**Unformatted text preview: **Exam
Course
Duration
Due Date
Instructor :
:
:
:
: 16s MATH 2451 quiz 08
Multivariable Calculus
25 min
2016-03-09 (Wed)
McCary T.C. Use (First Name) (Last Name) 2016-03-09 09:22 1. (10 points) Determine if each of the following functions is a parametrization of a manifold.
�
�
t
√
.
(a) γ1 : (−1, 1) → M given by γ1 (t) =
1 − t2
�
�
cos(π t)
.
(b) γ2 : (−1/2, 1/2) → M given by γ2 (t) =
− sin(π t3 ) �� �
x
2. (10 points) Let Xc =
y �
�
� 3
� x + y 3 − 3 y = c . Determine all values of c which make Xc a smooth curve.
� 3. (10 points)
(a) Several level sets of a function with signature R2 → R are plotted. For each plotted level set, determine if it is a
smooth manifold.
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�� �� � � � (b) Several level sets of a function with signature R2 → R are plotted. For each plotted level set, determine if it is a
smooth manifold.
� ��� ���� � ��� � ���� ����
�
��
��
� � ��� ���� ���� �� ����
�� �� � � � �� �� � � � (c) Is the following a smooth manifold? State your reasons.
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- Summer '09
- EYDELZON
- Multivariable Calculus, Manifold, smooth manifold