418 CHAPTER 22. CALCULUS OF VECTOR FIELDS

perhaps not the first thing you would think of.The main purpose of this presentation is to show how the divergence theorem is used

in a significant way to obtain balance laws and to indicate a very interesting direction forfurther study. To continue, one needs to specify T or S as an appropriate function of thingsrelated to the motion y. Often the thing related to the motion is something called the strainand such relationships are known as constitutive laws.

22.4.6 The Reynolds Transport FormulaThe Reynolds transport formula is another interesting application of the divergence theoremwhich is a generalization of the formula for taking the derivative under an integral.

ddt

∫ b(t)

a(t)f (x, t)dx =

∫ b(t)

a(t)

∂ f∂ t

(x, t)dx+ f (b(t) , t)b′ (t)− f (a(t) , t)a′ (t)

Of course there are difficult analytical questions connected with such a formal procedure,but these can be easily justified with sufficient machinery involving the Lebesgue integral.An elementary version of theorems necessary to justify this will be fussy and unpleasant soI am going to emphasize the derivation of the formula without worrying about interchangeof limit considerations and whether the divergence theorem holds for the region of interest.

First is an interesting lemma about the determinant. A p× p matrix can be thought ofas a vector in Cp2

. Just imagine stringing it out into one long list of numbers. In fact, away to give the norm of a matrix is just ∑i ∑ j

∣∣Ai j∣∣2 ≡ ∥A∥2. You might check to see that

this is the same as (trace(AA∗))1/2 = ∥A∥. It is called the Frobenius norm for a matrix.Also recall that det maps p× p matrices to C. It makes sense to ask for the derivative ofdet on the set of invertible matrices, an open subset of Cp2

with the norm measured as justdescribed. This is because A→ det(A) is continuous so the set where det(A) ̸= 0 wouldbe an open set. Recall that trace(AB) = trace(BA) whenever both products make sense.Indeed,

trace(AB) = ∑i

∑j

Ai jB ji = trace(BA)

This next lemma is a very interesting observation about the determinant of a matrixadded to the identity.

Lemma 22.4.2 det(I +U) = 1+ trace(U)+ o(U) where o(U) is defined in terms of theFrobenius norm for p× p matrices.

Proof: This is obvious if p = 1 or 2. Assume true for n− 1. Then for U an n× n,I +U = 

1+U11 U12 · · · U1n

U21 1+U22 · · · U2n...

. . . . . ....

Un1 · · · Un(n−1) 1+Unn

Expand along the last column and use induction.

(1+Unn)

(1+

n−1

∑k=1

Ukk +o(U)

)+o(U)