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

\[ abla^{2} = 2xy \]

Step-by-step solution

- Understand the Laplacian Operator

The Laplacian of a scalar field \( abla^{2} \) is given by \( abla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \). For a scalar field \( f(x, y, z) \), this means you need to compute the second partial derivatives with respect to \( x \), \( y \), and \( z \) and sum them up.

- Compute First Partial Derivatives

Find the first partial derivatives of \( f(x, y, z) = x y (x^{2} + y^{2} - 5 z^{2}) \) with respect to \( x \), \( y \), and \( z \) respectively. For \( x \): \[ \frac{\partial f}{\partial x} = y (3x^{2} + y^{2} - 5z^{2}) \] For \( y \): \[ \frac{\partial f}{\partial y} = x (x^{2} + 3y^{2} - 5z^{2}) \] For \( z \): \[ \frac{\partial f}{\partial z} = -10 x y z \]

- Compute Second Partial Derivatives

Find the second partial derivatives of each of the first partial derivatives. For \( x \): \[ \frac{\partial^{2} f}{\partial x^{2}} = y (6x) = 6xy \] For \( y \): \[ \frac{\partial^{2} f}{\partial y^{2}} = x (6y) = 6xy \] For \( z \): \[ \frac{\partial^{2} f}{\partial z^{2}} = -10 x y \]

- Sum the Second Partial Derivatives

To find the Laplacian, add up all the second partial derivatives: \[ abla^{2} f = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}} = 6xy + 6xy - 10xy = 2xy \]

Key concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They measure how a function changes as each variable changes, while keeping all other variables constant.

For example, if we have a scalar field function like \(f(x, y, z) = x y (x^{2} + y^{2} - 5 z^{2})\), we can find the partial derivatives with respect to each variable: \(x\), \(y\), and \(z\).

To find the partial derivative with respect to \(x\), we treat \(y\) and \(z\) as constants. This gives us: \[ \frac{\frac{\tau}{\tau x}} = y (3x^{2} + y^{2} - 5z^{2}) \]

Similarly, we find it for y by treating \(x\) and \(z\) as constants: \[ \frac{\tau f}{\tau y} = x (x^{2} + 3y^{2} - 5z^{2}) \]

Finally, for z: \[ \frac{\tau f}{\tau z} = -10 x y z \]

• Partial derivatives allow us to explore changes in the function in different directions.
• They are crucial for finding gradients, important in optimization, and essential in understanding physical phenomena like heat distribution and fluid flow.

Scalar Field

A scalar field assigns a scalar value (a single number) to every point in space. Scalars have magnitude but no direction.

For instance, consider a temperature map of a room. At each point in the room, you can measure the temperature. The temperature at any point is a scalar value, making temperature a scalar field.

In the given problem, our scalar field is defined by the function: \(f(x, y, z) = xy(x^{2} + y^{2} - 5z^{2})\).

• Scalar fields help model real-life phenomena such as temperature, pressure, and gravitational potential.
• The fact that scalar fields only have a magnitude makes them easier to handle than vector fields.

Scalar fields are used in fields like physics, engineering, and computer graphics to model various properties.

Second-Order Derivatives

Second-order derivatives provide information about the curvature or concavity of a function. They reflect how the rate of change of a function's derivative changes.

To calculate the second-order derivatives for our function, we first needed the first-order derivatives: \[ \frac{\tau^{2}f}{\tau x^{2}} = y(6x) = 6xy \] \[ \frac{\tau^{2}f}{\tau y^{2}} = x(6y) = 6xy \] \[ \frac{\tau^{2}f}{\tau z^{2}} = -10xy \]

Summing these gives us the Laplacian: \[ abla^{2} f = 6xy + 6xy - 10xy = 2xy \].

• Second-order derivatives help identify concavity and points of inflection in functions.
• They are used in optimization problems to determine maxima, minima, and saddle points.

The Laplacian, which involves these second-order derivatives, is widely used in physics, particularly in equations like the heat equation and wave equation.