MULTILINEAR FUNCTIONS AND THE AREA OF A PARALLELEPIPED - Line and Surface Integrals; Differential Forms and Stokes Theorem - Advanced Calculus of Several Variables

Advanced Calculus of Several Variables (1973)

Part V. Line and Surface Integrals; Differential Forms and Stokes' Theorem

Chapter 3. MULTILINEAR FUNCTIONS AND THE AREA OF A PARALLELEPIPED

In the first two sections of the chapter we have discussed 1-forms, which are objects that can be integrated over curves (or 1-dimensional manifolds) in Imagen. Our eventual goal in this chapter is a similar discussion of differential k-forms, which are objects that can be integrated over k-dimensional manifolds in Imagen. The definition of 1-forms involved linear functions on Imagen; the definition of k-forms will involve multilinear functions on Imagen.

In this section we develop the elementary theory of multilinear functions on Imagen, and then apply it to the problem of calculating the area of a k-dimensional parallelepiped in Imagen. This computation will be used in our study in Section 4 of k-dimensional surface area in Imagen.

Let (Imagen)k = Imagen × · · · × Imagen (k factors), and consider a function

Image

Then M is a function on k-tuples of vectors in Image if a1, . . . , ak are vectors in Imagen. Thus we can regard M as a function of k vector variables. The function M is called k-multilinear (or just multilinear) if it is linear in each of these variables separately, that is,

Image

We will often call M a k-multilinear function on Imagen, despite the fact that its domain of definition is actually (Imagen)k. Note that a 1-multilinear function on Imagen is just an (ordinary) linear function on Imagen.

We have seen that every linear function on Imagen is a linear combination of certain special ones, namely the projection functions dx1, . . . , dxn. Recall that, regarding Image as a column vector, dxi is the function that picks out the ith row of this vector, dxi(a) = ai.

We would like to have a similar description of multilinear functions on Imagen. Given a k-tuple i = (i1, . . . , ik) of integers (not necessarily distinct) between 1 and Image we define the function

Image

by

Image

That is, if A is the n × k matrix whose column vectors are a1, . . . , ak,

Image

and Ai denotes the k × k matrix whose rth row is the irth row of A, then

Image

Note that Ai is a k × k submatrix of A if i is an increasing k-tuple, that is, if Image.

It follows immediately from the properties of determinants that dxi is a k-multilinear function on Imagen and, moreover, is alternating. The k-multilinear function M on Imagen is called alternating if

Image

whenever some pair of the vectors a1, . . . , ak are equal, say ar = as (r ≠ s). The fact that dxi is alternating then follows from the fact that a determinant vanishes if some pair of its columns are equal.

It is easily proved (Exercise 3.1) that the k-multilinear function M on Imagen is alternating if and only if it changes signs upon the interchange of two of the vectors a1, . . . , ak, that is,

Image

The notation dxi generalizes the notation dxi for the ith projection function on Imagen, and we will prove (Theorem 3.4) that every alternating k-multilinear function M on Imagen is a linear combination of the dxi. That is, there exist real numbers ai such that

Image

the summation being over all k-tuples i of integers from 1 to n. This generalizes the fact that every linear function on Imagen is of the form Image Notice that every linear function on Imagen is automatically alternating.

Proposition 3.1 If M is an alternating k-multilinear function on Imagen, then

Image

if the vectors a1, . . . , ak are linearly dependent.

This follows immediately from the definitions and the fact that some one of the vectors a1, . . . , ak must be a linear combination of the others (Exercise 3.2) .

Corollary 3.2 If k > n, then the only alternating k-multilinear function on Imagen is the trivial one whose only value is zero.

The following theorem describes the nature of an arbitrary k-multilinear function on Imagen (not necessarily alternating).

Theorem 3.3 Let M be a k-multilinear function on Imagen. If a1, . . . , ak are vectors in Imagen, aj = (a1j, . . . , anj), then

Image

where

Image

Here ei denotes the ith standard unit basis vector in Imagen.

PROOF The proof is by induction on k, the case k = 1 being clear by linearity.

For each r = 1, . . . , n, let Nr denote the (k − 1)-multilinear function on Imagen defined by

Image

Then

Image

as desired.

Image

We can now describe the structure of M in terms of the dxi, under the additional hypothesis that M is alternating.

Theorem 3.4 If M is an alternating k-multilinear function on Image n, then

Image

where Image. Here the notation [i] signifies summation over all increasing k-tuples i = (i1, . . . , ik).

For the proof we will need the following.

Lemma 3.5 Let i = (i1, . . . , ik) and j = (j1, . . . , ik) be increasing k-tuples of integers from 1 to n. Then

Image

This lemma follows from the fact that Image is the determinant of the matrix (δij), where

Image

If i = j, then (δij) is the k × k identity matrix; otherwise some row of (δij) is zero, so its determinant vanishes.

PROOF OF THEOREM 3.4 Define the alternating k-multilinear function Image on Imagen by

Image

where Image. We want to prove that M = Image.

By Theorem 3.3 it suffices (because M and Image are both alternating) to show that

Image

for every increasing k-tuple j = (j1, . . . , jk). But

Image

immediately by Lemma 3.5 .

Image

The determinant function, considered as a function of the column vectors of an n × n matrix, is an alternating n-multilinear function on Imagen. This, together with the fact that the determinant of the identity matrix is 1, uniquely characterizes the determinant.

Corollary 3.6 D = det is the only alternating n-multilinear function on Image n such that

Image

PROOF Exercise 3.3.

