Unformatted text preview: 3 sin(θ) cos(θ)/r 0
Note again that the components have been expressed in the standard orthonormal Cartesian basis. For the spherical system one has that sin(ϕ) cos(θ)
g 1 → sin(ϕ) sin(θ) g 2 → cos(ϕ) sin(θ)/r cos(ϕ)
− sin(ϕ)/r (16)
− sin(θ)/r sin(ϕ)
g 3 → cos(θ)/r sin(ϕ) .
0 4 Gradient of a Scalar Function Consider a scalar function f . Its gradient is given as ∇f . This can be
converted through the use of the chain rule into curvilinear coordinates as:
∇f = ∂f i
e = k i ei = k g k .
∂z (17) Typically, however, results are expressed using the physical basis vectors and
not the natural basis vectors. For our two coordinates systems we have upon
∇f = 5 1 ∂f
r sin(ϕ) ∂θ (18)
(19) Gradient of a Vector To compute the gradient of a vector expressed in curvilinear coordinates
we need to be able to compute the gradient of the basis vectors as they are 4 functions of position (unlike in the Cartesian case). Importantly we will need
to know the derivatives
∂ ∂ xk
∂ 2 xk
= j i ek = j i ek .
∂z ∂z (20) The components of these vectors are usually expressed in the dual basis as
Γk = g k ·
ij ∂ gi
∂zj (21) where Γk is called the Christoﬀel symbol. For the cylindrical coordinate
system all of the Christoﬀel symbols are zero except
Γ1 = −...
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- Spring '10