- #1
geoduck
- 258
- 2
If you have I=∫∫dxdy [∇x∇y δ(x-y)] f(x)g(y)
where ∇x is the derivative with respect to x (and similarly for y), then doesn't it matter which order you take the derivatives? For example:
I=∫∫dxdy f(x) ∇x [∇y δ(x-y)] g(y)
=∫dx f(x) ∇x[-g'(x)]=∫dx f(x) [-g''(x)]
whereas if you take the other order:
I=∫∫dxdy ∇y [∇x δ(x-y)] f(x)g(y)
=∫∫dxdy g(y)∇y [∇x δ(x-y)] f(x)
=∫dx g(x) [-f''(x)]
where ∇x is the derivative with respect to x (and similarly for y), then doesn't it matter which order you take the derivatives? For example:
I=∫∫dxdy f(x) ∇x [∇y δ(x-y)] g(y)
=∫dx f(x) ∇x[-g'(x)]=∫dx f(x) [-g''(x)]
whereas if you take the other order:
I=∫∫dxdy ∇y [∇x δ(x-y)] f(x)g(y)
=∫∫dxdy g(y)∇y [∇x δ(x-y)] f(x)
=∫dx g(x) [-f''(x)]