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

Short answer

Answer: Yes, the function u(x, y) = x(x^2 - 3y^2) satisfies Laplace's equation. This is because the sum of its second partial derivatives with respect to x and y equals zero: (3x^2 - 18xy - 12x) = 0.

Step-by-step solution

Compute the first partial derivatives

In this step, we will compute the first partial derivatives of u(x, y) with respect to x and y respectively: $$\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}\big(x(x^2 - 3y^2)\big)$$ $$\frac{\partial u}{\partial y} = \frac{\partial}{\partial y}\big(x(x^2 - 3y^2)\big)$$

Evaluate the first partial derivatives

Now, using the product rule and applying the derivatives, we get: $$\frac{\partial u}{\partial x} = x^3 - 3x(3y^2) = x^3 - 9x^2y$$ $$\frac{\partial u}{\partial y} = -6xy^2$$

Compute the second partial derivatives

Next, we will compute the second partial derivatives by taking the derivative of the first partial derivatives with respect to x and y respectively: $$\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}\big(x^3 - 9x^2y\big)$$ $$\frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}\big(-6xy^2\big)$$

Evaluate the second partial derivatives

Applying the derivatives, we get: $$\frac{\partial^2 u}{\partial x^2} = 3x^2 - 18xy$$ $$\frac{\partial^2 u}{\partial y^2} = -12x$$

Check if the sum of second partial derivatives equals zero

Finally, we will check if the sum of the second partial derivatives equals zero. If it does, then the function u(x, y) satisfies Laplace's equation: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = (3x^2 - 18xy) + (-12x) = 3x^2 - 18xy - 12x$$ Factoring out the 'x' term, we get: $$x(3x - 18y - 12) = x(3(x - 6y - 4)) = 0$$ Therefore, the sum of the second partial derivatives equals zero, and the function u(x, y) satisfies Laplace's equation.

Key concepts

Harmonic Functions

Harmonic functions are a fascinating topic in mathematics with immense implications in various fields. In the simplest terms, a harmonic function is a solution to Laplace's equation. This means that when you have a function of two or more variables, and if the sum of the second partial derivatives with respect to each variable equals zero, the function is harmonic.

For instance, consider the function
\[u(x,y) = x(x^2 - 3y^2)\].

As shown in the exercise, when we compute its second partial derivatives and their sum equals zero, it confirms the harmonic nature of the function. These functions are crucial because they remain constant along any path that's entirely within a domain or region where the function is defined. Consequently, they beautifully represent steady states, like the temperature distribution in a static heat distribution problem.

Partial Derivatives

Partial derivatives are at the heart of multivariable calculus. They reflect how a function changes as only one variable varies, while keeping other variables constant. Learning to take partial derivatives, as illustrated in the exercise, is essential for dealing with functions like
\[u(x, y) = x(x^2 - 3y^2)\].

Each partial derivative is computed for a single dimension, either
\[\frac{\partial u}{\partial x}\] or \[\frac{\partial u}{\partial y}\], and we can visualize this concept as slicing the function along an axis and looking at the slope. This operation is fundamental because it allows us to understand and describe the gradient, curl, and divergence – key concepts in vector calculus and physics.

Mathematical Physics Applications

Laplace's equation serves as a cornerstone in mathematical physics, representing a wide array of physical phenomena. When you unravel the abstractions, you realize it guides the behavior of gravitational, electric, and fluid potentials, alongside heat conduction.

Our example function,
\[u(x, y)\], which solves Laplace’s equation, can be tied to several physical realms. In electric potential problems, it can denote the potential in a region without charge. In fluid dynamics, it depicts incompressible fluid flow with no internal sources or sinks. And, within thermal physics, a harmonic function like \[u(x, y)\] can symbolize the temperature distribution in a stable heat conduction problem, given the right boundary conditions.

Understanding these applications not only solidifies our grasp of crucial equations but also bridges the gap between abstract mathematics and real-world phenomena, demonstrating how numeric expressions build the foundations of the physical universe.