Page 137 - Intro to Tensor Calculus

132

~
~
and is such that the vectors E 1 , E 2 and bn form a right-handed system of coordinates.
If we transform from one set of curvilinear coordinates (u, v)toanother set(¯u, ¯), which are determined
v
by a set of transformation laws
v
v
u = u(¯u, ¯), v = v(¯u, ¯),
the equation of the surface becomes

r r v v v
~ = ~(¯u, ¯)= x(u(¯u, ¯),v(¯u, ¯)) b e 1 + y(u(¯u, ¯v),v(¯u, ¯v)) b e 2 + z(u(¯u, ¯v),v(¯u, ¯v)) b e 3
and the tangent vectors to the new coordinate curves are
r
r
r
∂~ ∂~ ∂u ∂~ ∂v ∂~ ∂~ ∂u ∂~ ∂v
r
r
r
= + and = + .
v
v
∂¯u ∂u ∂¯u ∂v ∂¯u ∂¯ ∂u ∂¯ ∂v ∂¯
v
Using the indicial notation this result can be represented as
i
∂y i ∂y ∂u β
= .
β
∂¯u α ∂u ∂¯u α
This is the transformation law connecting the two systems of basis vectors on the surface.
A curve on the surface is defined by a relation f(u, v) = 0 between the curvilinear coordinates. Another
way to represent a curve on the surface is to represent it in a parametric form where u = u(t)and v = v(t),
where t is a parameter. The vector
r
d~ r ∂~ du ∂~ dv
r
= +
dt ∂u dt ∂v dt
is tangent to the curve on the surface.
An element of arc length with respect to the surface coordinates is represented by
r
∂~ ∂~ α β α β
r
2
ds = d~r · d~r = · du du = a αβ du du (1.5.19)
∂u α ∂u β
r
∂~
∂~
r
where a αβ = ∂u α · ∂u β with α, β =1, 2 defines a surface metric. This element of arc length on the surface is
often written as the quadratic form
1 2 EG − F 2 2
2
2
2
A = ds = E(du) +2Fdu dv + G(dv) = (Edu + Fdv) + dv (1.5.20)
E E
2
and called the first fundamental form of the surface. Observe that for ds to be positive definite the quantities
2
E and EG − F must be positive.
The surface metric associated with the two dimensional surface is defined by
i
r
∂~ ∂~ ∂y ∂y i
r
~
~
a αβ = E α · E β = · = , α, β =1, 2 (1.5.21)
∂u α ∂u β ∂u ∂u β
α
α
with conjugate metric tensor a αβ defined such that a αβ a βγ = δ . Here the surface is embedded in a three
γ
dimensional space with metric g ij and a αβ is the two dimensional surface metric. In the equation (1.5.20)
the quantities E, F, G are functions of the surface coordinates u, v and are determined from the relations
i
∂~ r ∂~ r ∂y ∂y i
E =a 11 = · =
1
∂u ∂u ∂u ∂u 1
i
∂~ r ∂~ r ∂y ∂y i
F =a 12 = · = (1.5.22)
1
∂u ∂v ∂u ∂u 2
i
∂~ ∂~ ∂y ∂y i
r
r
G =a 22 = · =
2
∂v ∂v ∂u ∂u 2