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 steady-state 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. $$u(x, y)=e^{a x} \cos a y, \text { for any real number } a$$

Short answer

Question: Verify whether the given function u(x, y) = e^(ax) cos(ay) is harmonic by checking if it satisfies Laplace's equation. Answer: The given function u(x, y) = e^(ax) cos(ay) is harmonic, as the sum of its second derivatives with respect to x and y is equal to zero, satisfying Laplace's equation.

Step-by-step solution

Find the second derivative with respect to x

We have the given function \(u(x, y) = e^{ax} \cos ay\). First, we need to find the second derivative of u with respect to x: $$\frac{\partial u}{\partial x} = ae^{ax} \cos ay$$ Now, find the second derivative: $$\frac{\partial^2 u}{\partial x^2} = a^2e^{ax} \cos ay$$

Find the second derivative with respect to y

Similarly, we will find the second derivative with respect to y: $$\frac{\partial u}{\partial y} = -ae^{ax} \sin ay$$ Now, find the second derivative: $$\frac{\partial^2 u}{\partial y^2} = -a^2e^{ax} \cos ay$$

Check if the function is harmonic by fulfilling Laplace's equation

Now that we have both second derivatives, we can check if their sum is equal to zero: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = a^2e^{ax} \cos ay - a^2e^{ax} \cos ay = 0$$ Since the sum of the second derivatives of the given function with respect to x and y is equal to zero, the function is harmonic and satisfies Laplace's equation.

Key concepts

Harmonic Functions

Harmonic functions are fascinating mathematical entities. They are solutions to Laplace's equation, which is given in two dimensions as \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \). Having this balance means that the function's rate of change in one direction is perfectly balanced by the rate of change in the perpendicular direction.

To understand harmonic functions better, think about a soap film stretched across a wireframe. The shape it takes minimizes its potential energy, and this minimal surface can be described using harmonic functions.

The beauty of harmonic functions lies in their property of being infinitely differentiable and exhibiting the mean-value property. This property implies that the value at any point is the average of all surrounding points.

• This results in "smooth" and often visually pleasing surfaces, free of any abrupt changes.
• Harmonic functions are important in physics, helping to model phenomena in steady-state heat distribution, electrical potential, and fluid dynamics.

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They represent the rate of change of a function as described through one of the variables while holding the others constant.

For instance, consider the function \( u(x, y) = e^{ax} \cos ay \). To find \( \frac{\partial u}{\partial x} \), treat \( y \) as a constant. Differentiating \( e^{ax} \cos ay \) with respect to \( x \) results in \( ae^{ax} \cos ay \).

Partial derivatives provide deeper insight into functions with multiple variables by isolating the effect of each variable independently.

• This is crucial when modeling real-world phenomena governed by multiple changing factors, such as temperature fields or airflow.
• Understanding the interaction and contribution of each variable to the function's behavior is essential in engineering, physics, and economics.

Partial derivatives allow us to analyze Laplace's equation more meaningfully. By exploring changes in one direction and then another, we can check if functions like the given \( u(x, y) \) satisfy the equation, making them harmonic.

Ideal Fluid Flow

Ideal fluid flow refers to the theoretical motion of a fluid under perfect conditions, where the fluid is incompressible and has no viscosity. The flow is smooth and free from turbulence, making it an excellent model for many natural systems like atmospheric winds or ocean currents.

Laplace's equation often describes potential flows in ideal fluids. In these scenarios, the velocity potential function fulfills the harmonic property, ensuring the continuity and irrotational nature of ideal fluid flows.

Studying fluid dynamics under such simplified models helps develop foundational principles for more complex, real-world scenarios.

• Through this, engineers design efficient fluid systems and predict fluid behaviors in pipelines and aerodynamic structures.
• While real fluids exhibit complexities like viscosity and varying density, ideal fluid models provide starting points for deeper investigation.