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So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force. $$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$ $$\nabla \cdot \textbf{u}=0$$ We can apply the...

https://www.researchgate.net/publication/301874096_Emergent_behavior_in_active_colloids/link/5730bb3608ae08415e6a7c0a/download (expression 9 on this document derivation). I understand the need for substitution etc into the integral. What puzzles me is how the integral equals what it does. If...

Thank you very much

The K-L condition has projection operators onto the codespace for the error correction code, as I understand it. My confusion I think comes primarily from what exactly these projections are? As in, how would one find these projections for say, the Shor 9-qubit code?

In general, if R is the recovery channel of an error channel ε, with state ρ, then and according to these lecture slides, we get the final result highlighted in red for a bit flip error channel. I am simply asking how one reaches this final result. Thank you (a full-ish derivation can be found...

With free part L=-½(∂Φ)^2 -½m^2 Φ^2 and interaction term L=½gΦ^2 Any help would be appreciated, thank you.

There isn't much unfortunately. All we were given was the line element and range, the fact that its a two dimensional geometry and told to calculate what geometry that is.

Apologies for the notation. Yes you are right in what I meant. I don't think I recall being taught that the determinant being less than or equal to zero means the metric is lorentzian. Would 'Lorentzian' actually count as the geometry then?

I've tried spherical, but (I don't think) the line element given can be converted successfully. Also the its meant to be a 2-D geometry (unless it means a 2-D geometry embedded in 3 dimensions). It also doesn't specify whether or not the coordinate system is orthogonal. I'm at a bit of a loss...

i'm trying to find what sort of 2-d geometry this system is in, I've been given the line element 𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙 where 0≤𝜙<2𝜋 and 0≤𝜃<𝜋/2 Im just not sure where to start. I've tried converting the coordinates to cartesian to see if it yields a...

When parallel transporting a vector along a straight line on flat space, does the connection (when calculating the covariant derivative) always equal zero? Do things change at all when using an arbitrary connection, rather than Christoffel symbols?

Is there a general method to determine what geometry some line element is describing? I realize that you can tell whether a space is flat or not (by diagonalising the matrix, rescaling etc), but given some arbitrary line element, how does one determine the shape of the space? Thanks