Math 53. Multi-variable Calculus. Problems from past final exams
1. Find and classify all critical points of the function f(x, y) = y3+
3x2y−12yas local maxima, local minima or saddle points.
2. Find the maximum and minimum values of the function x+y+zon
the surface of the ellipsoid
x2
2+y2
4+z2
6= 12.
3. Change the order of integration in the triple integral:
Z1
0Zx
0Zy
0
f(x, y, z)dz dy dx =Z?
?Z?
?Z?
?
f(x, y, z)dy dx dz .
4. A homogeneous ball of radius Rand mass density µrotates with the
angular velocity ωabout a line passing through its center. Compute the
kinetic energy accumulated in the ball due to this rotation.
5. An ink spot of initially round shape (x−1)2+y2≤a2is carried by
the flow of a two-dimensional fluid with the velocity vector field
~
V= (x+e−x2cos y)
~
i+ (2y+ 2xe−x2sin y)~
j.
Find the area of the ink spot at the moment t.
6. Calculate the area of the surface
z=a(1 −2x2
a2−2y2
a2), z ≥0.
7. Compute the flux of the vector field ~
F=~r/|~r|3, where ~r =x
~
i+y~
j+z~
k,
across the surface (x+ 1)2=y2+z2, x ≤0, equipped with an orientation of
your choice.
8. Give an example of a divergence-free vector field which is not the curl
of any vector field. Justify your answer.
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