Let us do compare with respect to metric and with respect to Euclidean metric. For simplicity, let us try with function . Obviously,
Next, by definition of we obtain
Thus
So the question is why do they equal?
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Let us do compare with respect to metric and with respect to Euclidean metric. For simplicity, let us try with function . Obviously,
Next, by definition of we obtain
Thus
So the question is why do they equal?
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You can just put in the second computation to see what’s wrong. I think you want to compare the laplacian of functions in Euclidean space under different metrics right? If we denote the laplacian with respect to by and the corresponding gradient by , then your computation should read
etc. are, of course, not the same under different .
Comment by K — June 1, 2011 @ 21:45
Ah yes, thank for your kind explanation. So my computation was correct but
since the gradient and Laplacian depend strongly on metric. I hope you can continue contributing your opinion here.
Comment by Ngô Quốc Anh — June 1, 2011 @ 22:11