Page 148 - Intro to Tensor Calculus

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where a αβ is the metric of the surface. This same element when viewed as a spatial element is represented

2
j
i
ds = g ij dx dx . (1.5.60)
By equating the equations (1.5.59) and (1.5.60) we find that

i
∂x ∂x j
i j α β α β
g ij dx dx = g ij du du = a αβ du du . (1.5.61)
∂u ∂u β
α
The equation (1.5.61) shows that the surface metric is related to the spatial metric and can be calculated
i
∂x ∂x j
from the relation a αβ = g ij . This equation reduces to the equation (1.5.21) in the special case of
∂u ∂u β
α
α
β
Cartesian coordinates. In the surface coordinates we define the quadratic form A = a αβ du du as the first
fundamental form of the surface. The tangent vector to the coordinate curves defining the surface are given
i
∂x
by α and can be viewed as either a covariant surface vector or a contravariant spatial vector. We define
∂u
this vector as
∂x i
i
x = , i =1, 2, 3, α =1, 2. (1.5.62)
α α
∂u
Any vector which is a linear combination of the tangent vectors to the coordinate curves is called a surface
α
i
vector. A surface vector A can also be viewed as a spatial vector A . The relation between the spatial
i
α
α i
representation and surface representation is A = A x . The surface representation A ,α =1, 2and the
α
i
spatial representation A ,i =1, 2, 3 define the same direction and magnitude since
j
β j
α
β
i
α
β
α i
j
i
g ij A A = g ij A x A x = g ij x x A A = a αβ A A .
α β α β
i
i
α
α
Consider any two surface vectors A and B and their spatial representations A and B where
i
i
α i
α i
A = A x and B = B x . (1.5.63)
α α
These vectors are tangent to the surface and so a unit normal vector to the surface can be defined from the
cross product relation
j
n i AB sin θ = ijk A B k (1.5.64)
i
i
where A, B are the magnitudes of A ,B and θ is the angle between the vectors when their origins are made
to coincide. Substituting equations (1.5.63) into the equation (1.5.64) we find
β k
α j
n i AB sin θ = ijk A x B x . (1.5.65)
β
α
β
α
In terms of the surface metric we have AB sin θ = αβ A B so that equation (1.5.65) can be written in the
form
α
j
k
β
(n i αβ − ijk x x )A B =0 (1.5.66)
α β
which for arbitrary surface vectors implies
1 αβ j k
j
k
n i αβ = ijk x x or n i = ijk x x . (1.5.67)
α β α β
2
The equation (1.5.67) defines a unit normal vector to the surface in terms of the tangent vectors to the
coordinate curves. This unit normal vector is related to the covariant derivative of the surface tangents as