Image

As an application of Theorem 3.4 , we will next prove the Binet–Cauchy product formula, a generalization of the fact that the determinant of the product of two n × n matrices is equal to the product of their determinants. Recall that, if A is an n × k matrix, and i = (i1, . . . , ik), then Ai denotes that k × k matrix whose rth row is the irth row of A.

Theorem 3.7 Let A be an k × n matrix and B an n × k matrix, where Image. Then

Image

Here At denotes the transpose of A and, as in Theorem 3.4 , [i] signifies summation over increasing k-tuples.

Note that the case k = n, when A and B are both n × n matrices, is

Image

PROOF Let a1, . . . , ak be the row vectors of A, and b1, . . . , bk the column vectors of B. Since

Image

we see that, by holding fixed the matrix A, we obtain an alternating k-multilinear function M of the vectors b1, . . . , bk,

Image

Consequently by Theorem 3.4 there exist numbers αi (depending upon the matrix A) such that

Image

Then

Image

But

Image

so

Image

as desired.

Image

Taking A = Bt, we obtain

Corollary 3.8 If A is an n × k matrix, Image, then

Image

Our reason for interest in the Binet–Cauchy product formula stems from its application to the problem of computing the area of an k-dimensional parallelepiped in Imagen. Let a1, . . . , ak be k vectors in Imagen. By the k-dimensional parallelepiped P which they span is meant the set

Image

This is the natural generalization of a parallelogram spanned by two vectors in the plane. If k = n, then P is the image of the unit cube In, under the linear mapping Image : ImagenImagen such that L(ei) = ai, i = 1, . . . , n. In this case the volume of P is given by the following theorem.

Theorem 3.9 Let P be the parallelepiped in Imagen that is spanned by the vectors a1, . . . , an. Then its volume is

Image

where A is the n × n matrix whose column vectors are a1, . . . , an.

PROOF If the linear mapping L : Image nImage is defined by

Image

then L(ei) = ai, i = 1, . . . , n. Hence P = L(In). Therefore, by Theorem IV.5.1 ,

Image

since

Image

Now let X be a subset of a k-dimensional subspace V of Imagen (k < n). A choice of an orthonormal basis v1, . . . , vk for V (which exists by Theorem I.3.3) determines a 1-1 linear mapping φ : VImagek, defined by φ(vi) = ei. We want to define the k-dimensional area a(X) by

Image

However we must show that a(X) is independent of the choice of basis for V. If w1, . . . , wk is a second orthonormal basis for V, determining the one-to-one linear mapping Image : VImagek by Image(wi) = ei, then it is easily verified that

Image

is an orthogonal mapping (Exercise I.6.10) . It therefore preserves volumes, by Corollary IV.5.2. Since Image Image φ−1(φ(X)) = Image(X), we conclude that v(φ(X)) = v(Image(X)), so a(X) is well defined. The following theorem shows how to compute it for a k-dimensional parallelepiped.

Theorem 3.10 If P is a k-dimensional parallelepiped in Imagen (k < n) spanned by the vectors a1, . . . , ak, then

Image

where A is the n × k matrix whose column vectors are a1, . . . , ak.

Thus we have the same formula when k < n as when k = n.

PROOF Let V be a k-dimensional subspace of Imagen that contains the vectors a1, . . . , ak, and let v1, . . . , vn be an orthonormal basis for Imagen such that v1, . . . , vk generate V. Let φ : ImagenImagen be the orthogonal mapping defined by φ(vi) = ei, i = 1, . . . , n. If bi = φ(ai), i = 1, . . . , k, then φ(P) is the k-dimensional parallelepiped in Imagek that is spanned by b1, . . . , bk. Consequently, using Theorem 3.9 and the fact that the orthogonal mapping φ preserves inner products (Exercise I.6.10) , we have

Image

The following formula for a(P) now follows immediately from Theorem 3.10 and Corollary 3.8 .

Theorem 3.11 If P and A are as in Theorem 3.10 , then

Image

This result can be interpreted as a general Pythagorean theorem. To see this, let Image denote the k-dimensional coordinate plane in Imagen that is spanned by the unit basis vectors Image, where i = (i1, . . . , ik). If πi : ImagenImageik is the natural projection mapping, then

Image

Image

Figure 5.21

is the square of the k-dimensional area of the projection πi(P) of P into Image ik (see Exercise 3.4) . Thus Theorem 3.11 asserts that the area of the k-dimensional parallelepiped P is equal to the square root of the sum of the squares of the areas of all projections of P into k-dimensional coordinate planes of Imagen (see Fig. 5.21 for the case k = 2, n = 3). For k = 1, this is just the statement that the length of a vector is equal to the square root of the sum of the squares of its components.

Exercises

3.1Prove that the k-multilinear function M on Imagen is alternating if and only if the value M(a1, . . . , ak) changes sign when ar and as are interchanged. Hint: Consider M(a1, . . . , ar + as, . . . , ar + as, . . . , ak).

3.2Prove Proposition 3.1 .

3.3Prove Corollary 3.6 .

3.4Verify the assertion in the last paragraph of this section, that det Ait Ai is the square of the k-dimensional area of the projection πi(P), of the parallelepiped spanned by the column vectors of the n × k matrix A, into Imageik.

3.5Let P be the 2-dimensional parallelogram in Image3 spanned by the vectors a and b. Deduce from Theorem 3.11 the fact that its area is a(P) = Imagea × bImage.

3.6Let P be the 3-dimensional parallelepiped in Image3 that is spanned by the vectors a, b, c. Deduce from Theorem 3.9 that its volume is v(P) = Imagea · b × cImage.