Page 440 - Advanced engineering mathematics

420 CHAPTER 12 Vector Integral Calculus

We can also recognize ρ sin(ϕ) as the Jacobian
2
∂(x, y, z)
∂(ρ,θ,ϕ)
seen in the general expression for transformation of triple integrals.
Now let u i be a unit vector in the direction of increasing q i at the point
(x(q 1 ,q 2 ,q 3 ), y(q 1 ,q 2 ,q 3 ), z(q 1 ,q 2 ,q 3 )). In cylindrical coordinates, these unit vectors can be
written in terms of the standard i, j, and k as
u r = cos(θ)i + sin(θ)j,
u θ =−sin(θ)i + cos(θ)j,
u z = k.
In spherical coordinates,
u ρ = cos(θ)sin(ϕ)i + sin(θ)sin(ϕ)j + cos(ϕ)k,
u θ =−sin(θ)i + cos(θ)j,
u ϕ = cos(θ)cos(ϕ)i + sin(θ)cos(ϕ)j − sin(ϕ)k.
Unlike rectangular coordinates, where the standard unit vectors are constant, with orthogonal
curvilinear coordinates, the vectors u 1 , u 2 , u 3 are generally functions of the point.
A vector field in curvilinear coordinates has the form
F(q 1 ,q 2 ,q 3 ) = F 1 (q 1 ,q 2 ,q 3 )u 1 + F 2 (q 1 ,q 2 ,q 3 )u 2 + F 3 (q 1 ,q 2 ,q 3 )u 3 .
We want to write expressions for the gradient, Laplacian, divergence, and curl operations in
curvilinear coordinates.
Gradient

Let ψ(q 1 ,q 2 ,q 3 ) be a scalar-valued function. At any point, we want ∇ψ to be normal to the level
surface ψ = constant passing through that point, and we want this gradient to have magnitude
equal to the greatest rate of change of ψ from that point. Thus, the component of ∇ψ normal to
q 1 = constant must be ∂ψ/∂s 1 ,or
1 ∂ψ
.
h 1 ∂q 1
Arguing similarly for the other components, we have
1 ∂ψ 1 ∂ψ 1 ∂ψ
∇ψ(q 1 ,q 2 ,q 3 ) = u 1 + u 2 + u 3 .
h 1 ∂q 1 h 2 ∂q 2 h 3 ∂q 3
Divergence

We will use the flux interpretation of divergence to obtain an expression for the divergence of a
vector field in curvilinear coordinates. First write
F = F 1 u 1 + F 2 u 2 + F 3 u 3 .
Referring to Figure 12.31, the flux across the face abcd is approximately
F(q 1 + ds 1 ,q 2 ,q 3 ) · h 2 (q 1 + ds 1 ,q 2 ,q 3 )h 3 (q 1 + ds 1 ,q 2 ,q 3 )dq 2 dq 3 .
Across face ef gk the flux is
F(q 1 ,q 2 ,q 3 ) · u 1 h 2 (q 1 ,q 2 ,q 3 )h 3 (q 1 ,q 2 ,q 3 )dq 2 dq q .
Across both of these faces the flux is approximately
∂
(F 1 h 2 h 3 )dq 1 dq 2 dq 3 .
∂q 1

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October 14, 2010 14:53 THM/NEIL Page-420 27410_12_ch12_p367-424