Inverse and Implicit Mapping Theorems (Sections III.3-III.4)
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1 MATH W8 Daily Notes Preamble As an executive decision, I am skipping Section III.2. It is something like an 8-page lemma, with a number of concepts and results introduced at this stage mainly for the purpose of proving the big theorems in this section. As I also intend not to give proofs of these big theorems, I hope we are safe in skipping most of this material. To mention a result (or two), however, from the section, there is this theorem which is quite interesting. Theorem (III.2.8 Corollary): Let f : R n! R m be C at a. If df a, which maps R n! R m as well, is one-to-one, then f itself is one-to-one on some open set U containing a. To be one-to-one (or injective) means that no two distinct inputs x, in U have the same image. Given this, it is possible within f (U) to define an inverse function from y 2 f (U) back to the one U for which f (x) = y. This, almost, is the inverse mapping theorem. Big theorems Theorem 2 (III.3.3 Inverse Mapping Theorem): Suppose that the mapping f : R n! R n is C in a neighborhood W of the point a, with the matrix f (a) nonsingular. Then f is locally invertible i.e., there exist neighborhoods U W of a and V of b = f (a), and a one-to-one C mapping g: V! W such that g( f (x)) = x, for U, and f (g(y)) = y, for y 2 V. In particular, the local inverse g is the limit of the sequence (g k ) of successive approximations defined inductively by k= g (y) = a, g k+ (y) = g k (y) f (a) [ f (g k (y)) y], for y 2 V. Example : 2
2 MATH W8 Daily Notes Consider the C function f : R 2! R 2 given by f (x, y) = (xy, y 2 ), and suppose we want a local inverse near the point b = f (a) = (2, 3) with a = (2, ). We first check to see if f (a) is a nonsingular matrix. We have f (x, y) = y 2x a nonsingular matrix with inverse x 2y CA ) f 2 (2, ) = 4 2 CA, [ f (2, )] = By the inverse mapping theorem, the function f is locally invertible about the point f (2, ) = (2, 3). To see the beginning of the sequence of successive approximations, for y = (y, y 2 ) near (2, 3), from the theorem, we first take To obtain the first iterate, we have g (y) = a = (2, ). g (y) = g (y) f (a) [ f (g (y)) y] = a f (a) [(2, 3) (y, y 2 )] y = CA 4 CA 3 y CA = 2y + 2y y y Going one more step in the process is messy, as we must calculate f (g (y)), but software indicates g 2 (y) = 25 y2 25 y2 25 y y 2 5 y y + 3 y y y 2 + y y 3 2 y Even after just 2 iterates, at the point (.75, 3.25) = (2, 3) + (.25,.25) = b + y, f (g(3, 2)) (.7496, ), while at the point (3, 2) = (2, 3) + (, ) = b + y, f (g(3, 2)) (3.24, 2.27). Theorem 3 (III.3.4 Implicit Mapping Theorem): Let the mapping G: R m+n! R n be C in a neighborhood of the point (a, b) where G(a, b) =. If the partial derivative matrix D 2 G(a, b) (comprising the final n columns of G (a, b)) is nonsingular, then there exists a neighborhood U of a in R m, a neighborhood W of (a, b) in R m+n, and a C mapping h: U! R n, such that y = h(x) solves the equation G(x, y) = in W. 3
3 MATH W8 Daily Notes In particular, the implicitly defined mapping h is the limit of the sequence of successive approximations defined inductively by h (x) = b, h k+ (x) = h k (x) D 2 G(a, b) G(x, h k (x)), for U. Example 2: Consider just what the implicit mapping theorem has to say with regards to the function f : R 5! R 3 given by 2 4 G(x) = 2 3 CA x. 2 Find variables x, y and a satisfactory y = h(x). Note that, for the given matrix A, RREF reveals columns 2 and 4 are free, so we take x = (, ) and y = (x,, ). We can write G(x, y) = as 2 4 = 2 3 CA 2 2 = CA 2 x x CA 4 CA CA 2 4 = x CA + 2 CA + CA + 3 CA + CA 2 2 CA, where = 4, 2 so 2 CA 2 x CA = CA CA, ) y = 2 CA CA x = CA x. At this point, Edwards is ready to modify the name of what has heretofore been referred to as a manifold. I give the definition again for what he now calls a smooth manifold. 4
4 MATH W8 Daily Notes Definition : A set P in R n is called a smooth k-dimensional patch (or a C patch) if and only if there exists a permutation x i,...,x in of the coordinates x,...,x n in R n and a di erentiable function h: U! R n k on an open set U R k such that P = n R n (x i,...,x ik ) 2 U and (x ik+,...,x in ) = h(x i,...,x ik ) o. A set M is called a smooth k-dimensional manifold (or a C k-manifold) in R n if every point of M lies in an open subset V of R n such that V T M is an C k-dimensional patch. This next theorem appeared in Chapter II without proof. It can now be proved as a consequence of the Implicit Function Theorem (see the text). It has served to provide us one criterion for recognizing a (smooth) k-manifold. Corollary (Thm. II.5.7 = Thm. III.4.): Let G: R n! R m be a C function, where k = n m >. If M = n G () G (x) has rank m o, then M is a smooth k-manifold. Definition 2: Suppose : U R k! R n (k apple n) is of class C. Then is said to be regular if its n-by-k derivative matrix (u) has maximal rank k (i.e., its columns are linearly independent) for each u 2 U. The next theorem gives us another criterion for recognizing a k-manifold. Theorem 4: Let M be a subset of R n. Suppose that, given p 2 M, there exists an open set U R k (k < n) and a regular C mapping : U! R n such that p 2 (U), with (U ) being an open subset of M for each open set U U. Then M is a smooth k-manifold. Notes: The assertion that " (U ) is a subset of M" should be quite clear. But it requires clarification to interpret the meaning of " (U ) is an open subset of M". It is not the same as saying (U ) is 5
5 MATH W8 Daily Notes a subset of M that is open in R n. Rather, it means that there is an open set W R n for which (U ) = W T M. The sum total of these theorems is that smooth k-manifolds can arise in three di erent ways:. as the graph i.e., the set of points (x, F(x)) 2 R k R n k of a C mapping from F: U R k! R n k (as per the definition). 2. as the zero set of a C mapping from R n! R n k (established in Theorem III.4.), and 3. as the image of a C map from R k! R n established in Theorem III.4.2). It seems to come too fast too late in the Interim to define concepts such as coordinate patches, atlases and orientable manifolds, which appear at the end of Section III.4. We would make more of these if we were venturing into later chapters. Su ce it to say that, if in the future you study manifolds, you are likely to get a di erent definition than the one of this book, a definition that uses ideas like coordinate patches and an atlas. At that later date, Theorem III.4.3 will be useful in establishing that our definition of smooth manifold and the one you learn in a future course are the same (or, at least, ours is a special case of the other one). 6
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