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Chapter 9: Problem 18

Define vector fields \(X\) and \(Y\) on the plane by $$ X=x \frac{\partial}{\partial x}-y \frac{\partial}{\partial y}, \quad Y=x \frac{\partial}{\partial y}+y \frac{\partial}{\partial x} $$ Compute the flows \(\theta, \psi\) of \(X\) and \(Y\), and verify that the flows do not commute by finding explicit open intervals \(J\) and \(K\) containing 0 such that \(\theta_{s} \circ \psi_{t}\) and \(\psi_{t} \circ \theta_{s}\) are both defined for all \((s, t) \in J \times K\), but they are unequal for some such \((s, t)\).

Short Answer

Expert verified
Flows \( \theta_s \) and \( \psi_t \) do not commute; they are unequal for any \( s, t \neq 0 \).

Step by step solution

01

Set Up Differential Equations for X

The vector field \( X \) is given by \( X=x \frac{\partial}{\partial x}-y \frac{\partial}{\partial y} \). The flow \( \theta_t \) of \( X \) corresponds to solving the following system of ordinary differential equations (ODEs):\[ \frac{dx}{dt} = x, \quad \frac{dy}{dt} = -y \] with initial conditions \( x(0) = x_0, \ y(0) = y_0 \).
02

Solve ODEs for X's Flow

Solving the system of ODEs, we separate variables and integrate:- \( \frac{dx}{dt} = x \) gives \( x(t) = x_0 e^{t} \).- \( \frac{dy}{dt} = -y \) gives \( y(t) = y_0 e^{-t} \).Thus, the flow \( \theta_t \) of \( X \) is given by:\[ \theta_t(x_0, y_0) = (x_0 e^{t}, y_0 e^{-t}) \]
03

Set Up Differential Equations for Y

The vector field \( Y \) is given by \( Y=x \frac{\partial}{\partial y}+y \frac{\partial}{\partial x} \). The flow \( \psi_t \) of \( Y \) corresponds to solving:\[ \frac{dx}{dt} = y, \quad \frac{dy}{dt} = x \] with initial conditions \( x(0) = x_0, \ y(0) = y_0 \).
04

Solve ODEs for Y's Flow

Solving the system of ODEs, we use the structure suggesting rotating coordinates:- Solving \( \frac{d^2x}{dt^2} = -x \) with initial \( x(0) = x_0, \ \frac{dy}{dt}(0) = x_0 \) gives solutions: \[ x(t) = x_0 \cos(t) + y_0 \sin(t) \] - Similarly, \( y(t) = y_0 \cos(t) - x_0 \sin(t) \) Thus, the flow \( \psi_t \) is:\[ \psi_t(x_0, y_0) = (x_0 \cos(t) + y_0 \sin(t), y_0 \cos(t) - x_0 \sin(t)) \]
05

Compose Flows and Check Commutes

To check if \( \theta_s and \psi_t \) commute, compute composition:- \( \theta_s \circ \psi_t(x_0, y_0) = \theta_s(x_0 \cos(t) + y_0 \sin(t), y_0 \cos(t) - x_0 \sin(t)) = (e^s(x_0 \cos(t) + y_0 \sin(t)), e^{-s}(y_0 \cos(t) - x_0 \sin(t))) \)- \( \psi_t \circ \theta_s(x_0, y_0) = \psi_t(x_0 e^s, y_0 e^{-s}) = (x_0 e^s \cos(t) + y_0 e^{-s} \sin(t), y_0 e^{-s} \cos(t) - x_0 e^s \sin(t)) \)These are unequal unless \( s = 0 \) or \( t = 0 \).
06

Determine Definition Intervals J and K

Both flows \( \theta_t \) and \( \psi_t \) are defined for all \( t \in \mathbb{R} \), allowing us to choose any open interval \( J\) and \( K\) around 0. Evidently, choosing \( J = K = (-\epsilon, \epsilon) \) for small \( \epsilon > 0 \) satisfies this requirement.

