Question

Calculate the Laplacian \(\nabla^{2}\) of each of the following scalar fields. $$x y\left(x^{2}+y^{2}-5 z^{2}\right)$$

Short answer

\( abla^2 f = 2xy \)

Step-by-step solution

Identify the Scalar Field

The scalar field given is: \( f(x, y, z) = x y \big(x^2 + y^2 - 5z^2 \big)\).

Understand the Laplacian Operator

The Laplacian operator for a scalar field in Cartesian coordinates is defined as: \[abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\].

Compute the Second Partial Derivative with respect to x

First, compute the first partial derivative with respect to \( x \): \[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( x y (x^2 + y^2 - 5 z^2) \right) = y (3x^2 + y^2 - 5z^2). \]Then compute the second partial derivative: \[ \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} \left( y (3x^2 + y^2 - 5z^2) \right) = 6xy. \]

Compute the Second Partial Derivative with respect to y

First, compute the first partial derivative with respect to \( y \): \[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( x y (x^2 + y^2 - 5 z^2) \right) = x (x^2 + 3y^2 - 5z^2). \]Then compute the second partial derivative: \[ \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y} \left( x (x^2 + 3y^2 - 5z^2) \right) = 6xy. \]

Compute the Second Partial Derivative with respect to z

First, compute the first partial derivative with respect to \( z \): \[ \frac{\partial f}{\partial z} = \frac{\partial}{\partial z} \left( x y (x^2 + y^2 - 5 z^2) \right) = -10xyz. \]Then compute the second partial derivative: \[ \frac{\partial^2 f}{\partial z^2} = \frac{\partial}{\partial z} \left( -10xyz \right) = -10xy. \]

Sum the Second Partial Derivatives

Now, sum the second partial derivatives to obtain the Laplacian: \[ abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 6xy + 6xy - 10xy = 2xy. \]

Key concepts

Partial Derivatives

Partial derivatives are fundamental in understanding how functions change when you vary one variable while keeping others constant. If we have a function of multiple variables like $$ f(x, y, z) $$ and we want to see how it changes with respect to $$ x $$, we compute the partial derivative $$ \frac{\partial f}{\partial x} $$. This essentially focuses on the rate of change along the $$ x $$-axis, ignoring all changes in $$ y $$ and $$ z $$. Steps to compute partial derivatives include:

• Choose the variable of interest.
• Differentially treat other variables as constants.
• Apply standard differentiation rules.

In the exercise, we computed the partial derivatives of the field $$ x y (x^2 + y^2 - 5 z^2) $$ step-by-step, focusing individually on $$ x $$, $$ y $$, and $$ z $$.

Scalar Field

A scalar field is a mathematical function that associates a single scalar value to every point in space. For instance, in weather modeling, temperature distribution across a region can be considered a scalar field. In our problem, we dealt with the scalar field $$ f(x, y, z) = xy(x^2 + y^2 - 5z^2) $$.
This scalar field depends on spatial coordinates $$ x $$, $$ y $$, and $$ z $$. The goal was to compute how it changes by using the Laplacian, which measures the rate of change summed over all dimensions. Understanding scalar fields helps in visualizing how quantities distribute and vary in space, which is crucial for physics and engineering problems.

Cartesian Coordinates

Cartesian coordinates are a system that specifies points in space using numerical values along perpendicular axes, usually labeled $$ x $$, $$ y $$, and $$ z $$. For any point in 3D space, these coordinates give exact positions.
In our problem, we used Cartesian coordinates to define the positions in the scalar field. The formulas for partial derivatives and the Laplacian operate on these coordinates because they offer a straightforward way to represent spatial relations and changes. Understanding Cartesian coordinates is crucial when working with differential operators in multi-dimensional spaces such as in engineering and physics applications.

Differential Operators

Differential operators are operators defined as functions that take one function as an input and produce another function as an output by differentiating it according to certain rules. Key examples include the gradient, divergence, curl, and Laplacian. In this exercise, we used the Laplacian operator.
The Laplacian operator in Cartesian coordinates is defined as:
\[abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\]
To compute this for our scalar field $$ f(x, y, z) $$, we found the second partial derivatives $$ \frac{\partial^2 f}{\partial x^2} $$, $$ \frac{\partial^2 f}{\partial y^2} $$, and $$ \frac{\partial^2 f}{\partial z^2} $$, and summed them up. This final sum gave us the Laplacian of the scalar field, which turned out to be $$ 2xy $$.