Question

For \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},\) evaluate $$\nabla \cdot\left(\frac{\mathbf{r}}{|\mathbf{r}|}\right)$$

Short answer

\[abla \cdot\left(\frac{\mathbf{r}}{|\mathbf{r}|}\right) = 0\]

Step-by-step solution

- Express the radial vector

First, write the vector \(\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\). Its magnitude is given by \(|\mathbf{r}| = \sqrt{x^{2} + y^{2} + z^{2}}\).

- Construct the function to evaluate

We need to evaluate \( abla \cdot \left( \frac{{\mathbf{r}}}{{|\mathbf{r}|}}\right) \), which can be written as \( \mathbf{r} \cdot \abla \left( \frac{{1}}{{|\mathbf{r}|}}\right) + \left( \frac{{1}}{{|\mathbf{r}|}}\right) \abla \cdot \mathbf{r}\).

- Calculate the gradient

The gradient of \( \frac{{1}}{{|\mathbf{r}|}} \) can be obtained: \(abla\left(\frac{1}{|\mathbf{r}|}\right)=\frac{\partial}{\partial x}\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)\mathbf{i}+\frac{\partial}{\partial y}\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)\mathbf{j}+\frac{\partial}{\partial z}\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)\mathbf{k} \,\) which simplifies to \(\,-\frac{x}{(x^2+y^2+z^2)^{3/2}}\mathbf{i}-\frac{y}{(x^2+y^2+z^2)^{3/2}}\mathbf{j}-\frac{z}{(x^2+y^2+z^2)^{3/2}}\mathbf{k}\).

- Calculate the divergence

The divergence of \( \mathbf{r} \) is given by \(abla \cdot \mathbf{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3.\)

- Combine the results

Combine the gradient and divergence results:\(abla \cdot \left(\frac{\mathbf{r}}{|\mathbf{r}|}\right) =\mathbf{r} \cdotabla \(\frac{1}{|\mathbf{r}|}\) + \(\frac{1}{|\mathbf{r}|}\)abla \cdot\mathbf{r} =\mathbf{r} \cdot (-\frac{\mathbf{r}}{(x^2+y^2+z^2)^{3/2}}) + \(\frac{3}{|\mathbf{r}|}\)\) \(abla \cdot \left(\frac{\mathbf{r}}{|\mathbf{r}|}\right) = -\frac{x^2+y^2+z^2}{(x^2+y^2+z^2)^{3/2}} + \(\frac{3}{|\mathbf{r}|}\) = -\frac{1}{(x^2+y^2+z^2)^{1/2}} + \(\frac{3}{|\mathbf{r}|}\) = 0\)\.

Key concepts

Gradient of Vector Fields

In vector calculus, the gradient of a vector field plays a crucial role. It gives us the rate of change of a scalar field in space. For example, if you have a temperature field, the gradient tells you in which direction the temperature increases most rapidly and how steep the increase is.
In our exercise, to find the gradient of the function \(\frac{1}{|\textbf{r}|}\), we need to take the partial derivatives with respect to x, y, and z. The formula for the gradient is \(abla\frac{1}{|\textbf{r}|} = -\frac{x}{(x^2 + y^2 + z^2)^{3/2}}\textbf{i} - \frac{y}{(x^2 + y^2 + z^2)^{3/2}}\textbf{j} - \frac{z}{(x^2 + y^2 + z^2)^{3/2}}\textbf{k}\).
This expression shows how the magnitude of the vector field inversely influences its rate of change.

Divergence of Vector Fields

Divergence measures how much a vector field spreads out from a point, like a fluid flowing away from or towards a source. Mathematically, the divergence of a vector field \(\textbf{F} = P\textbf{i} + Q\textbf{j} + R\textbf{k}\) is given by \(abla \bullet \textbf{F} = \frac{\text{d}P}{\text{d}x} + \frac{\text{d}Q}{\text{d}y} + \frac{\text{d}R}{\text{d}z}\).
In our case, the divergence of \(\textbf{r} = x\textbf{i} + y\textbf{j} + z\textbf{k}\) is straightforward: \(abla \bullet \textbf{r} = \frac{\text{d}x}{\text{d}x} + \frac{\text{d}y}{\text{d}y} + \frac{\text{d}z}{\text{d}z} = 3\).
This result appears from the fact that each component contributes a value of 1, adding up to 3.

Radial Vector

A radial vector points from the origin to any point in space. For a point \((x, y, z)\), the radial vector is \(\textbf{r} = x\textbf{i} + y\textbf{j} + z\textbf{k}\). Its magnitude is the distance from the origin, calculated as \(|\textbf{r}| = \sqrt{x^2 + y^2 + z^2}\).
Using radial vectors simplifies many vector calculus problems, as they naturally express positions in a symmetric and straightforward form. For our exercise, expressing other vectors and their magnitudes in terms of the radial vector streamlines our calculations.

Vector Calculus

Vector calculus is a branch of mathematics focusing on vector fields and operations like differentiation and integration. It is essential in physics and engineering for describing physical phenomena in a spatial domain.
The key concepts in vector calculus include:

• Gradient: Provides the direction and rate of the fastest increase of a scalar field.
• Divergence: Measures the flux density of a vector field from a point.
• Curl: Describes the rotation or swirling around a point.

These operations enable us to analyze and solve complex problems involving vector quantities in fields such as electromagnetism, fluid dynamics, and more.