Question

.\( Is it possible to rectify the direction field of the equation \)\dot{x}=v(t, x)\( on the whole extended phase space \)\boldsymbol{R} \times \boldsymbol{R}^{n}$ when the right-hand side is smooth and defined on this entire space?

Short answer

To rectify the direction field of the given equation \(\dot{x}=v(t,x)\), we need to see if it's possible to find a coordinate transformation \((t,x)\to(\tau(t,x),y(t,x))\) that simplifies the equation so that the local behaviors of its solutions are more easily understood. This depends on the specific vector field \(v(t,x)\), and a definitive answer cannot be provided without analyzing the details of the vector field.

Step-by-step solution

Definition of Rectifying the Direction Field

To rectify a direction field means to simplify it in such a way that the local behavior of solutions is more easily understood. Let's introduce a transformation that simplifies the given equation, if possible.

Transformation

Consider the coordinate transformation \((t,x)\to(\tau(t,x),y(t,x))\) such that the new vector field has the form \(\dot{y} = B\in\boldsymbol{R}^{n}\). That is, we look for a transformation such that the new equation has a constant right-hand side.

Finding Partial Derivatives

Compute the partial derivatives of the transformation with respect to \(t\) and \(x\), namely \(\frac{\partial\tau}{\partial t}\), \(\frac{\partial\tau}{\partial x}\), \(\frac{\partial y}{\partial t}\), and \(\frac{\partial y}{\partial x}\).

Apply the Chain Rule

In the new coordinates, the equation reads as: \[ \frac{d}{d\tau}y = \frac{dy}{d\tau} = B \] Applying the chain rule to express the derivatives in the original coordinates yields: \[ \frac{dy}{d\tau} = \frac{dy}{dt}\frac{dt}{d\tau} = \frac{\partial y}{\partial t} + \frac{\partial y}{\partial x}\dot{x} \]

Equate the Components

Equate components of the new equation with the components of the original equation: \[ \frac{\partial y}{\partial t} + \frac{\partial y}{\partial x}v(t,x) = B\frac{dt}{d\tau} \]

Analyze the Conditions

If it is possible to rectify the direction field, we must be able to find a coordinate transformation \((t,x)\to(\tau(t,x),y(t,x))\) that satisfies the above equation. We must find the partial derivatives of \(\tau\) and \(y\) with respect to \(t\) and \(x\) that make the equation true. If we cannot find any transformation that satisfies these conditions, it means that the direction field is not rectifiable, and it is not possible to transform the equation in such a way that its behavior becomes simpler.

Final Answer

Whether it is possible to rectify the direction field of the equation \(\dot{x}=v(t,x)\) on the whole extended phase space \(\boldsymbol{R} \times \boldsymbol{R}^{n}\) when the right-hand side is smooth and defined on this entire space depends on the specific vector field \(v(t,x)\), and whether we can find a coordinate transformation with the required properties. We cannot provide a definitive answer without analyzing the specific details of the vector field.

Key concepts

Understanding Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. When a problem involves rate of change or time-evolving processes, differential equations come into play. In our exercise, the equation \( \dot{x}=v(t,x) \) is a way to represent how the state of the system (given by \( x \) evolves over time, as \( t \) progresses.

The solutions to differential equations help us understand the behavior of dynamic systems, whether it be population growth, the swinging of a pendile, or even the complex interactions within an ecosystem. The goal in rectifying the direction field of a differential equation is to transform it into a simpler form, where the behavior of the system is clearer and more easily analyzed.

Coordinate Transformation

Coordinate transformation is a powerful tool in mathematics used to simplify complex equations or to convert an equation into a more convenient form for analysis. In this context, transforming the coordinates means changing from the original set of variables \( (t,x) \) to a new set \( (\tau(t,x),y(t,x)) \) that potentially makes solving the differential equation easier.

The effectiveness of the transformation hinges on finding a new way to express the complex relationships in a form where, ideally, the variables are decoupled and the rate of change, represented by a derivative, is constant or simpler to integrate. This is what the question asks about the equation \( \dot{x}=v(t,x) \) - is there a transformation that simplifies its direction field?

Chain Rule in Calculus

The chain rule is a fundamental principle in calculus used to calculate the derivative of a composite function. When functions are composed with each other, the rate of change in one variable depends not only on its direct relationship, but also on how it is indirectly linked through other variables.

In our exercise, after transforming the coordinates, we deal with a new function of \( \tau \) and \( y \) that's derived from the original variables \( t \) and \( x \. By applying the chain rule, as shown in the solution steps, we can relate the derivatives in the original coordinates to those in the new coordinates. This is essential in understanding how changes in \( t \) and \( x \) propagate to our transformed variables \( \tau \) and \( y \).

Partial Derivatives

Partial derivatives represent how a multivariable function changes as only one of the variables is varied, holding the others constant. This concept is crucial when studying systems where multiple variables contribute independently to an end result.

In the context of our problem, to apply our coordinate transformation, we need the partial derivatives of \( \tau \) and \( y \) with respect to \( t \) and \( x \). This step unravels the effect of each original variable on the new ones. Ultimately, by equating the components that changed due to the partial derivatives with the constant \( B \) as seen in the solution steps, we are trying to express the transformed system in terms of a simple, constant rate of change, which makes the analysis of the system’s behavior more straightforward.