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3.
Prove Equation (4.35).
4. Let N be the normal vector to a surface at a point P , and let S and T be tan-
gent vectors at the point P such that
ST N . Given an invertible 3
×=
×
3
ma-
) (
)
(
)
(
)
(
1T
(
)
trix M , show that
MS MT M M S T , supporting the fact
that normals are correctly transformed by the inverse transpose of the matrix
M . [ Hint. Use Equation (2.25) to write the cross product
×
=
det
×
(
)
(
)
MS
×
MT as
(
)
(
)
0
MS
MS
z
y
(
)
(
)
(
)
(
)
.
MS
×
MT
=
MS
0
MS
MT
z
x
(
)
(
)
MS
MS
0
y
x
Then find a matrix G such that
(
)
(
)
0
SS
0
MS
MS
 
z
y
z
y
 
(
)
(
)
G
S
0
−=
S
MS
0
MS
M
,
z
x
 
z
x
(
)
(
)
SS
0
 
MS
MS
0
 
y
x
y
x
) (
)
(
1T
and finally use Equation (3.65) to show that
G
=
det
MM
.]
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