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OK, I'm currently reading Hughes' Finite Element Method book, and I'm stuck on a chapter the goal of which is to prove that the Galerkin solution to a boundary value problem is exact at the nodes.
So, the author first speaks about the Dirac delta function: "Let [itex]\delta_{y}(x) = \delta(x-y)[/itex] denote the Dirac delta function." Now, what exactle does this mean? Is it simply an operator, where [itex]\delta[/itex] can be any function, and y any real number?
Further on, the author points out that, for a continuous function w on [0, 1], we write: "[itex](w, \delta_{y}) = \int_{0}^1 w(x)\delta(x-y)dx = w(y)[/itex]", so "[itex]\delta_{y}[/itex] sifts out the value of w at y". I don't understand where this result comes from.
Any help is highly appreciated.
So, the author first speaks about the Dirac delta function: "Let [itex]\delta_{y}(x) = \delta(x-y)[/itex] denote the Dirac delta function." Now, what exactle does this mean? Is it simply an operator, where [itex]\delta[/itex] can be any function, and y any real number?
Further on, the author points out that, for a continuous function w on [0, 1], we write: "[itex](w, \delta_{y}) = \int_{0}^1 w(x)\delta(x-y)dx = w(y)[/itex]", so "[itex]\delta_{y}[/itex] sifts out the value of w at y". I don't understand where this result comes from.
Any help is highly appreciated.