Problem 76 - Calculus Early Transcendentals

Question

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, $$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{-x} \sin y$$

Step-by-Step Solution

Answer

Answer: Yes, the function u(x, y) = e^{-x} sin(y) satisfies Laplace's equation, and it is harmonic.

Step 1: Find the first partial derivatives of u(x, y) with respect to x and y

To start, we will find the first partial derivatives of the given function u(x, y) = e^{-x} sin(y) with respect to x and y. $$ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(e^{-x} \sin y) = -e^{-x} \sin y $$ $$ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(e^{-x} \sin y) = e^{-x} \cos y $$

Step 2: Find the second partial derivatives of u(x, y) with respect to x and y

Now, let's find the second order partial derivatives of u(x, y) with respect to x and y. $$ \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}\left(-e^{-x} \sin y\right) = e^{-x} \sin y $$ $$ \frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}\left(e^{-x} \cos y\right) = -e^{-x} \sin y $$

Step 3: Substitute the second partial derivatives in the Laplace's equation

Now we substitute the second partial derivatives we computed in Step 2 into Laplace's equation. $$ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = e^{-x} \sin y - e^{-x} \sin y $$

Step 4: Determine if the function u(x, y) is harmonic

Finally, we check whether the sum of the second partial derivatives equals zero, which would indicate that the function u(x, y) is harmonic. $$ e^{-x} \sin y - e^{-x} \sin y = 0 $$ Since the sum of the second derivatives equals zero, we conclude that the function u(x, y) = e^{-x} sin(y) is indeed harmonic and satisfies Laplace's equation.

Key Concepts

Harmonic Functions

Harmonic functions are special solutions to Laplace's equation. This equation is fundamental in various fields, representing steady-state phenomena where sources or sinks are absent. In simpler words, they describe smooth behaviors without peaking. Harmonic functions have some fascinating properties, including:

• Smoothness: They are generally very smooth, which means they are continuously differentiable.
• Mean Value Property: The value at any point is the average of its values over any surrounding disk in two dimensions.
• Maximum Principle: Harmonic functions attain their maximum and minimum values typically on the boundary of a domain.

These functions serve as models for many physical systems that are in equilibrium. Laplace's equation, which harmonic functions satisfy, is given by:\[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\] This form highlights how changes in one direction are perfectly balanced by changes in another direction, maintaining a neutral or stable state.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, particularly in multivariable calculus. They refer to the derivatives of a function of several variables with respect to one variable at a time, keeping the others constant. When taking a partial derivative:

• Focus on just one variable, treating others as constant.
• Think of slicing a multidimensional surface parallel to the axis of the chosen variable.

For a function \(u(x, y) = e^{-x} \sin y\), to find the partial derivative with respect to \(x\), treat \(y\) as constant and vice versa:

• \(\frac{\partial u}{\partial x} = -e^{-x} \sin y\)
• \(\frac{\partial u}{\partial y} = e^{-x} \cos y\)

These derivatives inform us about how the function changes locally with respect to each variable independently. Understanding partial derivatives is key to interpreting Laplace's equation since it involves second-order derivatives of functions in terms of both \(x\) and \(y\). Partial derivatives reveal the function's internal behavior and are essential for analyzing its harmonic nature.

Ideal Fluid Flow

In fluid dynamics, ideal fluid flow is a theoretical construct that simplifies real fluid behavior. An ideal fluid is incompressible and has no viscosity, meaning it flows without friction. These simplifications allow us to solve complex problems using equations like Laplace's equation. Ideal fluid flows are:

• Irrational: There are no swirls or spins in the flow.
• Steady: Its speed and direction remain constant over time at any given point.

Harmonic functions appear as potential functions in these flows. That's because potential functions for ideal flows satisfy Laplace's equation, ensuring the flow is consistent and stable without sudden changes.
This connection between harmonic functions and ideal fluid flow aids in visualizing how such a flow behaves, confirming no net forces acting on the fluid mass. This relationship is particularly useful in engineering for designing smooth-flowing systems, such as pipes and aerodynamic surfaces.

Electrostatic Potentials

Electrostatic potentials describe the potential energy per unit charge at a point within an electric field. They are central to understanding electrostatic phenomena, which occur when charges remain at rest. These potentials can be represented as harmonic functions: solutions to Laplace’s equation.

• Potentials are scalar functions describing the energy landscape.
• In regions with no charges, such potentials follow Laplace’s equation.

The Laplace’s equation assures that the electrostatic potential remains stable, not varying abruptly, as it satisfies:\[\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0\] This equation tells us the potential at any point is influenced equally by its surroundings. The concept is crucial for designing safe and efficient electronic components, grounding systems, and insulation materials. Understanding how Laplace's equation governs electrostatic potentials helps in predicting and manipulating how charges interact within a given space, ensuring consistent performance in electrical systems.