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Key Concepts

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

Differential Equations
Differential equations form the basis of analyzing the flows of vector fields. They are mathematical equations that relate a function with its derivatives, helping to describe how a quantity changes over time. For vector fields, these equations are crucial as they allow us to understand how objects move within those fields.
The main types of differential equations include:
  • Ordinary Differential Equations (ODEs): These involve functions of only one variable and its derivatives. For example, in the exercise, the differential equations for vector field \(X\) are ODEs given by \( \frac{dx}{dt} = x \) and \( \frac{dy}{dt} = -y \). Solving these gives us the flows, or the paths, that objects follow.
  • Partial Differential Equations (PDEs): These involve multiple independent variables. Though not directly used in this exercise, PDEs are valuable when dealing with more complex systems where variables interrelate in multiple dimensions.
Overall, solving differential equations is crucial for predicting the behavior of systems modeled by vector fields, providing insights into how a system evolves over time.
Flow of Vector Fields
The flow of a vector field represents the trajectory that a point in the field follows over time. Visualizing these flows can help understand the dynamics of the system described by the vector fields. In this problem, computing the flows involves integrating the differential equations for each vector field.
For the vector field \( X \), the flow \( \theta_t \) is derived by solving the system of ODEs, resulting in:
  • \[ x(t) = x_0 e^t \]
  • \[ y(t) = y_0 e^{-t} \]
This flow illustrates exponential growth in one direction and decay in the other.
For the vector field \( Y \), the flow \( \psi_t \) depicts rotations, described by:
  • \[ x(t) = x_0 \,\cos(t) + y_0 \,\sin(t) \]
  • \[ y(t) = y_0 \,\cos(t) - x_0 \,\sin(t) \]
Understanding these flows helps in visualizing how points move through the vector field, offering a dynamic view of their behavior across time.
Coordinate Transformations
Coordinate transformations allow us to change the way we describe points in space. They are essential for simplifying problems or converting solutions from one form to another to better fit the situation. By transforming coordinates, we often find it easier to solve differential equations or analyze geometric properties.
In the context of the exercise, coordinate transformations manifest through the rotation seen in the flow of vector field \( Y \). The flow \( \psi_t \) rotates points around the origin, illustrating how transformations can be pivotal in visualizing and understanding the movement in vector fields. Compared to the flow \( \theta_t \) of \( X \), which represents scaling rather than rotation, we see how different transformations embody different physical actions in the field.
By using coordinate transformations strategically, one can reveal symmetries, simplify equations, and offer clearer insights into the behavior of systems influenced by vector fields. In this way, they help bridge the gap between abstract mathematical solutions and physical interpretations.

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Most popular questions from this chapter

Suppose \(M\) is a smooth manifold and \(S \subseteq M\) is an embedded hypersurface (not necessarily compact). Suppose further that there is a smooth vector field \(V\) defined on a neighborhood of \(S\) and nowhere tangent to \(S\). Show that \(S\) has a neighborhood in \(M\) diffeomorphic to \((-1,1) \times S\), under a diffeomorphism that restricts to the obvious identification \(\\{0\\} \times S \approx S\). [Hint: using the notation of the flowout theorem, show that \(\mathcal{O}_{\delta} \approx \mathcal{O}_{1}\).]

Suppose \(M\) is a smooth, compact manifold that admits a nowhere vanishing smooth vector field. Show that there exists a smooth map \(F: M \rightarrow M\) that is homotopic to the identity and has no fixed points.

Prove the converse to Euler's homogeneous function theorem (Problem 8-2): if \(f \in C^{\infty}\left(\mathbb{R}^{n} \backslash\\{0\\}\right)\) satisfies \(V f=c f\), where \(V\) is the Euler vector field and \(c \in \mathbb{R}\), then \(f\) is positively homogeneous of degree \(c\).

Give an example of smooth vector fields \(V, \tilde{V}\), and \(W\) on \(\mathbb{R}^{2}\) such that \(V=\tilde{V}=\partial / \partial x\) along the \(x\)-axis but \(\mathscr{L}_{V} W \neq \mathscr{L}_{\tilde{V}} W\) at the origin. [Remark: this shows that it is really necessary to know the vector field \(V\) to compute \(\left(\mathscr{L}_{V} W\right)_{p}\); it is not sufficient just to know the vector \(V_{p}\), or even to know the values of \(V\) along an integral curve of \(V .]\)

Suppose \(M\) is a compact smooth manifold and \(V: J \times M \rightarrow T M\) is a smooth time-dependent vector field on \(M\). Show that the domain of the time-dependent flow of \(V\) is all of \(J \times J \times M\).
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