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Chapter 8: Problem 5

Show that the rank of a smooth mapping \(f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is a lower semicontinuous function, that is rank \(f(x) \geq \operatorname{rank} f\left(x_{0}\right)\) in a neighborhood of a point \(x_{0} \in \mathbb{R}^{m}\).

Short Answer

Expert verified
Answer: For the rank of a smooth mapping to be lower semicontinuous means that, for a given point x_0 in the domain, the rank of the mapping at any point x in a neighborhood around x_0 is greater than or equal to the rank of the mapping at x_0. In other words, the rank of the mapping does not decrease as we move from x_0 to x in the neighborhood.

Step by step solution

01

Definitions and notations

Recall that the rank of a smooth mapping \(f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is the maximum number of linearly independent columns (or rows) of its Jacobian matrix, denoted by \(J_f\). The Jacobian matrix \(J_f(x)\) is an \(n \times m\) matrix, with its components being the partial derivatives of the components of \(f\) with respect to the components of \(x\). That is, the \((i,j)\)-th entry of \(J_f(x)\) is \(\frac{\partial f_i}{\partial x_j}(x)\), where \(f_i\) is the \(i\)-th component of \(f\).
02

Open neighborhood

To show that rank\(f(x) \geq \operatorname{rank}f(x_0)\) in a neighborhood of \(x_0\), consider an open neighborhood \(U\) around \(x_0\) and take any point \(x\) in this neighborhood. Our goal is to show that the rank of \(J_f(x)\) is greater than or equal to the rank of \(J_f(x_0)\).
03

Rank of Jacobian matrix

Let \(A\) be a subset of \(J_f(x_0)\) such that \(\operatorname{rank}(A)=\operatorname{rank}(J_f(x_0))\). Without loss of generality, assume that \(A\) is the maximal collection of linearly independent rows in the Jacobian matrix \(J_f(x_0)\).
04

Linear independence/differentiability of functions

Since \(f\) is a smooth mapping, its component functions are differentiable. Also, note that the functions in the set \(A\) are linearly independent, as a consequence of the properties of the rank. Since these functions are differentiable, we can find an open neighborhood \(V\) around \(x_0\) such that these functions remain linearly independent for all \(x\) in \(V\).
05

Intersection of neighborhoods

Now, consider the open neighborhood \(W = U \cap V\). We know that \(x_0 \in W\) and \(W\) is an open set, so every point in \(W\) belongs to both neighborhoods \(U\) and \(V\). Since these functions are linearly independent in \(V\), they must also be linearly independent in \(W\).
06

Consequence of the intersection

As the linear independence is preserved in the neighborhood \(W\), it implies that \(\operatorname{rank}(J_f(x)) \geq \operatorname{rank}(A) = \operatorname{rank}(J_f(x_0))\) for all points \(x\) in the neighborhood \(W\). Therefore, the rank of \(f(x)\) can only increase as we move from \(x_0\) to \(x\) in this neighborhood.
07

Conclusion

In conclusion, we have shown that the rank of a smooth mapping \(f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is a lower semicontinuous function. We have accomplished this by demonstrating that there exists a neighborhood around \(x_0\) such that the rank of the Jacobian matrix \(J_f(x)\) doesn't decrease.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Jacobian Matrix
Imagine you have a multi-dimensional space where every point can move and stretch in different directions, kind of like a spider web in the wind. The Jacobian matrix is like a snapshot of the web at a specific point, showing us which way things are moving and how much they're stretching. It鈥檚 an array filled with numbers that represent the rate of change of a multi-variable function. Each entry in this matrix is a partial derivative, which tells us how sensitive the function is to a small nudge in each direction. When we're looking at a map of a city, we want to see how the streets and landmarks are arranged; the Jacobian matrix does the same for functions, giving us a layout of all the tiny changes happening inside our function's 'landscape'.
Linear Independence
To get the gist of linear independence, think about different flavored strings of a guitar. When you pluck them, each string vibrates independently, creating a unique sound. The concept of linear independence in mathematics is similar. It鈥檚 about having a set of functions (or vectors) where no one can be replicated by mixing others together. Just like no single guitar string can mimic the sound of another, in linear independence, no function in the set is a duplicate of another created by combining the rest. This uniqueness is crucial when we鈥檙e dealing with the functions in the Jacobian matrix as they tell us about different 'sounds' or directions our multi-variable function can take.
Differentiable Functions
When out for a drive, a differentiable function is like smooth road that allows for a comfortable, bump-free ride. In mathematics, it means we have a function where we can compute the derivative at each point. A derivative provides us a snapshot of the function's rate of change at any given moment鈥攊t's like looking at your car's speedometer to see how fast you're going. If a function is smoothly differentiable, you're able to say, 'Ah! This is exactly how quickly things are changing right here and now!' without worrying about running into any mathematical potholes.
Lower Semicontinuity
Imagine a staircase where each step represents the rank of our smooth mapping at different points. Lower semicontinuity assures us that as we take steps forward, we won't suddenly drop to a lower step. In other words, if our function is lower semicontinuous, the rank at a new point is at least as high as it was at our starting point鈥攜ou can go up the stairs or stay put, but not step down. This concept is crucial for understanding that certain properties of our functions do not deteriorate abruptly as we look around a neighborhood of points; they stay the same or get 'better' in a specific sense. This idea helps mathematicians ensure stability and predictability in the behaviors of functions when things are changing in a space.

