimas gracias profesor por sus videos.. Magnífico trabajo!!.. Un saludo desde España 🇪🇸

[rockrock]

Thank You eigenchris. You are a very good lecturer. I completely understand the concept.

[deepanjan]

You are just great and your teaching style is just amazing . Thank you so much.

[VERONICA-fxwd]

Bless you & your patience. We all owe you a drink, or tuition ^_^

[vidstige]

Great stuff! Wrote a small program in rust that can integrate geodesic along a sphere and other cool surfaces thanks to this video series.

[uisqebaugh]

Excellent video.

May I suggest that you can also test the sphere geodesic by solving the intersection of a plane passing through the origin with the sphere equation?

[shreeshree]

This lecture series offers crystal-clear explanations and comes as a great contribution towards explaining tensor calculus to students. Thank you EigenChris.

My question: If we want a great circle other than the equator, shouldn't it come from the equations? At 16:29, we write the geodesic equation. If we try v=lambda, and look for a non-equatorial great circle, then we get stuck at the equator. Because by the 2nd equation, cot(u) du/dLambda = 0, So u cannot vary, and we get u=pi/2. A solution like u=cot^{-1}sin( Lambda) [which we get by setting Y=Z] would give a 45-degree great circle. But it gets "lost" from this equation of 16:29. Could you explain?

[vitorsousa]

Astonishing teaching ! thanks a lot, you make differential geometry a simple subject.

[michaelmorgan]

Thank you for providing these friendly and easy-to-follow lectures. FWIW, one very minor correction to the geodesic equation for the sphere: both symmetric Cristoffel terms Gamma^2_12=Gamma^2_21=cot(u) need to be added to the D2v/D(lambda)^2 differential equation. This doubles the cot(u) term, giving 2 cot(u) in the DE.

[Amplituhedron]

Amazing Chris, your explanations are fantastic 👏

[amirkbagheri]

Thank you for these awesome videos. A quick question, you've written the Christoffel symbols with one upstairs index and two downstairs indexes. Do these tie in with the usual vector/covector index positions? Is there a connection there? Thanks!

[sabya]

Your videos are beautiful! Thanks for doing what you do!

[bankaikun]

I've been binge watching all your videos, and I have a few questions about the metric tensor. Why exactly is the metric tensor a 2x2 matrix? Shouldn't it be in the form of [[g11 g21][g12 g22]]? That seems to be more aligned with the indices of the components.

Cause it seems that a 2x2 matrix would imply a (1, 1) tensor, which the metric tensor isn't.

[NeonNotch]

Question that’s probably been answered and if so I’m very sorry. At 15:30 your lower Christophel symbol shows 212=212=... but in the slide previously you said 212=221=... so should the indices on the middle symbol be 221?

[quaereverum]

Something that I was wondering when you made the distinction between intrinsic and extrinsic geometry, and rears its head again now, is the following:
At 8:00 is it not in some sense "cheating" to express the derivative w.r.t. u and v in terms of X, Y and Z? I would imagine that if we were doing intrinsic geometry, we would have no notion of X, Y and Z. Did I get this wrong?

[leandrocarg]

your videos! Please make a series on complex analysis