This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 3 ) = (r, ϕ, θ) in the spherical case. For the basis vectors we
will introduce for two types of basis vectors. The natural basis vectors and 1 x3 x3 ϕ
r x2 x2 z
θ θ x1 x1 Figure 1: Deﬁnition of the cylindrical and spherical coordinate systems.
the physical basis vectors. Both bases are orthogonal but the physical basis
has the additional property of orthonormality. The basic deﬁnitions are
g k gk = (5) ek (6) To diﬀerentiate between the physical basis vectors and the usual Cartesian
ones we typically write er , eθ , · · · etc. For the cylindrical coordinate system
one has: cos(θ)
0 sin(θ) r cos(θ) 0
eθ = g 2
ez = g 3 .
Note that the components have been expressed in the standard orthonormal
Cartesian basis. For the spherical system one has that sin(ϕ) cos(θ)
r cos(ϕ) cos(θ)
g 1 → sin(ϕ) sin(θ) g 2 → r cos(ϕ) sin(θ) cos(ϕ)
−r sin(ϕ) (9)
−r sin(ϕ) sin(θ)
g 3 → r sin(ϕ) cos(θ) 0
er = g 1 2 and
er = g 1 2.1 1
eϕ = g 2
r eθ = 1
r sin(ϕ) 3 (10) Physical and Natural Components As with all bases we can express the components of vectors and tensors with
respect to our new curvilinear bases. In this regard, it is very important
with the curvilinear coordinates to know whether or not the com...
View Full Document
- Spring '10