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Most popular questions from this chapter

a) Let \(I^{m}=\left\\{x=\left(x^{1}, \ldots, x^{m}\right) \in \mathbb{R}^{m}|| x^{i} \mid \leq c^{i}, i=1, \ldots, m\right\\}\) be an \(m\)-dimensional closed interval and \(I\) a closed interval \([a, b] \subset \mathbb{R}\). Show that if the function \(f(x, y)=f\left(x^{1}, \ldots, x^{m}, y\right)\) is defined and continuous on the set \(I^{m} \times I\), then for any positive number \(\varepsilon>0\) there exists a number \(\delta>0\) such that \(\left|f\left(x, y_{1}\right)-f\left(x, y_{2}\right)\right|<\) \(\varepsilon\) if \(x \in I^{m}, y_{1}, y_{2} \in I\), and \(\left|y_{1}-y_{2}\right|<\delta\) b) Show that the function $$ F(x)=\int_{a}^{b} f(x, y) \mathrm{d} y $$ is defined and continuous on the closed interval \(I^{m}\). c) Show that if \(f \in C\left(I^{m} ; \mathbb{R}\right)\), then the function $$ \mathcal{F}(x, t)=f(t x) $$ is defined and continuous on \(I^{m} \times I^{1}\), where \(I^{1}=\\{t \in \mathbb{R}|| t \mid \leq 1\\} .\) d) Prove Hadamard's lemma: If \(f \in C^{(1)}\left(I^{m} ; \mathbb{R}\right)\) and \(f(0)=0\), there exist functions \(g_{1}, \ldots, g_{m} \in C\left(I^{m} ; \mathbb{R}\right)\) such that $$ f\left(x^{1}, \ldots, x^{m}\right)=\sum_{i=1}^{m} x^{i} g_{i}\left(x^{1}, \ldots, x^{m}\right) $$ in \(I^{m}\), and in addition $$ g_{i}(0)=\frac{\partial f}{\partial x^{i}}(0), \quad i=1, \ldots, m $$

Let \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be a smooth mapping satisfying the Cauchy-Riemann equations $$ \frac{\partial f^{1}}{\partial x^{1}}=\frac{\partial f^{2}}{\partial x^{2}}, \quad \frac{\partial f^{1}}{\partial x^{2}}=-\frac{\partial f^{2}}{\partial x^{1}} $$ a) Show that the Jacobian of such a mapping is zero at a point if and only if \(f^{\prime}(x)\) is the zero matrix at that point. b) Show that if \(f^{\prime}(x) \neq 0\), then the inverse \(f^{-1}\) to the mapping \(f\) is defined in a neighborhood of \(f\) and also satisfies the Cauchy- Riemann equations.

