Question

Prove that for a real number \(p,\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}}=\frac{3-p}{|\mathbf{r}|^{p}}\)

Short answer

Answer: The divergence of the vector field is \(\frac{3-p}{|\mathbf{r}|^{p}}\).

Step-by-step solution

Given vector field

We are given a vector field \(\mathbf{r}=\langle x, y, z\rangle,\) and we are supposed to find the divergence of \(\frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}}\). First, let's calculate \(|\mathbf{r}|\). \(|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}\) Now let's rewrite the given vector field as follows: \(\frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}} = \langle \frac{x}{|\mathbf{r}|^p}, \frac{y}{|\mathbf{r}|^p}, \frac{z}{|\mathbf{r}|^p}\rangle\)

Calculating the divergence

Now we must compute the divergence \(\nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}}\), which for Cartesian coordinates is given by: \(\nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}} = \frac{\partial}{\partial x}\left(\frac{x}{|\mathbf{r}|^p}\right) + \frac{\partial}{\partial y}\left(\frac{y}{|\mathbf{r}|^p}\right) + \frac{\partial}{\partial z}\left(\frac{z}{|\mathbf{r}|^p}\right)\)

Computing partial derivatives

Now we will compute each partial derivative. We'll start with \(\frac{\partial}{\partial x}\left(\frac{x}{|\mathbf{r}|^p}\right)\): To make the calculation easier, let's denote \(R = |\mathbf{r}| = \sqrt{x^2+y^2+z^2}\), so we need to compute the derivative of \(\frac{x}{R^p}\). Using the chain rule: \(\frac{\partial}{\partial x}\left(\frac{x}{R^p}\right) = \frac{1}{R^p} - px\frac{\partial}{\partial x}\left(\frac{1}{R^{p-1}}\right)\) Applying the chain rule again: \(\frac{\partial}{\partial x}\left(\frac{1}{R^{p-1}}\right) = -(p-1)\frac{x}{R^{p+1}}\) So we have: \(\frac{\partial}{\partial x}\left(\frac{x}{R^p}\right) = \frac{1}{R^p} - px\left(-(p-1)\frac{x}{R^{p+1}}\right) = \frac{1}{R^p} + (p-1)\frac{x^2}{R^{p+1}}\) Similarly, we can compute the other two partial derivatives, obtaining: \(\frac{\partial}{\partial y}\left(\frac{y}{R^p}\right) = \frac{1}{R^p} + (p-1)\frac{y^2}{R^{p+1}}\) \(\frac{\partial}{\partial z}\left(\frac{z}{R^p}\right) = \frac{1}{R^p} + (p-1)\frac{z^2}{R^{p+1}}\)

Summing the partial derivatives

Now let's sum up all the partial derivatives to get the divergence: \(\nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}} = \frac{1}{R^p} + (p-1)\frac{x^2}{R^{p+1}} + \frac{1}{R^p} + (p-1)\frac{y^2}{R^{p+1}} + \frac{1}{R^p} + (p-1)\frac{z^2}{R^{p+1}}\) Grouping the terms, we get: \(\nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}} = \frac{3-p}{R^p} + (p-1)\frac{x^2+y^2+z^2}{R^{p+1}}\) But we know that \(x^2+y^2+z^2 = R^2\), so we have: \(\nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}} = \frac{3-p}{R^p} + (p-1)\frac{R^2}{R^{p+1}} = \frac{3-p}{R^p}\) Thus, we have proven that the divergence of the given vector field is \(\frac{3-p}{|\mathbf{r}|^{p}}\), as required.

Key concepts

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiation and integration of vector fields. It's where multiple variables and vectors come together and where one can use operations such as gradient, divergence, and curl. A vector field assigns a vector to every point in a subset of space, showcasing directions and magnitudes, such as the flow of a fluid or the strength of a magnetic field.

In our exercise, we dealt with the concept of divergence, which measures the magnitude of a source or sink at a given point in a vector field. Essentially, it tells us if something is emanating out of or converging into a point. For a vector field \( \mathbf{r}=\langle x, y, z\rangle \), divergence captures how the vector field spreads out from or converges towards each point.

Partial Derivatives

Partial derivatives come into play when a function depends on multiple variables. They measure the change in the function as you vary one of its variables, keeping the others constant. In vector calculus, partial derivatives are used to compute the rate of change of a component of the vector field relative to one axis.

For instance, the calculation of the divergence of our vector field \( \frac{\langle x, y, z\rangle}{|\mathbf{r}|^p} \) required us to find the partial derivatives with respect to \(x\), \(y\), and \(z\). These derivatives reveal how slight changes in the coordinates can influence the entire field at a specific point.

Chain Rule

The chain rule is an essential tool in calculus for finding the derivative of a composite function. When functions are entangled together, the chain rule enables us to differentiate them in steps. In the context of our problem, we see the chain rule at work when differentiating the components of the vector field.

During the calculation of \( \frac{\partial}{\partial x}\left(\frac{x}{R^p}\right) \) where \(R = |\mathbf{r}| \), we applied the chain rule twice. It allowed us to handle the more complex expressions involving the partial derivatives with respect to \(x\), \(y\), and \(z\) one step at a time, teasing apart the layers of the function composition to reach the final derivative.

Gradient Operator

The gradient operator, denoted by \(abla\), is a vector operator that represents the collection of partial derivatives with respect to all spatial variables. It can be used to calculate the gradient of a scalar field, the divergence of a vector field, or the curl of a vector field.

When applying the gradient operator in computing the divergence, as in our exercise, we take the dot product of \(abla\) with the vector field. Here, \(abla \) acts as a ‘guide’ that, when dotted with the vector field, quantifies the spread of the field's vectors at each point. Thus, \(abla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^p}\) challenges us to find out how the components of the vector change in sync with one another as we move through space.