Title
First Fundamental Form and Arc Length
MATH 2040
October 9, 2016
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First Fundamental Form
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Readings
Readings
Readings: 4.3
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First Fundamental Form
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First Fundamental Form
Definition of the first fundamental form
Definition
Let M be a surface in R3 . The tangent space at p of M is the set of
tangent vectors to M at p. We denote it by Tp M.
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First Fundamental Form
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First Fundamental Form
Definition of the first fundamental form
Definition
Let M be a surface in R3 . The tangent space at p of M is the set of
tangent vectors to M at p. We denote it by Tp M.
Definition
Let M be a surface in R3 . The First Fundamental Form of M is the
map
Ip : Tp M × Tp M → R
given by
Ip (X, Y) = hX, Yi .
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First Fundamental Form
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First Fundamental Form
The first fundamental form in coordinates
The coefficients of the first fundamental form (called “metric
coefficients”) with respect to a coordinate patch x : U → M at (u 1 , u 2 )
are
D
E
gik (u 1 , u 2 ) = xi (u 1 , u 2 ), xk (u 1 , u 2 ) = Ix(u 1 ,u 2 ) (xi (u 1 , u 2 ), xk (u 1 , u 2 )).
These are functions on U.
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First Fundamental Form
October 9, 2016
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First Fundamental Form
The first fundamental form in coordinates
The coefficients of the first fundamental form (called “metric
coefficients”) with respect to a coordinate patch x : U → M at (u 1 , u 2 )
are
D
E
gik (u 1 , u 2 ) = xi (u 1 , u 2 ), xk (u 1 , u 2 ) = Ix(u 1 ,u 2 ) (xi (u 1 , u 2 ), xk (u 1 , u 2 )).
These are functions on U.
You can use them to compute the inner product of tangent vectors
X = X 1 x1 + X 2 x2 and Y = Y 1 x1 + Y 2 x2
by
I(X, Y) =
2
X
gik X i Y k .
i,k =1
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First Fundamental Form
October 9, 2016
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First Fundamental Form
Length of a curve
Consider a curve α(t) = x(α1 (t), α2 (t)) in a surface M.
The length of the curve between a and b can be computed as follows:
v

Z bu
2
u X
dαi dαk 
u
t
gik
dt.
length =
dt dt
a
i,k =1
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First Fundamental Form
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First Fundamental Form
Inverses and determinants of first fundamental form
Let gik denote the coefficients of the FFF with respect to a
parametrization x of a surface M.
Notation:
• Let g = det(gik )
• Let g ik denote the inverse matrix to gik .
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First Fundamental Form
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First Fundamental Form
Inverses and determinants of first fundamental form
Let gik denote the coefficients of the FFF with respect to a
parametrization x of a surface M.
Notation:
• Let g = det(gik )
• Let g ik denote the inverse matrix to gik .
Theorem
Given a coordinate patch x the coefficients of the FFF satisfy
1
2
3
g = |x1 × x2 |2
g 11 = g22 /g, g 12 = g 21 = −g12 /g, g 22 = g11 /g.
P
For all i, k, k g ik gkj = δji .
Here δji = 1 if i = j or 0 otherwise.
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First Fundamental Form
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First Fundamental Form
Transformation of the first fundamental form
• Let x(u 1 , u 2 ) and y(v 1 , v 2 ) be parametrizations.
• Let gik be the coefficients of the FFF with respect to x, and let ḡik be
the coefficients with respect to y.
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First Fundamental Form
October 9, 2016
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First Fundamental Form
Transformation of the first fundamental form
• Let x(u 1 , u 2 ) and y(v 1 , v 2 ) be parametrizations.
• Let gik be the coefficients of the FFF with respect to x, and let ḡik be
the coefficients with respect to y.
gik =
X
m,n
ḡmn
∂v m ∂v n
∂u i ∂u k
ḡik =
X
m,n
gmn
∂u m ∂u n
∂v i ∂v k
and
2
∂v i
g = det j
ḡ,
∂u
MATH 2040
ḡ =
First Fundamental Form
∂u i
det j
∂v
2
g
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First Fundamental Form
Matrix form for transformation of FFF, I
We could also write this as
∂v 1
g11 g12
∂u 1
=
∂v 1
g21 g22
2
∂u
MATH 2040
∂v 2
∂u 1
∂v 2
∂u 2
!
ḡ11 ḡ12
ḡ21 ḡ22
First Fundamental Form
∂v 1
∂u 1
∂v 2
∂u 1
∂v 1
∂u 2
∂v 2
∂u 2
!
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First Fundamental Form
Matrix form for transformation of FFF, I
We could also write this as
∂v 1
g11 g12
∂u 1
=
∂v 1
g21 g22
2
!
ḡ11 ḡ12
ḡ21 ḡ22
∂v 1
∂u 1
∂v 2
∂u 1
∂v 1
∂u 2
∂v 2
∂u 2
!
∂u
∂v 2
∂u 1
∂v 2
∂u 2
∂u 1
∂v 1
∂u 1
∂v 2
∂u 2
∂v 1
∂u 2
∂v 2
!
g11 g12
g21 g22
∂u 1
∂v 1
∂u 2
∂v 1
∂u 1
∂v 2
∂u 2
∂v 2
!
and
ḡ11 ḡ12
ḡ21 ḡ22
MATH 2040
=
First Fundamental Form
October 9, 2016
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First Fundamental Form
Matrix form for transformation of FFF, II
Denoting
J=
MATH 2040
∂u 1
∂v 1
∂u 2
∂v 1
∂u 1
∂v 2
∂u 2
∂v 2
!
J −1 =
First Fundamental Form
∂v 1
∂u 1
∂v 2
∂u 1
∂v 1
∂u 2
∂v 2
∂u 2
!
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First Fundamental Form
Matrix form for transformation of FFF, II
Denoting
∂u 1
∂v 1
∂u 2
∂v 1
J=
∂u 1
∂v 2
∂u 2
∂v 2
!
J −1 =
∂v 1
∂u 1
∂v 2
∂u 1
∂v 1
∂u 2
∂v 2
∂u 2
!
we have
and
g11 g12
g21 g22
MATH 2040
ḡ11 ḡ12
ḡ21 ḡ22
= (J
−1 t
=J
)
t
ḡ11 ḡ12
ḡ21 ḡ22
g11 g12
g21 g22
First Fundamental Form
J −1
J.
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