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In the lecture you mentioned that the Laplacian is invariant to your choice of direction -- what exactly does that mean? Is it like, we can take any set of n linearly independent vectors in R^n to compute the directional derivatives and it still gives the same output?


What does the coordinates don't matter mean?


How does the laplacian conceptually represent an average of neighbors? Are we using the derivatives in each direction as an approximation?


Can this be thought of intuitively as 'how fast the gradient changes as we follow it'?