Question

Calculate the Laplacian \(\nabla^{2}\) of each of the following scalar fields.\(\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}\)

Short answer

abla^2\big(\frac{1}{\sqrt{x^2 + y^2 + z^2}}\big) = -4\pi \delta(\mathbf{r})

Step-by-step solution

Identify the Given Scalar Field

The scalar field given is \[\frac{1}{\sqrt{x^2 + y^2 + z^2}}\].

Understand the Laplacian Operator

The Laplacian of a scalar field is given by \[abla^2 = \frac{abla \bullet abla}{\text{sum of second partial derivatives}} = \frac{abla^2}{\text{coordinate system}}\].

Calculate Partial Derivatives

First, calculate the partial derivatives of the field with respect to x, y, and z. Let's find \[\frac{\text{d}}{\text{dx}}\frac{1}{\sqrt{x^2 + y^2 + z^2}}\]. Using the chain rule: \[\frac{\text{d}}{\text{dx}}\frac{1}{\sqrt{x^2 + y^2 + z^2}}= -\frac{1}{2}(x^2 + y^2 + z^2)^{-\frac{3}{2}}(2x) = -x(x^2 + y^2 + z^2)^{-\frac{3}{2}}\].

Sum Second Partial Derivatives

Now, calculate the second partial derivatives: \[\frac{\text{d}^2}{\text{d}x^2} \frac{1}{\sqrt{x^2 + y^2 + z^2}}= \frac{\text{d}}{\text{dx}}\big(-x(x^2 + y^2 + z^2)^{-\frac{3}{2}}\big)\]. Simplify this to get the second derivative.

Compute Second Derivatives for y and z

Repeat the previous step for y and z: \[\frac{\text{d}}{\text{dy}} \frac{1}{\sqrt{x^2 + y^2 + z^2}} = -y(x^2 + y^2 + z^2)^{-\frac{3}{2}}\] and \[\frac{\text{d}}{\text{dz}} \frac{1}{\sqrt{x^2 + y^2 + z^2}} = -z(x^2 + y^2 + z^2)^{-\frac{3}{2}}\]. Then calculate the respective second derivatives.

Sum All Second Derivatives

Sum all the second derivatives obtained from x, y, and z components to get the Laplacian: \[abla^2\big(\frac{1}{\text{√}{x^2 + y^2 + z^2}}\big) = \text{second derivative of x} + \text{second derivative of y} + \text{second derivative of z} \].

Simplify Laplacian

Simplify the expression obtained to achieve the final answer for the Laplacian.

Key concepts

scalar field

A scalar field is a function that assigns a single scalar value to every point in a space. Imagine a 3D weather map where each point represents a temperature value; this is a type of scalar field. In mathematics and physics, scalar fields are often used to represent distributions of quantities like temperature, pressure, or gravitational potentials. The given exercise deals with the scalar field \(\frac{1}{\text{√}{x^2 + y^2 + z^2}}\) which can be thought of as a potential field centered at the origin of a 3D space.

partial derivatives

Partial derivatives measure how a function changes as one of its input variables changes, while keeping the other input variables constant. For a function of several variables, like \(\frac{1}{\text{√}{x^2 + y^2 + z^2}}\), the partial derivative with respect to x, written as \( \frac{\text{∂}}{\text{∂x}} \), describes how the function value changes when only x varies.

In solving the given exercise, we calculate the first-order partial derivatives with respect to x, y, and z. For example, using the chain rule, we find \( \frac{\text{∂}}{\text{∂x}} \frac{1}{\text{√}{x^2 + y^2 + z^2}} = -x(x^2 + y^2 + z^2)^{-\frac{3}{2}} \). Calculating partial derivatives is an essential step in finding the Laplacian of a scalar field.

chain rule

The chain rule is a fundamental theorem used to calculate the derivative of composite functions. It is particularly useful when dealing with functions that have been composed or nested.

For example, in the exercise, we need to differentiate \(\frac{1}{\text{√}{x^2 + y^2 + z^2}} \), which is a composite function. By applying the chain rule, we can break down the differentiation process into simpler parts.

As shown: \(\frac{\text{d}}{\text{dx}} \frac{1}{\text{√}{x^2 + y^2 + z^2}}= -x(x^2 + y^2 + z^2)^{-\frac{3}{2}} \). By using the chain rule, we efficiently find the needed partial derivatives that help in computing the Laplacian.

Laplacian operator

The Laplacian operator, denoted by \( abla^2 \), is a differential operator that combines second-order partial derivatives. It is a measure of the rate at which a scalar field changes at a point, concerning its neighboring values. For a function \(f(x, y, z)\), the Laplacian is given by \( abla^2f = \frac{\text{∂}^2 f}{\text{∂x}^2} + \frac{\text{∂}^2 f}{\text{∂y}^2} + \frac{\text{∂}^2 f}{\text{∂z}^2}\).

In our exercise, after calculating first and second partial derivatives for each variable, we sum them to get the final Laplacian. Calculating the Laplacian of \( \frac{1}{\text{√}{x^2 + y^2 + z^2}} \) involves doing this for each variable x, y, and z, then summing all the second-order partial derivatives.