Multivariable Calculus Unit 22: Curl, Div and Flux

We look at the curl and divergence for vector fields in three space and then define the flux integral which will be needed for Stokes theorem.

[danielprovder]

Could you elaborate on how one would extend the construction to higher dimensions? My mathematical maturity tells me to induct on pascals construction, but it’s unclear to me how I could use the previous n operators to get the n+1th. Is there some kind of hockey stick identity for the nth level differential operators to generate the n+1’s? This might be dubious, but what does the symmetric properties of the triangle like nCk = nCn-k imply about the corresponding differential operators? I’d argue that each row represents a sequence of finite dimensional vector spaces, each with dim(v) = nCk. Then for 1 3 3 1, the map between the two 3-dim spaces should be given by some 3x3 matrix. What about in other dimensions?