PROBLEM SET VIII
DUE FRIDAY,  APRIL
Exercise . Use the Poincaré lemma to show that there exists a smooth function g : R .
R such that
′
(sin x + x exp(g(x)) − 1)g (x) + g(x) cos x + 2x exp(g(x)) = 0.
2
Definition. For any real numbers 0 ≤ r ≤ s, let T2 (r, s) ⊂ R3 denote the torus
{
}
T2 (r, s) := (x, y, z) ∈ R3 (x2 + y2 + z2 + s2 − r2 )2 = 4s2 (x2 + y2 ) .
∪
Let us write ST(r, s) for the union ρ∈[0,r] T2 ( ρ, s).
Exercise . What is the volume of ST(r, s)?
Exercise . What is the surface area of T2 (r, s)?
Exercise⋆ . Consider the following vector field on R3 − {(0, 0, 0)}:
1
φ(x, y, z) = 2
(−y, x, (z − 1/4)(z + 1/4) − x2 − y2 ).
x + y2 + z2
(a) Make a good faith attempt to draw this vector field (or some flow lines).
(b) Is this vector field a gradient vector field?
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