Problem 52 - Calculus with Concepts in Calculus

Question

A function \(z\) satisfies Laplace's equation if $$ \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 $$ In Exercises \(52-53\) show that the function satisfies Laplace's equation. $$ z=x^{4}-6 x^{2} y^{2}+y^{4} $$

Step-by-Step Solution

Answer

The function satisfies Laplace's equation.

Step 1: Compute the First Partial Derivatives

To verify if the function satisfies Laplace's equation, first, compute the first derivatives with respect to both variables, \( x \) and \( y \).For \( z = x^4 - 6x^2y^2 + y^4 \):* \( \frac{\partial z}{\partial x} = 4x^3 - 12xy^2 \)* \( \frac{\partial z}{\partial y} = -12x^2y + 4y^3 \)

Step 2: Compute the Second Partial Derivatives

Now, compute the second partial derivatives for both \( x \) and \( y \) from the first derivatives.* Second derivative with respect to \( x \): \( \frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x}(4x^3 - 12xy^2) = 12x^2 - 12y^2 \)* Second derivative with respect to \( y \): \( \frac{\partial^2 z}{\partial y^2} = \frac{\partial}{\partial y}(-12x^2y + 4y^3) = -12x^2 + 12y^2 \)

Step 3: Verify Laplace's Equation

Substitute the second derivatives into Laplace's equation:\[ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = (12x^2 - 12y^2) + (-12x^2 + 12y^2) \]The sums simplify to:\[ 12x^2 - 12y^2 - 12x^2 + 12y^2 = 0 \]Since the equation simplifies to zero, the function satisfies Laplace's equation.

Key Concepts

Partial Derivatives

In the realm of multivariable calculus, partial derivatives are an essential concept. They help us understand how a function changes when one variable changes, while the other variables remain constant.
Partial derivatives allow us to explore the behavior of functions with more than one variable, like the function given in the exercise. When computing a partial derivative with respect to a specific variable, we treat other variables as constants.
For the function \( z = x^4 - 6x^2y^2 + y^4 \), we found the first partial derivatives as follows:

• Partial derivative with respect to \( x \): \( \frac{\partial z}{\partial x} = 4x^3 - 12xy^2 \)
• Partial derivative with respect to \( y \): \( \frac{\partial z}{\partial y} = -12x^2y + 4y^3 \)

These derivatives show how the function \( z \) is changing as \( x \) or \( y \) change individually, providing insights into the behavior of the surface described by the function.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions of more than one variable. This field allows us to study complex systems and phenomena by exploring relationships between multiple variables concurrently.
When dealing with multiple variables, we often find ourselves calculating partial derivatives and using concepts like gradients, divergence, and curl.
In our exercise, we work with a function of two variables, \( x \) and \( y \), where the changes in \( z \) depend simultaneously on both. Through the exercise's solution, we compute the second partial derivatives, which involve taking the derivative of a derivative:

• Second derivative with respect to \( x \): \( \frac{\partial^2 z}{\partial x^2} = 12x^2 - 12y^2 \)
• Second derivative with respect to \( y \): \( \frac{\partial^2 z}{\partial y^2} = -12x^2 + 12y^2 \)

These calculations are central in determining how the function behaves regarding curvature and how it might behave as a surface.

Partial Differential Equations

Partial differential equations (PDEs) like Laplace's equation play a vital role in modeling various physical phenomena, where changes occur with respect to multiple dimensions or conditions.
Laplace's equation is a specific type of PDE represented as:\[ \frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial y^{2}} = 0 \]This equation is ubiquitous in physics, appearing in electromagnetism, fluid dynamics, and heat conduction problems.
Our task is to show that a given function satisfies this equation by substituting the computed second partial derivatives back into it. In this exercise, after substituting the second partial derivatives of \( z \):

• \( 12x^2 - 12y^2 + (-12x^2 + 12y^2) = 0 \)

The resulting expression simplifies to zero, confirming that \( z \) indeed satisfies Laplace's equation, showing that the function is harmonic.