Page 438 - Advanced engineering mathematics

418 CHAPTER 12 Vector Integral Calculus

In this equation, equate coefficients of (dq i ) for i = 1,2,3 to obtain
2
2 2 2

∂x ∂y ∂z
2
h = + + ,
11
∂q 1 ∂q 1 ∂q 1
2 2 2

∂x ∂y ∂z
2
h = + + ,
22
∂q 2 ∂q 2 ∂q 2
2 2 2

∂x ∂y ∂z
2
h = + + .
33
∂q 3 ∂q 3 ∂q 3
We left the cross product terms within the summation because all such terms are zero for
orthogonal coordinates. For example,
∂x ∂x ∂y ∂y ∂z ∂z

2
h = 2 + +
12
∂q 1 ∂q 2 ∂q 1 ∂q 2 ∂q 1 ∂q 2
=∇x(q 1 ,q 2 ,q 3 ) ·∇y(q 1 ,q 2 ,q 3 ) = 0
by virtue of the orthogonality of the curvilinear coordinates. Similarly, each h ij = 0for i = j.To
simplify the notation, write h ii = h i for i = 1,2,3. Finally we can write
2
2
2
2
(ds) = (h 1 dq 1 ) + (h 2 dq 2 ) + (h 3 dq 3 ) . (12.13)
with h 1 ,h 2 ,h 3 given in terms of partial derivatives of x, y, and z in terms of q 1 ,q 2 and q 3 .
EXAMPLE 12.31
We will put these ideas into the context of cylindrical coordinates. Now q 1 =r, q 2 =θ, and q 3 = z.
Compute

2 2 2
∂x ∂y ∂z

h r = + + = 1,
∂r ∂r ∂r

2 2 2

∂x ∂y ∂z
h θ = + + =r,
∂θ ∂θ ∂θ

2 2 2
∂x ∂y ∂z

h z = + + = 1.
∂z ∂z ∂z
In the plane, cylindrical coordinates are polar coordinates and the differential element dx dy of
area in rectangular coordinates corresponds to
dx dy = ds 1 ds 2 = h r h θ dr dθ =rdrdθ.
This accounts for the change of variables formula for transforming a double integral from
rectangular to polar coordinates:

f (x, y)dx dy = f (r cos(θ),r sin(θ))rdrdθ.
D D
We can also recognize r as the Jacobian

∂(x, y) cos(θ) −r sin(θ)
= =r.
∂(r,θ) sin(θ) r cos(θ)
In 3-space,
dx dy dz = h r h θ h z dr dθ dz =rdrdθ dz.

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October 14, 2010 14:53 THM/NEIL Page-418 27410_12_ch12_p367-424