Problem 8
Question
Suppose that the mapping \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is continuously differentiable. Suppose also that \(\mathbf{F}(\mathbf{0})=\mathbf{0}\) and that the derivative matrix \(\mathbf{D F}(\mathbf{0})\) has the property that there is some positive number \(c\) such that $$ \|\mathbf{D F}(\mathbf{0}) \mathbf{h}\| \geq c\|\mathbf{h}\| $$ for all \(\mathbf{h}\) in \(\mathbb{R}^{n}\). Prove that there is some positive number \(r\) such that $$ \|\mathbf{F}(\mathbf{h})\| \geq c / 2\|\mathbf{h}\| \quad \text { if }\|\mathbf{h}\| \leq r $$
Step-by-Step Solution
Verified Answer
There is a positive number \(r\) such that for any \(\mathbf{h}\) with \(\|\mathbf{h}\| \leq r\), the inequality \(\|\mathbf{F}(\mathbf{h})\| \geq \frac{c}{2}\|\mathbf{h}\|\) holds due to the continuity of \(\mathbf{D F}\) and the Mean Value Theorem.
1Step 1: Understanding the Given Information
First, interpret the information provided. We have a continuously differentiable map \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) such that \(\mathbf{F}(\mathbf{0}) = \mathbf{0}\), and its derivative matrix \(\mathbf{D F}(\mathbf{0})\) satisfies \(\|\mathbf{D F}(\mathbf{0}) \mathbf{h}\| \geq c\|\mathbf{h}\|\) for a positive constant \(c\) and for all vectors \(\mathbf{h}\) in \(\mathbb{R}^{n}\). The goal is to prove that there exists some positive number \(r\) such that \(\|\mathbf{F}(\mathbf{h})\| \geq \frac{c}{2}\|\mathbf{h}\|\) whenever \(\|\mathbf{h}\| \leq r\).
2Step 2: Using Mean Value Theorem
Apply the Mean Value Theorem in several variables for \(\mathbf{F}\) on the segment from \(\mathbf{0}\) to \(\mathbf{h}\), which implies there exists a point \(\mathbf{t}\) on the segment such that \[\mathbf{F}(\mathbf{h}) - \mathbf{F}(\mathbf{0}) = \mathbf{D F}(\mathbf{t})(\mathbf{h} - \mathbf{0}) = \mathbf{D F}(\mathbf{t})\mathbf{h}\] Since \(\mathbf{F}(\mathbf{0}) = \mathbf{0}\), the equation simplifies to \(\mathbf{F}(\mathbf{h}) = \mathbf{D F}(\mathbf{t})\mathbf{h}\).
3Step 3: Estimating the Difference of Derivatives
We know \(\mathbf{D F}\) is continuous at \(\mathbf{0}\) and \(\mathbf{D F}(\mathbf{0})\mathbf{h}\) is bounded below by \(c\|\mathbf{h}\|\). There must exist a distance \(r\) for which \(\|\mathbf{D F}(\mathbf{t}) - \mathbf{D F}(\mathbf{0})\| < \frac{c}{2}\) whenever \(\|\mathbf{t}\| \leq r\), due to the continuity of \(\mathbf{D F}\).
4Step 4: Bounding the Norm of \(\mathbf{F}(\mathbf{h})\)
We can estimate \(\|\mathbf{F}(\mathbf{h})\|\) using the relation \(\mathbf{F}(\mathbf{h}) = \mathbf{D F}(\mathbf{t})\mathbf{h}\) and the previous step, such that \[\|\mathbf{F}(\mathbf{h})\| = \|\mathbf{D F}(\mathbf{t})\mathbf{h}\| \geq \|\mathbf{D F}(\mathbf{0})\mathbf{h}\| - \|\mathbf{D F}(\mathbf{t})\mathbf{h} - \mathbf{D F}(\mathbf{0})\mathbf{h}\|\] Using the reverse triangle inequality and the estimates above, we get \(\|\mathbf{F}(\mathbf{h})\| \geq c\|\mathbf{h}\| - (c/2)\|\mathbf{h}\| = (c/2)\|\mathbf{h}\|\) for \(\|\mathbf{h}\| \leq r\).
5Step 5: Concluding the Proof
We conclude that for \(\|\mathbf{h}\| \leq r\) it holds that \(\|\mathbf{F}(\mathbf{h})\| \geq \frac{c}{2}\|\mathbf{h}\|\), which completes the proof.
Key Concepts
Mean Value TheoremDerivative matrixNorm of a vector
Mean Value Theorem
The Mean Value Theorem is a cornerstone of calculus that connects the value of a derivative to the slope of a secant line over an interval. In the context of a mapping from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\), it states that for a continuously differentiable function \(\mathbf{F}\), there exists at least one point within a chosen interval where the instantaneous rate of change matches the average rate of change over that interval.
