Exercise 2.5.2

[Graphics:HTMLFiles/Ex2.5.2-03_1.gif]

<<Calculus`VectorAnalysis`

SetCoordinates[Spherical[r, θ, ϕ]]

Spherical(r, θ, ϕ)

Overscript[r,^] = {Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ], Cos[θ]} Overscript[& ... bsp;             Cos[ϕ], 0}

{cos(ϕ) sin(θ), sin(θ) sin(ϕ), cos(θ)}

{cos(θ) cos(ϕ), cos(θ) sin(ϕ), -sin(θ)}

{-sin(ϕ), cos(ϕ), 0}

∂_r {Overscript[r,^], Overscript[θ,^], Overscript[ϕ,^]}

( 0   0   0 )            0   0   0            0   0   0

∂_θ {Overscript[r,^], Overscript[θ,^], Overscript[ϕ,^]}

( cos(θ) cos(ϕ)    cos(θ) sin(ϕ)    -sin(θ)             ɯ ... 52;) sin(ϕ)   -cos(θ)            0                          0                          0

∂_ϕ {Overscript[r,^], Overscript[θ,^], Overscript[ϕ,^]}

( -sin(θ) sin(ϕ)   cos(ϕ) sin(θ)    0                        ɯ ...    cos(θ) cos(ϕ)    0            -cos(ϕ)               -sin(ϕ)               0

e1 = Laplacian[ψ[r, θ, ϕ]]

1/r^2 (csc(θ) (sin(θ) ψ^(2, 0, 0)(r, θ, ϕ) r^2 + 2 sin(θ) ψ ... os(θ) ψ^(0, 1, 0)(r, θ, ϕ) + sin(θ) ψ^(0, 2, 0)(r, θ, ϕ)))

e2 = Div[Grad[ψ[r, θ, ϕ]]]

1/r^2 (csc(θ) (sin(θ) ψ^(2, 0, 0)(r, θ, ϕ) r^2 + 2 sin(θ) ψ ... os(θ) ψ^(0, 1, 0)(r, θ, ϕ) + sin(θ) ψ^(0, 2, 0)(r, θ, ϕ)))

Simplify[e1e2]

True

Mgrad[f_] := Overscript[r,^] ∂_rf + Overscript[θ,^]/r ∂_θ f + Overscript[ϕ,^]/(r Sin[θ]) ∂_ϕ f

MLapl[f_] := Overscript[r,^] . ∂_rMgrad[f] + Overscript[θ,^]/r . ∂_θMgrad[f]    + Overscript[ϕ,^]/(r Sin[θ]) . ∂_ϕ Mgrad[f]

e3 = Simplify[MLapl[ψ[r, θ, ϕ]]]

1/r^2 (ψ^(2, 0, 0)(r, θ, ϕ) r^2 + 2 ψ^(1, 0, 0)(r, θ, ϕ) r + csc ... , ϕ) + cot(θ) ψ^(0, 1, 0)(r, θ, ϕ) + ψ^(0, 2, 0)(r, θ, ϕ))

Simplify[e1e3]

True

Three ways of solving part (b) are shown.  The first one is a simple use of built in ∇^2operator but does not really follow the text's required path.  The second one is closer to what the text want us to do.  The third one requires more effort but does exactly what the text wants.
To have outputs in the nice matrix form choose: Cell → Default Output Format Type → Traditional.
To write Overscript[r,^] enter: r [CTRL]-&^[CTRL]_ (the last character denotes spacebar).
To write ∂_r use ∂_  on the basic palette.


Created by Mathematica  (September 29, 2003)