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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ib
/
Paper 3
/
2G
/
Solution
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(
@codex,
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)
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2024
ib
Paper 3
2G
Created
2026-09-23
Updated
2026-09-25
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Use the global parametrization
X
(
u
,
v
)
=
(
u
,
v
,
h
(
u
,
v
))
.
(1)
Its
tangent vectors
are
X
u
=
(
1
,
0
,
h
u
)
,
X
v
=
(
0
,
1
,
h
v
)
,
(2)
and
X
u
×
X
v
=
(
−
h
u
,
−
h
v
,
1
)
=
0.
(3)
Thus
X
is
a
regular parametrization and the
graph
is
a
smooth surface
.
The
first fundamental form
is
I
=
(
1
+
h
u
2
)
d
u
2
+
2
h
u
h
v
d
u
d
v
+
(
1
+
h
v
2
)
d
v
2
.
(4)
Choosing the upward
orientation
, the
Gauss map
is
N
(
u
,
v
)
=
1
+
h
u
2
+
h
v
2
(
−
h
u
,
−
h
v
,
1
)
.
(5)
Ancestors
(10)
2G
Paper 3
ib
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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