The Metric Tensor: Definition and Properties

[@prosimion]

That was frickin awesome….👏 Warmed my math heart ❤️‍🔥

[@altissimo4158]

Great intro to the topic

[@barcabashar]

Thank you your explanation is so good.

[@FacultyofKhan]

Glad it was helpful!

[@n0tthemessiah]

Yup, that's a metric tensor alright

[@eulefranz944]

Classy as always:)

[@FacultyofKhan]

Appreciate it! Nice to hear from you again!

[@I.H.R07]

Nice

[@philp4684]

8:00 That bilinearity definition confuses me. It's just ordinary multiplication & addition of numbers (matrix components, vector components, and a scalar), isn't it? So won't those linearity relations always be true?

[@benburdick9834]

I'm still confused about what "fundamentally different" means here. Also, what does a non-euclidean metric tensor look like?

[@FacultyofKhan]

@@benburdick9834 In the intuitive sense, it means that in different spaces, you calculate distances differently. In Euclidean space, you’re probably familiar with using a straight line to calculate distances between two points. But take a non Euclidean space like the surface of a sphere: if you want to calculate the distance between two points, you can’t just take a line that penetrates the sphere and use that, you have to go along the surface of the sphere to calculate distance so your metric tensor becomes fundamentally different from your Euclidean metric tensor. Hope that clarifies things!

[@karrisrivenkatakrishnaredd1212]

@@FacultyofKhan but we are using the spherical coordinates for that only right

[@FacultyofKhan]

@@karrisrivenkatakrishnaredd1212 Spherical coordinates are the most convenient way of expressing the metric tensor of a sphere, yes. However, it's not the same as the metric tensor in Euclidean space because on a sphere, your distance from the origin R is constrained and can only be a constant since you're stuck on the sphere's surface.

[@martinmadrazzi8629]

I think theres an error in the thumbnail, it should be g_{ij}=e_i • e_j

[@FacultyofKhan]

Fixed! Thanks for letting me know!