Question

For any real positive number \(n\) show that the vector field \(X=x^{n} \partial_{x}\) is differentiable on the manifold \(\mathbb{R}^{+}\)consisting of the positive real line \([x \in \mathbb{R} \mid x>0]\). Why is this not true in general on the enture real line \(\mathrm{R}\) ? As done for the case \(n=2\) in Example 15,13, find the maximal one-parameter subgroup \(\sigma_{i}\) generated by this vector field at any point \(x>0\).

Short answer

The vector field is differentiable on \(\mathbb{R}^{+}\) because \(x^n\) is continuously differentiable for any positive real number \(x\). It's not differentiable on the entire real line \(\mathbb{R}\) because the derivative at \(x=0\) doesn't exist if \(n<1\). The maximal one-parameter subgroups generated by this vector field at any \(x>0\) are \(((1-n)t + x_0^{1-n})^{1/(1 - n)}\) for \( n ≠ -1\) and \(x_0e^{t}\) for \( n=-1\).

Step-by-step solution

Prove the given vector field is differentiable

The given vector field is \(X=x^{n} \partial_{x}\). To show that it's differentiable on the manifold \(\mathbb{R}^{+}\), we need to show its smoothness at every point of the manifold. Since for any positive real number \(x>0\), \(x^n\) is also continuously differentiable, thus the vector field \(X=x^{n} \partial_{x}\) is smooth or differentiable on the manifold \(\mathbb{R}^{+}\).

Explain why it's not differentiable on the entire real line

The vector field \(X=x^{n} \partial_{x}\) is not smooth at \(x=0\), because the derivative doesn't exist at this point if \(n<1\) . This is why the vector field is not differentiable on the entire real line \(\mathbb{R}\).

Finding the maximal one-parameter subgroup

The maximal one-parameter subgroup generated by a vector field \(X\) at any point \(x_0\) is the integral curve of \(X\) with initial condition \(\sigma_i(0) = x_0\). We start by solving the ordinary differential equation related to \(X\) vector field. The ODE here is \(\frac{dx}{dt} = x^n\). When separating and integrating the variables we find that if \( n ≠ -1\), \(\sigma_i(t) = ((1-n)t + x_0^{1-n})^{1/(1 - n)}\) and when \( n=-1\), \(\sigma_i(t) = x_0e^{t}\). Hence, these are the maximal one-parameter subgroups for each condition.

Key concepts

Vector Field

A vector field can be thought of as an association of a vector to every point in a particular space. Imagine a gentle breeze blowing across a field, with the wind direction and force varying from point to point; this is akin to a vector field. In the context of differential geometry, vector fields are essential as they help in defining flows or movements across different spaces and are core to understanding many of the dynamical systems that occur in nature and engineering.

• The vector field in our exercise is expressed through equations such as \(X = x^{n}\partial_{x}\). This notation specifies that for every positive real number \(x\), the vector field assigns a tangent vector \(x^n\partial_x\).
• The functionality of vector fields extends beyond intuitive visualizations, delving into differential equations and their solutions.

The vector field helps in tracing the paths or curves known as integral curves, which reveal the flow patterns generated by the field.

Manifold

Manifolds provide a way to generalize curves and surfaces, extending them to higher dimensions where the intuitive notions of geometry might not apply. Think of them as spaces that locally resemble Euclidean space but can have different global properties.

• For example, in our exercise, the manifold considered is the positive real line \(\mathbb{R}^+\), meaning it includes all real numbers greater than zero.
• This manifold allows exploring differentiability in a piecemeal manner as if each small part were flat and akin to segments of our familiar Euclidean space.
• Manifolds serve as the underlying settings to define vector fields and explore their properties.

Understanding manifolds is crucial as they form the backbone of many complex structures in mathematics, including the study of topology and calculus on more abstract spaces.

Real Numbers

Real numbers are the foundation of almost all of classical mathematics and are essential in various applications from physics to finance. They consist of all the numbers that can be found on the number line.

• Real numbers include both rational numbers (like fractions, which can be expressed as ratios) and irrational numbers (like \(\sqrt{2}\) or \(\pi\), which cannot be expressed as simple fractions).
• In our problem, the vector field is defined on positive real numbers, represented as \(\mathbb{R}^+\), thereby restricting our consideration to numbers greater than zero.
• Real numbers allow us the flexibility to perform calculus, engage with continuous data, and solve a wide variety of equations.

Understanding the properties and structure of real numbers is pivotal for grasping more advanced mathematical topics.

Differentiability

Differentiability is a core concept in calculus and differential geometry, referring to the presence of derivatives or smooth changes. If a function is differentiable, it means you can draw a tangent line to its graph at every point in its domain.

• In the context of the exercise provided, the differentiability of the vector field \(X = x^{n}\partial_{x}\) is highlight as it is differentiable on the manifold of positive real numbers \(\mathbb{R}^+\).
• However, differentiability fails on the entire real line \(\mathbb{R}\) when \(x = 0\), as derivatives do not exist there for \(n < 1\).
• Differentiable functions exhibit smoothness, allowing the accurate prediction of behavior, which is crucial in approximating solutions near points of interest.

Grasping the notion of differentiability allows mathematicians and scientists to better understand and predict the behavior of systems modeled by complex functions.