Kernel approximation of derivatives ↩

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created: 2022-04-25 16:20:19
modified: 2022-04-25 16:33:50

Where is the nabla operator, derivatives with respect to .

Proof

We substitute into the normal form of kernel approximation:

Where contains derivatives with respect to , not .

Now we can transform the first integral using the divergence theorem, so it becomes a surface integral on , but since has compact support, it vanishes:

Then can flip the sign of the other part by switching to , because the sign of and is opposite inside :

QED