Figure 7-27.
Coordinate surfaces and coordinate curves for cylindrical coordinates.
The partial derivative vectors
∂r
∂ρ = sin θcos φˆe^1 + sin θsin φˆe^2 + cos θˆe^3
∂r
∂θ =ρcos θcos φˆe^1 +ρcos θsin φˆe^2 −ρsin θˆe^3
∂r
∂φ=−ρsin θsin φˆe 1 +ρsin θcos φˆe 2are tangent vectors to the coordinate curves and the scaled vectors
eˆρ=∂r
∂ρ= sin θcos φˆe 1 + sin θsin φˆe 2 + cos θˆe 3ˆeθ=^1
ρ∂r
∂θ= cos θcos φˆe 1 + cos θsin φˆe 2 −sin θˆe 3ˆeφ=^1
ρsin θ=−sin φˆe 1 + cos φˆe 2(7 .108)are unit vectors tangent to the coordinate curves. The spherical coordinate system
is a right-handed orthogonal coordinate system because
ˆeρ·eˆθ= 0, ˆeρ·ˆeφ= 0, ˆeθ·ˆeφ= 0, ˆeρ׈eθ=eˆφThe above unit vectors are sometimes expressed in the matrix form^10 as the column
vectors.
ˆeρ=
sin θcos φ
sin θsin φ
cos θ
, ˆeθ=
cos θcos φ
cos θsin φ
−sin θ
, ˆeφ=
−sin φ
cos φ
0
(^10) See chapter 10 for a discussion of matrices.