91Ó°ÊÓ

Partial derivatives are a way of measuring how a function changes as its variables change individually. In the given exercise, we explore the changes in the function \( f(x, y, z) = 2z^3 - 3(x^2 + y^2)z \) relative to each variable \( x \), \( y \), and \( z \).When we talk about partial derivatives:

• We differentiate the function with respect to one variable while keeping the other variables constant.
• This helps us understand how the function behaves along different dimensions or directions in space.

For example, to find the partial derivative of \( f \) with respect to \( x \), we only differentiate the terms in the function that contain \( x \), treating \( y \) and \( z \) as constants. This can be seen in the intermediate step:\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left(2z^3 - 3(x^2 + y^2)z\right) = -6xz.\]The process is repeated similarly for \( y \) and \( z \). By calculating second partial derivatives, as done here, we deepen our understanding of the function's curvature and its multi-dimensional behavior.

A scalar function is a function that associates a scalar value with every point in its domain. In our current context, the function \( f(x, y, z) \) is a scalar function, as it maps the coordinates \( x, y, z \) to a single output value.Here are some important properties of scalar functions:

• They do not depend on the direction, meaning that their value at any given point doesn't change no matter how you look at it.
• They are particularly useful in fields like physics and engineering where scalar quantities like temperature or pressure are of interest.

In the exercise, \( f \) transforms the three coordinates to a value determined by the equation \( 2z^3 - 3(x^2 + y^2)z \). This means at every point \( (x, y, z) \), the function gives us a single real number, characterizing some property of that point as per the specific context, such as potential energy in physics.

The Laplacian is a differential operator that takes a scalar function and returns another scalar. This operator helps us recognize how a function behaves in space. Mathematically, it is denoted as \( \Delta f \) and for a three-dimensional function \( f(x, y, z) \), it's computed by adding the second partial derivatives of \( f \) with respect to each variable:\[\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.\]The exercise provided solves this for function \( f \), showing that

• \( \frac{\partial^2 f}{\partial x^2} = -6z \)
• \( \frac{\partial^2 f}{\partial y^2} = -6z \)
• \( \frac{\partial^2 f}{\partial z^2} = 12z \)

Summing these gives \( \Delta f = 0 \), as required by the Laplace equation. This simplifies our understanding by showing that such a function has a balanced spatial distribution, often relating to steady-state solutions like potentials.