Homogeneous functions and Euler's identity. A function \(f: G \rightarrow \mathbb{R}\) defined in some domain \(G \subset \mathbb{R}^{m}\) is called homogeneous (resp. positive-homogeneous) of degree \(n\) if the equality $$ f(\lambda x)=\lambda^{n} f(x) \quad\left(\text { resp. } f(\lambda x)=|\lambda|^{n} f(x)\right) $$ holds for any \(x \in \mathbb{R}^{m}\) and \(\lambda \in \mathbb{R}\) such that \(x \in G\) and \(\lambda x \in G\) A function is locally homogeneous of degree \(n\) in the domain \(G\) if it is a homogeneous function of degree \(n\) in some neighborhood of each point of \(G\). a) Prove that in a convex domain every locally homogeneous function is homogeneous. b) Let \(G\) be the plane \(\mathbb{R}^{2}\) with the ray \(L=\left\\{(x, y) \in \mathbb{R}^{2} \mid x=2 \wedge y \geq 0\right\\}\) removed. Verify that the function $$ f(x, y)= \begin{cases}y^{4} / x, & \text { if } x>2 \wedge y>0 \\ y^{3}, & \text { at other points of the domain, }\end{cases} $$ is locally homogeneous in \(G\), but is not a homogeneous function in that domain. c) Determine the degree of homogeneity or positive homogeneity of the following functions with their natural domains of definition: $$ \begin{aligned} f_{1}\left(x^{1}, \ldots, x^{m}\right) &=x^{1} x^{2}+x^{2} x^{3}+\cdots+x^{m-1} x^{m} \\ f_{2}\left(x^{1}, x^{2}, x^{3}, x^{4}\right) &=\frac{x^{1} x^{2}+x^{3} x^{4}}{x^{1} x^{2} x^{3}+x^{2} x^{3} x^{4}} \\ f_{3}\left(x^{1}, \ldots, x^{m}\right) &=\left|x^{1} \cdots x^{m}\right|^{l} \end{aligned} $$ d) By differentiating the equality \(f(t x)=t^{n} f(x)\) with respect to \(t\), show that if a differentiable function \(f: G \rightarrow \mathbb{R}\) is locally homogeneous of degree \(n\) in a domain \(G \subset \mathbb{R}^{m}\), it satisfies the following Euler identity for homogeneous functions: $$ x^{1} \frac{\partial f}{\partial x^{1}}\left(x^{1}, \ldots, x^{m t}\right)+\cdots+x^{m} \frac{\partial f}{\partial x^{m}}\left(x^{1}, \ldots, x^{m}\right) \equiv n f\left(x^{1}, \ldots, x^{m}\right) $$ e) Show that if Euler's identity holds for a differentiable function \(f: G \rightarrow \mathbb{R}\) in a domain \(G\), then that function is locally homogeneous of degree \(n\) in \(G\). Hint: Verify that the function \(\varphi(t)=t^{-n} f(t x)\) is defined for every \(x \in G\) and is constant in some neighborhood of \(1 .\)

Let \(x^{1}, \ldots, x^{m}\) be Cartesian coordinates in \(\mathbb{R}^{m}\). The differential operator $$ \Delta=\sum_{i=1}^{m} \frac{\partial^{2}}{\partial x^{i^{2}}} $$ acting on functions \(f \in C^{(2)}(G ; \mathbb{R})\) according to the rule $$ \Delta f=\sum_{i=1}^{m} \frac{\partial^{2} f}{\partial x^{i^{2}}}\left(x^{1}, \ldots, x^{m}\right) $$ is called the Laplacian. The equation \(\Delta f=0\) for the function \(f\) in the domain \(G \subset \mathbb{R}^{m}\) is called Laplace's equation, and its solutions are called harmonic functions in the domain \(G\). a) Show that if \(x=\left(x^{1}, \ldots, x^{m}\right)\) and $$ \|x\|=\sqrt{\sum_{i=1}^{m}\left(x^{i}\right)^{2}} $$ then for \(m>2\) the function $$ f(x)=\|x\|^{-\frac{2-m}{2}} $$ is harmonic in the domain \(\mathbb{R}^{m} \backslash 0\), where \(0=(0, \ldots, 0)\). b) Verify that the function $$ f\left(x^{1}, \ldots, x^{m}, t\right)=\frac{1}{(2 a \sqrt{\pi t})^{m}} \cdot \exp \left(-\frac{\|x\|^{2}}{4 a^{2} t}\right) $$ which is defined for \(t>0\) and \(x=\left(x^{1}, \ldots, x^{m}\right) \in \mathbb{R}^{m}\), satisfies the heat equation $$ \frac{\partial f}{\partial t}=a^{2} \Delta f $$ that is, verify that \(\frac{\partial f}{\partial t}=a^{2} \sum_{i=1}^{m} \frac{\partial^{2} f}{\partial x^{i^{2}}}\) at each point of the domain of definition of the function.

a) Show that the functions \(\pi^{i}(x)=x^{i}(i=1, \ldots, m)\), regarded as functions of the point \(x=\left(x^{1}, \ldots, x^{m}\right) \in \mathbb{R}^{m}\), form an independent system of functions in a neighborhood of any point of \(\mathbb{R}^{m}\). b) Show that, for any function \(f \in C\left(\mathbb{R}^{m} ; \mathbb{R}\right)\) the system \(\pi^{1}, \ldots, \pi^{m}, f\) is functionally dependent. c) If the system of smooth functions \(f^{1}, \ldots, f^{k}, k
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