Question

Show that the functions defined are harmonic functions. \(f(x, y)=x^{3} y-x y^{3}\)

Short answer

The function is harmonic because it satisfies Laplace's equation \(\nabla^2 f = 0\).

Step-by-step solution

Understanding Harmonic Functions

A function is considered harmonic if it satisfies Laplace's equation, which is given as \(abla^2 f = 0\). For a function \(f(x, y)\), this translates to the condition: \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0\). Our goal is to prove this for the given function \(f(x, y) = x^3 y - xy^3\).

Compute the First Order Derivatives

Compute the first partial derivatives of the function. We have: - \(\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^3 y - xy^3) = 3x^2 y - y^3\) - \(\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^3 y - xy^3) = x^3 - 3xy^2\)

Compute the Second Order Derivatives

Compute the second partial derivatives needed for Laplace's equation: - \(\frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}(3x^2 y - y^3) = 6xy\) - \(\frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}(x^3 - 3xy^2) = -6xy\)

Verify Laplace's Equation

Sum the second order derivatives to verify Laplace's equation:\[\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 6xy - 6xy = 0\]Since the sum equals zero, \(f(x, y)\) satisfies Laplace's equation and is therefore harmonic.

Key concepts

Laplace's Equation

Laplace's equation is a crucial concept in mathematics, particularly in the field of potential theory and physics. It is commonly used to identify functions known as harmonic functions.

At its core, Laplace's equation is expressed as \[ abla^2 f = 0 \] which means that the second partial derivatives of a function must sum to zero. For functions of two variables, \(x\) and \(y\), it is structured as: \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \]

• This looks simple but is immensely powerful as it describes phenomena like steady-state heat distribution and electric potentials.
• Solutions to Laplace's equation, known as harmonic functions, have special properties, such as possessing mean value properties that allow the function's value at a point to be averaged from nearby values.

Verifying harmonic functions involves computing their second partial derivatives and checking if these satisfy the equation, i.e., if they sum to zero.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus used to understand functions of multiple variables. They measure how the function changes as one of its input variables changes, while keeping other variables constant.

For the function given in the exercise, \( f(x, y) = x^3 y - x y^3 \), we compute two key partial derivatives:

• \( \frac{\partial f}{\partial x} = 3x^2 y - y^3 \) calculates the rate of change of \( f \) with respect to \( x \).
• \( \frac{\partial f}{\partial y} = x^3 - 3xy^2 \) measures the change in \( f \) concerning \( y \).

• The process involves treating other variables as constants temporarily while deriving.
• Each partial derivative provides information on how the function behaves in relation to the changes of each specific variable.

Mastering partial derivatives is essential for tackling more complex topics, as seen with their role in solving Laplace's equation.

Second Order Derivatives

Second order derivatives are obtained by taking the derivative of a derivative. In the context of functions of several variables, second order derivatives help further explore the curvature and behavior of functions. This is crucial for understanding harmonic functions via Laplace's equation.

For the given function \( f(x, y) = x^3 y - x y^3 \), we explore:

• The second derivative with respect to \( x \), \( \frac{\partial^2 f}{\partial x^2} = 6xy \), shows how \( \frac{\partial f}{\partial x} \) changes with \( x \).
• The second derivative with respect to \( y \), \( \frac{\partial^2 f}{\partial y^2} = -6xy \), captures the change in \( \frac{\partial f}{\partial y} \).

• The key idea behind these is examining whether their sum equals zero to satisfy Laplace’s equation.
• Second order derivatives thus give a deeper look into how variations in each direction interact within a function.

Understanding these concepts makes it easier to identify harmonic functions, an essential task in many fields of mathematics and science.