1) Compute the following integrals involving the Dirac delta function
1.1) R5
−2δ(x)f(x)dx
1.2) R5
−1δ(x−3)f(x)dx
1.3) R4
−1δ(2x−3)f(x)dx
1.4) R3
−2δ(x2−1)f(x)dx
1.5) R1
−5δ(x2−4)f(x)dx
2) Compute the following integrals involving the 3dDirac delta function
2.1) R2
−2dx R1
−3dy R5
0dzδ(~x −(ˆx+ 2ˆz))f(x, y, z)
2.2) R2
−2dx R1
−3dy R5
0dzδ(~x −(ˆx+ 2 ˆy+ 2ˆz))f(x, y, z )
2.3) R2
−2dx R3
−3dy R5
0dzδ(~x −(ˆx+ ˆy+ 2ˆz))f(x, y, z )
3) Consider the vector field ~
F= 3xˆx+ (2x+z)ˆy+ 3yˆz
3.1) Compute ~
∇ · ~
F
3.2) Compute ~
∇ × ~
F
3.3) Can you split ~
Finto ~
F=~
F1+~
F2such that ~
∇ × ~
F1=~
0 and ~
∇ · ~
F2= 0?
3.4) Find a scalar Vsuch that ~
∇V=~
F1
3.5) Find a vector ~
Asuch that ~
∇ × ~
A=~
F2
4) Optional Exercise: There is an identity for the most general delta function (in one
dimension) one can write down
Zx2
x1
δ(g(x))dx =X
iZx2
x1
δ(x−xi)
|g0(xi)|dx
where the sum above is over all the points where g(x) is zero in the interval [x1, x2] i.e.
g(xi) = 0 ∀i.
4.1) Split the integral over a sum over the zeros of the function g(x)
Zx2
x1
δ(g(x))dx =X
iZxi+
xi−
δ(g(x))dx
In the above you are only integrating over small intervals. Why can you discard the rest?
Can you give an idea on what size the intervals should be so as not to run into trouble?
4.2) In the region around each xiyou can write g(x)≈(x−xi)g0(xi). Why? You thus have
Zx2
x1
δ(g(x)) = X
iZxi+
xi−
δ(xg0(xi)−g0(xi)xi)dx
4.3) Prove that Rx2
x1δ(ax−xi)dx =1
|a|R
1
|a|x2
1
|a|x1δ(x−xi
|a|)dx. Why is the absolute value important?
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