In the given exercise, the Mean Value Theorem is applied to the vector-valued function \(\mathbf{F}\), which tells us that between the origin (\(\mathbf{0}\)) and any other point \(\mathbf{h}\), there exists some point at which the rate of change of \(\mathbf{F}\) is exactly as it is at the origin, under the derivative matrix \(\mathbf{D F}(\mathbf{0})\). This fact is essential for proving the relationship between the norms of \(\mathbf{F}(\mathbf{h})\) and the vector \(\mathbf{h}\), as sought in the exercise.
In the given exercise, the Mean Value Theorem is applied to the vector-valued function \(\mathbf{F}\), which tells us that between the origin (\(\mathbf{0}\)) and any other point \(\mathbf{h}\), there exists some point at which the rate of change of \(\mathbf{F}\) is exactly as it is at the origin, under the derivative matrix \(\mathbf{D F}(\mathbf{0})\). This fact is essential for proving the relationship between the norms of \(\mathbf{F}(\mathbf{h})\) and the vector \(\mathbf{h}\), as sought in the exercise.
Derivative matrix
The derivative matrix, often denoted as \(\mathbf{D F}(\mathbf{x})\), is a matrix that represents all the first-order partial derivatives of a vector-valued function. It is fundamental in analyzing how a function stretches, shrinks, or rotates vectors. In higher dimensions, this matrix is the generalization of the derivative of a function in single-variable calculus.
In our exercise, the derivative matrix at the origin, \(\mathbf{D F}(\mathbf{0})\), plays a pivotal role. It provides a linear approximation to our function \(\mathbf{F}\) near \(\mathbf{0}\). The condition given, that \(||\mathbf{D F}(\mathbf{0}) \mathbf{h}|| \geq c||\mathbf{h}||\), implies the derivative matrix is invertible at \(\mathbf{0}\), guaranteeing a certain 'strength' of response in the function's output compared to the input, within a neighborhood around \(\mathbf{0}\). This property directly influences the bound we seek to prove for \(||\mathbf{F}(\mathbf{h})||\).
In our exercise, the derivative matrix at the origin, \(\mathbf{D F}(\mathbf{0})\), plays a pivotal role. It provides a linear approximation to our function \(\mathbf{F}\) near \(\mathbf{0}\). The condition given, that \(||\mathbf{D F}(\mathbf{0}) \mathbf{h}|| \geq c||\mathbf{h}||\), implies the derivative matrix is invertible at \(\mathbf{0}\), guaranteeing a certain 'strength' of response in the function's output compared to the input, within a neighborhood around \(\mathbf{0}\). This property directly influences the bound we seek to prove for \(||\mathbf{F}(\mathbf{h})||\).
Norm of a vector
The norm of a vector, written as \(||\mathbf{h}||\), quantifies the 'length' or 'magnitude' of the vector in the vector space \(\mathbb{R}^{n}\). It is a measurement of the distance from the origin to the point represented by the vector. In the provided problem, we encounter the norm frequently used to compare sizes of vectors and their transformations under the function \(\mathbf{F}\).
For instance, the condition \(||\mathbf{D F}(\mathbf{0}) \mathbf{h}|| \geq c||\mathbf{h}||\) asserts a lower bound on how the magnitude of vectors is altered by the derivative matrix at the origin, ensuring that no vector's length is diminished below a certain factor of its original length. This concept of the norm is critical when we evaluate inequalities involving \(\mathbf{F}(\mathbf{h})\) and \(\mathbf{h}\) themselves, as it helps us understand and establish the desired lower bound on the function \(\mathbf{F}\) relative to the input vector \(\mathbf{h}\). Understanding the norm and its properties assists in deriving the conclusion needed to effectively solve the original exercise.
For instance, the condition \(||\mathbf{D F}(\mathbf{0}) \mathbf{h}|| \geq c||\mathbf{h}||\) asserts a lower bound on how the magnitude of vectors is altered by the derivative matrix at the origin, ensuring that no vector's length is diminished below a certain factor of its original length. This concept of the norm is critical when we evaluate inequalities involving \(\mathbf{F}(\mathbf{h})\) and \(\mathbf{h}\) themselves, as it helps us understand and establish the desired lower bound on the function \(\mathbf{F}\) relative to the input vector \(\mathbf{h}\). Understanding the norm and its properties assists in deriving the conclusion needed to effectively solve the original exercise.
Other exercises in this chapter
Problem 7
Suppose that the function \(u: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is harmonic. Let \(a, b, c,\) and \(d\) be real numbers such that $$ a^{2}+b^{2}=1, \quad
View solution Problem 8
Suppose that the functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) have continuous second-order partial derivative
View solution Problem 8
Define \(A=\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right) .\) Find all \(2 \times 2\) matrices \(\mathbf{B}\) with the property that $$ \mathbf{A B}=\m
View solution Problem 9
Let \(\mathcal{O}=\left\\{(x, y, z)\right.\) in \(\left.\mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}>0\right\\}\) and define the function \(u: \mathcal{O} \rightarrow
View solution