Question

Suppose all second partial derivatives of \(F=F(x, y)\) are continuous and \(F_{x x}+F_{y y}=0\) on an open rectangle \(R\). (A function with these properties is said to be harmonic; see also Exercise 42.) Show that \(-F_{y} d x+F_{x} d y=0\) is exact on \(R,\) and therefore there's a function \(G\) such that \(G_{x}=-F_{y}\) and \(G_{y}=F_{x}\) in \(R .\) (A function \(G\) with this property is said to be a harmonic conjugate of \(\underline{F} .)\)

Short answer

Question: Prove that if a function F is harmonic, then the differential equation -F_y dx + F_x dy = 0 is exact, and there exists a harmonic conjugate G such that G_x = -F_y and G_y = F_x. Answer: In order to show that if F is harmonic, the given differential equation is exact and there exists a harmonic conjugate G, we followed the steps below: 1. We demonstrated that -F_y dx + F_x dy is exact by showing that the mixed second partial derivatives of the two functions are equal. 2. After proving the exactness of the differential equation, we found the harmonic conjugate G that satisfies the conditions G_x = -F_y and G_y = F_x by integrating G_x and relating G_y to F_x through Schwarz's theorem. This relationship between F and G confirms the existence of a harmonic conjugate G.

Step-by-step solution

Show that -F_y dx + F_x dy is exact

To show that the given differential equation is exact, we must show that the mixed second partial derivatives of the two functions are equal. That is, we need to show that (d/dx)(-F_y) = (d/dy)(F_x). By applying the second partial derivatives, we get: $$ -\frac{\partial F_{y}}{\partial x}=\frac{\partial F_{x}}{\partial y} $$ Now, since F is given to be harmonic, we know that F_xx + F_yy = 0. This implies that F_xx = -F_yy. Since all second partial derivatives are continuous, we can use Schwarz's theorem which states that the mixed partial derivatives are equal: $$ F_x = F_{yx} = F_{xy} $$ Similarly, we have: $$ F_y = F_{xy} = F_{yx} $$ Therefore, we have: $$ -\frac{\partial F_{y}}{\partial x} = \frac{\partial F_{yx}}{\partial x} = \frac{\partial F_{xy}}{\partial y} = \frac{\partial F_{x}}{\partial y} $$ Thus, -F_y dx + F_x dy is exact.

Find the harmonic conjugate G

Now that we have shown our differential equation is exact, we can now find the harmonic conjugate G. We are given that G_x = -F_y and G_y = F_x. Let's integrate G_x = -F_y with respect to x: $$ G(x, y) = -\int F_y(x, y)dx + C_1(y) $$ where C_1(y) is an arbitrary function of y. Now, let's differentiate G(x, y) w.r.t. y: $$ \frac{\partial G(x, y)}{\partial y} = -\frac{\partial (\int F_y (x, y) dx)}{\partial y} + C_1'(y) $$ By Schwarz's theorem, we can switch the order of differentiation on the left side: $$ \frac{\partial(\int F_y (x, y) dx)}{\partial y} = \int \frac{\partial F_y (x, y)}{\partial y} dx = \int F_{yy} (x, y) dx $$ Substituting this back into our equation for G_y: $$ G_{y}(x, y) = - \int F_{yy}(x, y) dx + C_1'(y) $$ By definition, we want G_y = F_x. Replacing G_y with F_x, we get: $$ F_x(x, y) = - \int F_{yy}(x, y) dx + C_1'(y) $$ To satisfy this equation, we choose C_1'(y) = 0, which gives us a constant C. Thus, we have: $$ F_x(x, y) = - \int F_{yy}(x, y) dx + C $$ Now that we have found the relationship between F and G, we can declare G as a harmonic conjugate of F since it satisfies the conditions G_x = -F_y and G_y = F_x.

Key concepts

Exact Differential Equations

Exact differential equations are a type of differential equation that are structured in such a way that they can be expressed as a total derivative. This means the equation can be written as a derivative of a function with respect to a single variable, making it simpler to solve.

In the context of this exercise, we aim to show that the given form \(-F_y dx + F_x dy\) is exact. This involves proving that the mixed partial derivatives are equal. Specifically, this means that \(\frac{\partial(-F_y)}{\partial x} = \frac{\partial F_x}{\partial y}\).

This condition arises due to the fact that if a differential equation is exact, there exists some function \(G\) in which the components of the equation correspond to the gradients of \(G\). Thus, verifying exactness is crucial for finding solutions.

Harmonic Conjugate

A harmonic conjugate is a function that pairs with a given harmonic function to establish a pair of functions that are used in complex analysis and potential theory. When paired, these functions help define analytic functions in the complex plane.

In terms of this exercise, if \(F\) is the given harmonic function, the goal is to find a function \(G\) such that \(G_x = -F_y\) and \(G_y = F_x\). This relationship is crucial because it allows us to construct a complex analytic function using \(F\) and \(G\) as real and imaginary parts respectively.

The importance of finding a harmonic conjugate lies in its utility in solving and simplifying problems in mathematical physics and engineering, especially those involving potentials and flows.

Partial Derivatives

Partial derivatives are derivatives where we hold some variables constant while differentiating with respect to another. These are essential in multivariable calculus, allowing us to understand how a function changes in each direction of its variables.

In this context, partial derivatives are used to demonstrate the conditions for exactness of the differential equation and to establish the relationships necessary to find a harmonic conjugate. By calculating \(F_x\), \(F_y\), and the mixed derivatives like \(\frac{\partial F_x}{\partial y}\), we verify that the equation is exact.

Mastering partial derivatives helps in determining the behavior of surfaces and curves in higher-dimensional spaces and is fundamental in optimization and other mathematical applications.

Schwarz's Theorem

Schwarz's Theorem, also known as Clairaut's Theorem, is a fundamental result in calculus concerning mixed partial derivatives. It states that if the second partial derivatives of a function are continuous, the order of differentiation doesn't matter. In mathematical terms, \(F_{xy} = F_{yx}\).

This theorem is applied in the exercise to confirm that the mixed partial derivatives required for the equation to be exact are indeed equal. Because all second partial derivatives in our problem are continuous, Schwarz's Theorem assures us that \(\frac{\partial(-F_y)}{\partial x} = \frac{\partial F_x}{\partial y}\), facilitating the proof of exactness.

Understanding Schwarz's Theorem is crucial for anyone dealing with differential equations or multivariable calculus, as it guarantees certain symmetries in partial derivatives that are often necessary in problem-solving.