Problem 31 - Advanced Engineering Mathematics

Question

If \(z=f(x, y)\) is a function with continuous first partial derivatives in a region \(R\), then a flow \(\mathbf{V}(x, y)=(P(x, y), Q(x, y))\) in \(R\) may be defined by letting \(P(x, y)=-\frac{\partial f}{\partial y}(x, y)\) and \(Q(x, y)=\frac{\partial f}{\partial x}(x, y) .\) Show that if \(\mathbf{X}(t)=(x(t), y(t))\) is a solution of the plane autonomous system $$ \begin{aligned} &x^{\prime}=P(x, y) \\ &y^{\prime}=Q(x, y) \end{aligned} $$ then \(f(x(t), y(t))=c\) for some constant \(c .\) Thus a solution curve lies on the level curves of \(f\). [Hint: Use the Chain Rule to compute \(\left.\frac{d}{d t} f(x(t), y(t)) .\right]\)

Step-by-Step Solution

Answer

The solution curve lies on the level curves of \(f\) as \(f(x(t), y(t)) = c\) for some constant \(c\).

Step 1: Write down the given autonomous system equations

The given system of differential equations is:\[\begin{aligned}&x^{\prime} = P(x, y), \&y^{\prime} = Q(x, y)\end{aligned}\]

Step 2: Use the chain rule to differentiate f

The function \(f(x(t), y(t))\) is a composition of functions with respect to \(t\). Use the chain rule to find \(\frac{d}{dt} f(x(t), y(t))\):\[\frac{d}{dt} f(x(t), y(t)) = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}\]

Step 3: Substitute expressions for partial derivatives

Replace \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) with \(x'\) and \(y'\) respectively:\[\frac{d}{dt} f(x(t), y(t)) = \frac{\partial f}{\partial x} P(x, y) + \frac{\partial f}{\partial y} Q(x, y)\]

Step 4: Substitute the values of P(x,y) and Q(x, y)

Using the definitions \(P(x, y) = -\frac{\partial f}{\partial y}\) and \(Q(x, y) = \frac{\partial f}{\partial x}\), substitute back into the equation:\[\frac{d}{dt} f(x(t), y(t)) = \frac{\partial f}{\partial x} \left(-\frac{\partial f}{\partial y} \right) + \frac{\partial f}{\partial y} \frac{\partial f}{\partial x}\]

Step 5: Simplify the expression

Simplify the expression:\[\frac{d}{dt} f(x(t), y(t)) = - \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} + \frac{\partial f}{\partial y}\frac{\partial f}{\partial x}\]Now observe that the two terms on the right-hand side are equal in magnitude but opposite in sign, thus:\[\frac{d}{dt} f(x(t), y(t)) = 0\]

Step 6: Conclude the function's behavior

Since \(\frac{d}{dt} f(x(t), y(t)) = 0\), the function \(f(x(t), y(t))\) is constant with respect to \(t\). This implies \(f(x(t), y(t)) = c\) for some constant \(c\). Thus, each solution curve \(\mathbf{X}(t)\) lies on a level curve of \(f\).

Key Concepts

Partial Derivatives

Understanding partial derivatives is crucial when dealing with functions of multiple variables. A partial derivative of a function of several variables is its derivative with respect to one of those variables, while holding the others constant.
In the context of this exercise, if we have a function \( f(x,y) \), its partial derivatives are denoted by \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \). These represent the rate of change of the function \( f \) with respect to \( x \) and \( y \) respectively, considering all other variables as constants during differentiation.
For example, \( \frac{\partial f}{\partial x} \) describes how \( f \) changes as \( x \) changes, while \( y \) remains fixed.Partial derivatives are fundamental in distinguishing the variations in multivariable systems, and they are essential when constructing flows and analyzing the behavior of functions over different parameters.

Chain Rule

The chain rule is a fundamental theorem in calculus, instrumental for differentiating composite functions.
When a function is a composition of other functions, the chain rule allows us to find its derivative by multiplying the derivatives of the inner and outer functions. In mathematical terms, if \( z = f(x, y) \) and \( x \) and \( y \) are functions of \( t \), then the chain rule gives:

• \( \frac{d}{dt} f(x(t), y(t)) = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} \)

This rule becomes especially useful when the functions are implicit rather than directly dependent on \( t \). In the given system of differential equations, it helps us track how the function \( f \) changes as \( x \) and \( y \) change with time \( t \). Applying the chain rule reveals that the total derivative equates to 0, implying constancy along specific paths in the \( xy \)-plane.

Autonomous Systems

Autonomous systems are systems of differential equations where the time variable \( t \) does not explicitly appear, making the system time-independent.
This means that the equations' structure does not change over time, allowing for analysis of the system's behavior purely based on the states within it. For the given autonomous system:

• \( x' = P(x, y) \)
• \( y' = Q(x, y) \)

We can understand the system as describing a continuous change of \( x \) and \( y \) as time progresses.
Autonomous systems can reveal the stability and behavior of the system by analyzing the equilibrium points and trajectories, or flows, in the phase plane. In our exercise, these flows are defined by the partial derivatives of the function \( f(x,y) \). Understanding autonomous systems can help explore the long-term behavior or attractors within the system, crucial for many applications in physics and engineering.

Level Curves

Level curves, or contour lines, are curves along which a function of two variables is constant.
In the exercise context, level curves of \( f \) are described by \( f(x, y) = c \), where \( c \) is a constant. These curves visually illustrate the "shape" of the function \( f \) in terms of its constant values.
When we demonstrate that \( f(x(t), y(t)) = c \) for a solution curve \( \mathbf{X}(t) \), we are proving that the solution remains on the same level curve as time progresses.Level curves are essential in understanding how functions behave in multi-dimensional spaces because they provide insight into the relationship between variables and the consistency of outcomes.
They help us understand the topology and gradient of the function, and are extremely useful in fields such as meteorology for weather maps, in geography for elevation plots, and in economics for utility functions.