Eliminating Arbitrary Functions in Partial Differential Equations

Partial differential equations (PDEs) are a crucial part of mathematical analysis. They are used to describe a wide variety of phenomena in physics, engineering, and other sciences. This article focuses on the case where an equation involving an arbitrary function needs to be simplified by eliminating that function. Specifically, we aim to remove the arbitrary function f from the equation fz - xy x^2 y^2 0 using a systematic approach. We will explore the process through new variables, partial derivatives, and the Jacobian determinant, thereby transforming the initial equation into a more tractable form.

Introduction to Partial Differential Equations

Partial differential equations (PDEs) are equations that involve partial derivatives of an unknown function of several variables. These equations are pivotal in modeling phenomena in physics, engineering, and other fields. In this context, the equation fz - xy x^2 y^2 0 involves an arbitrary function f of the variables x, y, z. Our goal is to eliminate f from the equation without losing information about the relationship between x, y, and z.

Introducing New Variables

To simplify the equation, we will introduce new variables defined by the arguments of the arbitrary function f. Define the new variables as follows:

u z - xy v x^2 y^2

With these new variables, the original equation can be rewritten as fu v 0. The process of elimination involves studying the partial derivatives of the functions u and v with respect to the original variables x, y, z.

Partial Derivatives of the New Variables

The partial derivatives of u and v are given by:

Partial derivatives of u: #8706;u/#8706;x -y #8706;u/#8706;y -x #8706;u/#8706;z 1

and for v:

#8706;v/#8706;x 2x #8706;v/#8706;y 2y #8706;v/#8706;z 0

These partial derivatives are essential for the next steps, as they form the basis for the Jacobian determinant and further manipulations.

Constructing the Jacobian Determinant

Construct the Jacobian matrix as:

J [ -y, -x, 1; 2x, 2y, 0 ]

The Jacobian determinant is calculated as:

det(J) -y(0) - (-x)(2y) 2xy

The Jacobian determinant is vital for understanding the transformation of variables and for applying implicit function theorem techniques.

Applying the Implicit Function Theorem

The implicit function theorem allows us to operate on the equation fu v 0 by relating the derivatives of fu and fv. This is done by setting up the following relations:

#8706;f/#8706;u (-y) #8706;f/#8706;v (2x) 0 #8706;f/#8706;u (-x) #8706;f/#8706;v (2y) 0 #8706;f/#8706;u (1) #8706;f/#8706;v (0) 0

From these relations, we can solve for the partial derivatives:

2xy(#8706;f/#8706;v) y(#8706;f/#8706;u)

2xy(#8706;f/#8706;v) x(#8706;f/#8706;u)

This simplifies to:

v(#8706;f/#8706;v) 0

This final partial differential equation v(#8706;f/#8706;v) 0 indicates that the function v is constant, leading to the conclusion that z - xy and x^2 y^2 are the same in form.

In summary, the resulting form of the PDE after eliminating the arbitrary function f from the equation fz - xy x^2 y^2 0 is:

z - xy 0 x^2 y^2 0

These equations define the relationship between the variables x, y, z in 3D space, providing a clearer understanding of the underlying mathematical structure of the initial PDE.