Practice Problems
Easy problems:
1. The unit sphere in R3is the set S2of all points (x, y, z) such that
x2+y2+z2= 1.
For what points (x0,y0, z0) is it possible to find a C1function z(x, y) defined near
(x0, y0) such that z(x0, y0) = z0and
x2+y2+z(x, y)2= 1?
2. Let f(x, y) = (excos(y), exsin(y)). Show that fis locally invertible everywhere,
but not invertible.
3. Find the equation of the plane tangent to the surface x3+2xy2−7z3+ 3y+ 1 = 0
at the point (1,1,1).
Problems of average difficulty:
4. Prove that if F:Rn→Rmis differentiable at x0, then the partial derivative
∂fi
∂xj
(x0)
is equal to the (i, j) entry of DF (x0). (where fi, as usual, denotes the i-th output
of the function F).
5. Problems 6 and 7 in Section 11.3 of Wade.
6. Let Vbe an open ball in Rmand suppose that F:V→Rnis a continuously
differentiable function such that DF (x) = A, where Ais a fixed n×mmatrix, for
all x∈V. Show that there exists c∈Rnsuch that
F(x) = Ax +c
for all x∈V(Hint: there is no nice MVT for functions Rm→Rn).
Harder problems:
7. We define the parabolic coordinate system in R2via the formulas
y(u, v) = uv
x(u, v)=1/2(v2−u2)
(1)
(a) Show that the mapping defined by (1) is locally invertible for all (u,v )6= (0,0)
but not globally invertible.
(b) Show that the mapping (1) is invertible if we restrict its domain to v > 0.
(c) Let the function f:R2→R2be defined by the formula (in parabolic coordi-
nates)
f(u, v) = v−u2.
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