11.6. EXERCISES 275
Notation 11.5.3 It is customary to write the line integral∫
C f ·dr as∫C
f1 (x)dx1 + f2 (x)dx2 + · · ·+ fq (x)dxq
which is called differential form notation.
11.6 Exercises1. A random vector X, with values in Rp has a multivariate normal distribution written
as X ∼ Np (m,Σ) if for all Borel E ⊆ Rp,
λX (E) =∫Rp
XE (x)1
(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dmp
Here Σ is a positive definite symmetric matrix. Recall that λX (E) ≡ P(X ∈ E) .Using the change of variables formula, show that λX defined above is a probabilitymeasure. One thing you must show is that∫
Rp
1
(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dmp = 1
Hint: To do this, you might use the fact from linear algebra that Σ = Q∗DQ where Dis a diagonal matrix and Q is an orthogonal matrix. Thus Σ−1 =Q∗D−1Q. Maybe youcould first let y = D−1/2Q(x−m) and change the variables. Note that the changeof variables formula works fine when the open sets are all of Rp. You don’t needto confine your attention to finite open sets which would be the case with Riemannintegrals which are only defined on bounded sets.
2. Consider the surface z = x2 for (x,y) ∈ (0,1)× (0,1) . Find the area of this sur-face. Hint: You can make do with just one chart in this case. Let R−1 (x,y) =(x,y,x2
)T,(x,y) ∈ (0,1)× (0,1). Then
DR−1 =
(1 0 2x0 1 0
)T
It follows that DR−1∗DR−1 =
(4x2 +1 0
0 1
).
3. A parametrization for most of the sphere of radius a > 0 in three dimensions is x =asin(φ)cos(θ) ,y = asin(φ)sin(θ) ,z = acos(φ). where we will let φ ∈ (0,π) ,θ ∈(0,2π) so there is just one chart involved. As mentioned earlier, this includes all ofthe sphere except for the line of longitude corresponding to θ = 0. Find a formulafor the area of this sphere. Again, we are making do with a single chart.
4. Let V be such that the divergence theorem holds. Show that∫
V ∇ · (v∇u) dV =∫∂V v ∂u
∂n dA where n is the exterior normal. Here ∂u∂n ≡ ∇u ·n.
5. To prove the divergence theorem, it was shown first that the spacial partial deriva-tive in the volume integral could be exchanged for multiplication by an appropriatecomponent of the exterior normal. This problem starts with the divergence theoremand goes the other direction. Assuming the divergence theorem, holds for a regionV , show that
∫∂V nudA =
∫V ∇udV . Note this implies
∫V
∂u∂x dV =
∫∂V n1udA.