Question

For \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) and \(r=|\mathbf{r}|\), simplify the following. a. \(\nabla \times(\mathbf{k} \times \mathbf{r})\). b. \(\nabla \cdot\left(\frac{\mathrm{r}}{r}\right)\). c. \(\nabla \times\left(\frac{\mathbf{r}}{r}\right)\). d. \(\nabla \cdot\left(\frac{\mathrm{r}}{r^{3}}\right)\). e. \(\nabla \times\left(\frac{\mathrm{r}}{r^{3}}\right)\).

Short answer

The answers for the exercise are: (a) \( \mathbf{i} \), (b) 3, (c) 0, (d) 0, and (e) 0.

Step-by-step solution

Solution for (a)

To get the result of \(\nabla \times(\mathbf{k} \times \mathbf{r})\), notice that \( \mathbf{k} \times \mathbf{r} = - y \mathbf{i} + x \mathbf{j} \). Now apply the curl operator \( \nabla \times \) to get the result as \( (0 - (-1)) \mathbf{i} + ((0 - 0) \mathbf{j} + (0 - 0) \mathbf{k} = \mathbf{i} \).

Solution for (b)

For \(\nabla \cdot \left( \frac{\mathbf{r}}{r} \right)\), notice that \( \frac{\mathbf{r}}{r} \) is a unit vector. Now apply the divergence operator (\( \nabla \cdot \)) on it, thus \( \nabla \cdot ( \frac{\mathbf{r}}{r} ) = 3 \).

Solution for (c)

For \(\nabla \times \left( \frac{\mathbf{r}}{r} \right)\), notice that \( \frac{\mathbf{r}}{r} \) is a unit vector. Considering \(\nabla \times \) is a curl operator, it will yield 0 when applied to an uniform vector field, thus \(\nabla \times \left( \frac{\mathbf{r}}{r} \right) = 0\).

Solution for (d)

In \(\nabla \cdot\left( \frac{\mathrm{r}}{r^{3}} \right)\), remember \(\nabla \cdot \) is the divergence operator and \( \frac{r}{r^{3}} \) simplifies to \( \frac{1}{r^{2}} \). Thus, the divergence operator applied to \( \frac{1}{r^{2}} \) yields 0.

Solution for (e)

Just like in previous steps, for \(\nabla \times \left( \frac{\mathbf{r}}{r^{3}} \right)\), considering that \(\nabla \times\) is the curl operator and it applied to \( \frac{\mathbf{r}}{r^{3}} \), we will also get the result as 0.

Key concepts

Curl Operator

The curl operator is a key concept in vector calculus used to measure the rotation of a vector field in three-dimensional space. To understand the curl, consider a vector field that assigns a vector to each point in space. The curl at any point measures how much the field tends to rotate around that point. The formula for the curl of a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is:\[abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}\]

• If the curl is zero in a region, the field is said to be irrotational there.
• Calculating the curl is useful in fluid dynamics and electromagnetic fields.

In the original exercise, you see how the curl operator is applied. For example, applying the curl to a cross product of vectors can yield a straightforward result, such as in the solution for part (a).

Divergence

Divergence is another fundamental operator in vector calculus that measures the magnitude of a source or sink at a given point in a vector field. Simply put, divergence quantifies how much a vector field is spreading out or drawing in at a particular point. If you imagine a field of arrows depicting airflow, divergence tells us if the air is compressing or expanding at any point.The mathematical expression for the divergence of a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is:\[abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]

• A positive divergence indicates a source, where field vectors are moving away.
• A negative divergence indicates a sink, where field vectors converge.

In example (b) and (d) of the exercise, you encounter divergence by observing how a vector field like \(\frac{\mathbf{r}}{r}\) behaves. Understanding how to calculate divergence helps in modeling and solving physical problems like fluid flow and heat transfer.

Unit Vector

A unit vector is a vector with a magnitude of one. It serves as a directional component, often used to indicate direction without regard to length. Unit vectors are essential in breaking down vectors into their components because each vector can be expressed as a combination of unit vectors.To find a unit vector in the direction of a given vector \(\mathbf{a}\), you divide \(\mathbf{a}\) by its magnitude \(|\mathbf{a}|\):\[\mathbf{u} = \frac{\mathbf{a}}{|\mathbf{a}|}\]

• Unit vectors are commonly represented with the hat symbol, such as \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), denoting unit vectors in the x, y, and z directions respectively.
• They are useful in expressing vector quantities and simplifying complex equations.

In the solutions provided, identifying \(\frac{\mathbf{r}}{r}\) as a unit vector is crucial in understanding the nature of the problems and simplifies calculations, making it easier to apply operators like the curl or divergence.

Gradient

The gradient is a vector that shows the direction and rate of fastest increase of a scalar function. Think of it as the slope of a hill at a given point, indicating which direction you would go to ascend the steepest path.Mathematically, if \(f(x, y, z)\) is a scalar function, the gradient of \(f\) is given by:\[abla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}\]

• The gradient is perpendicular to level surfaces of the function.
• The magnitude of the gradient vector represents the rate of change.

Understanding gradients is useful in optimizing functions and in fields such as thermodynamics and electromagnetics. While the original exercise directly applies curl and divergence, having a grasp on the gradient concept can deepen the comprehension of how vector fields behave.