Problem 37 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

2-37. (a) Let \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be a continuously differentiable function. Show that \(f\) is not 1-1. Hint: If, for example, \(D_{1} f(x, y) \neq 0\) for all \((x, y)\) in some open set \(A\), consider \(g: A \rightarrow \mathbf{R}^{2}\) defined by \(g(x, y)=\) \((f(x, y), y)\) (b) Generalize this result to the case of a continuously differentiable function \(f: \mathbf{R}^{n} \rightarrow \mathbf{K}^{m}\) with \(m

Step-by-Step Solution

Answer

If \( f: \mathbf{R}^{2} \rightarrow \mathbf{R} \) is continuously differentiable, it is not 1-1 because locally invertible functions \( g \) imply there exist distinct inputs mapping to the same output. Similarly, \( f: \mathbf{R}^n \rightarrow \mathbf{K}^m \) with \( m < n \) cannot be 1-1.

Step 1: Understand the Problem Statement

Given a continuously differentiable function \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\), we need to show that \(f\) is not one-to-one (1-1). Additionally, we must generalize this result for \(f: \mathbf{R}^{n} \rightarrow \mathbf{K}^{m}\), where \(m < n\). The hint suggests considering the derivative \(D_{1} f(x, y) eq 0\).

Step 2: Define the Function g

Use the hint to define a function \( g: A \rightarrow \mathbf{R}^{2} \) where \( g(x, y) = (f(x, y), y) \). Notice that \( g \) maps \( (x, y) \) to the output of \( f(x, y) \) and \( y \).

Step 3: Compute the Jacobian Matrix of g

Compute the Jacobian matrix \( J_g \) of \( g \). The Jacobian matrix for \( g \) is: \[J_g = \begin{bmatrix}D_1 f & D_2 f\0 & 1\ \end{bmatrix} \]Because \( D_1 f(x, y) eq 0 \), the determinant of \( J_g \) is non-zero, making the matrix invertible.

Step 4: Local Invertibility of g

Since \(J_g\) is invertible, by the Inverse Function Theorem, \( g \) is locally invertible. This means there exists an open set \( U \subset A \) where \( g \) is a bijection onto its image.

Step 5: Local Invertibility Implies Non 1-1 of f

Given \( g(x_1, y_1) = g(x_2, y_2) \) in \( U \), then \( (f(x_1, y_1), y_1) = (f(x_2, y_2), y_2) \). This implies \( y_1 = y_2 \) and hence \( f(x_1, y_1) = f(x_2, y_2) \). Therefore, \( x_1 eq x_2 \), meaning \( f \) is not 1-1.

Step 6: Generalize to n Dimensions

For \( f: \mathbf{R}^n \rightarrow \mathbf{K}^m \) with \( m < n \), we can similarly define \( g: \mathbf{R}^n \rightarrow \mathbf{R}^m \times \mathbf{R}^{n-m} \) where \( g(x) = (f(x), x_{m+1}, \ldots, x_n) \). The Jacobian matrix of \( g \) will be an \( n \times n \) block matrix which is invertible due to \( n > m \), making \( g \) locally invertible, which implies \( f \) is not 1-1.

Key Concepts

Inverse Function Theorem

The Inverse Function Theorem is a powerful tool in multivariable calculus. It gives sufficient conditions for a function to have a local inverse. Let's break it down. Suppose you have a function \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n \) that is continuously differentiable. The theorem states that if the Jacobian matrix of \( f \) at a point \( a \) is invertible, then there exist neighborhoods around \( a \) and \( f(a) \) where \( f \) is one-to-one and onto. This means, locally, \( f \) behaves like an invertible function. In simpler terms, an invertible Jacobian matrix guarantees that \( f \) has an inverse function nearby.

Jacobian Matrix

The Jacobian matrix is essential when dealing with multivariable functions. For a function \( f: \mathbb{R}^n \rightarrow \mathbb{R}^m \), the Jacobian is an \( m \times n \) matrix of its first partial derivatives. Each entry of the Jacobian is a partial derivative of one of the output functions with respect to one of the input variables. For example, if \( f(x, y) \) is a function with two variables and two outputs, the Jacobian matrix \( J_f \) would look like this:

\[ J_f = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix} \]

An invertible Jacobian matrix suggests that the function \( f \) is locally invertible around a point, which is crucial for applying the Inverse Function Theorem.

Local Invertibility

Local invertibility is a concept that ensures a function can be 'undone' in a small region around a point. If a function \( f \) satisfies certain conditions, such as having an invertible Jacobian matrix at a point, it means \( f \) behaves like an invertible function in a neighborhood around that point. This does not imply global invertibility but guarantees that there is a small region where there is a one-to-one correspondence between the inputs and the outputs. This concept is crucial because it bridges differential calculus with real-world applications, where functions often only need to be invertible in specific contexts, not everywhere.