Problem 77 - 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. $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$

Step-by-Step Solution

Answer

Answer: No, the function is not harmonic because its sum of second-order partial derivatives, $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$$, is equal to $$-2x$$, which is not equal to zero.

Step 1: Compute the first partial derivatives

(With respect to x and y) First, we need to find the first order partial derivatives of the function with respect to x and y. Let's now find \(\frac{\partial u}{\partial x}\) and \(\frac{\partial u}{\partial y}\). $$\frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left[x\left(x^{2}-3 y^{2}\right)\right] = (x^{2}-3 y^{2})+2x^2$$ $$\frac{\partial u}{\partial y} = \frac{\partial}{\partial y} \left[x\left(x^{2}-3 y^{2}\right)\right] = -6xy$$

Step 2: Compute the second partial derivatives

(With respect to x and y) Now, we need to find the second-order partial derivatives with respect to x and y. Let's now find \(\frac{\partial^{2} u}{\partial x^{2}}\) and \(\frac{\partial^{2} u}{\partial y^{2}}\). $$\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} \left[(x^{2}-3 y^{2})+2x^2\right] = 4x$$ $$\frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y} \left[-6xy\right] = -6x$$

Step 3: Verify Laplace's equation

(Sum of second partial derivatives) Now, let's check if the sum of the second-order partial derivatives is equal to zero, which is required for the function to be harmonic. Let's calculate \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\). $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} = 4x - 6x$$ The sum of the second-order partial derivatives is equal to the following: $$4x - 6x = -2x$$ Since the sum of the second-order partial derivatives is not equal to zero, the given function $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$ does not satisfy Laplace's equation, and thus it is not a harmonic function.

Key Concepts

Harmonic Functions

Harmonic functions are an essential concept in calculus, particularly for solving Laplace's Equation. They are functions that satisfy Laplace's equation, meaning the sum of their second partial derivatives equals zero. In two dimensions, this condition is expressed by the equation:\[ \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0 \]Harmonic functions often appear in various scientific fields, including physics and engineering. They model situations like ideal fluid flow, electrostatic potentials, and steady-state heat distribution. Understanding whether a function is harmonic involves calculating these second partial derivatives and checking if their sum is zero.To determine if a given function is harmonic, one needs to understand and apply partial differentiation effectively. This leads us to the concept of partial derivatives, which play a critical role in finding whether a function satisfies Laplace's equation.

Partial Derivatives

Partial derivatives allow us to analyze functions with multiple variables. Unlike ordinary differentiation, where we differentiate with respect to one variable, partial derivatives involve treating other variables as constants. This process helps us understand how the function changes along different axes, which is crucial in fields like physics and engineering.To compute partial derivatives, follow these steps:

• Choose the variable to differentiate with respect to, while treating other variables as constants.
• Apply the differentiation rules to find the first partial derivative.
• If needed, repeat the process to obtain higher-order derivatives, such as the second partial derivative.

For example, in the given problem of a function \(u(x, y)=x(x^2-3y^2)\), calculating \(\frac{\partial u}{\partial x}\) and \(\frac{\partial u}{\partial y}\) is necessary to determine if the function is harmonic. A function is declared harmonic if the sum of its second partial derivatives amounts to zero, satisfying Laplace's equation. Understanding partial derivatives equips us with the mathematical tools required to tackle more complex problems, including those found in differential equations and their applications.

Theory and Applications of Differential Equations

Differential equations are mathematical equations that involve derivatives, and they describe various phenomena in the natural and engineering sciences. Laplace's equation is a type of differential equation called a partial differential equation (PDE). Understanding it can help solve many physical problems where variables change continuously in time or space. Laplace's equation is fundamental in several areas:

• Electrostatics, where it governs the electric potential field in a charge-free region.
• Fluid dynamics, helping to describe ideal fluid flow.
• Thermal processes, particularly in heat conduction analysis where the temperature distribution reaches a steady state.

By applying the theory of differential equations, scientists and engineers can predict and analyze real-world systems' behavior. Solving these equations often involves finding functions that satisfy specific conditions — such as being harmonic, as in the case of Laplace's equation. In summary, understanding and applying differential equations allow us to model complex systems accurately and solve real-life problems efficiently, aligning with the essential theories underpinning much of modern science and engineering.