Channel: Andrew Dotson

Category: Education

Tags: odusphericalgeneral relativitycalculusaffinespherical coordinatesnmsumajorphysicslaplaciandotsonmetric tensorcontravariantgradientdegreegradschoolgrtensorphdandrewcovarianttensor notationdivergencegradstudentderivationchristoffel

Description: Today we generalize the concept of divergence and the laplacian into their covariant forms by substituting the usual del operator with covariant derivatives. We start with a refresher on how covariant and contravariant vectors are related to "ordinary vectors". Once we derive the covariant divergence, we do an example of recovering the equation for the divergence in spherical coordinates. Lastly, with divergence defined, we can immediately write down the covariant laplacian. This series is based off "Tensor Calculus for Physics" by Dwight Neuenschwander which can be found at: amazon.com/gp/product/1421415658/ref=as_li_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=1421415658&linkCode=as2&tag=andrewdotson-20&linkId=b45243268a957a6cfdfc854a8f677a58 Episode 12 on Christoffel Symbols: youtube.com/watch?v=OJ-MoYqz9QY&t=9s&ab_channel=AndrewDotson Video on covariant/contravariant: youtube.com/watch?v=lJYNw-nGDoM&t=1639s&ab_channel=AndrewDotson