Problem 265 - Calculus Volume 3

Question

Suppose that \(\nabla \cdot \mathbf{F}=0\) and \(\nabla \cdot \mathbf{G}=0\). Does \(\mathbf{F} \times \mathbf{G}\) necessarily have zero divergence?

Step-by-Step Solution

Answer

No, it does not necessarily have zero divergence.

Step 1: Understand the Concept of Divergence

The divergence of a vector field \( \mathbf{A} \), denoted as \( abla \cdot \mathbf{A} \), measures the rate of flow of a vector field out of an infinitesimal volume in 3-dimensional space. If \( abla \cdot \mathbf{A} = 0 \), the field \( \mathbf{A} \) is said to be divergence-free, meaning it doesn't "expand" or "compress" at any point.

Step 2: Recall the Divergence of a Cross Product

The rule for the divergence of a cross product \( \mathbf{F} \times \mathbf{G} \) is given by the vector calculus identity: \[abla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (abla \times \mathbf{F}) - \mathbf{F} \cdot (abla \times \mathbf{G})\]. This tells us how to compute the divergence of the cross product of two vector fields.

Step 3: Apply Given Conditions

Since both \( abla \cdot \mathbf{F} = 0 \) and \( abla \cdot \mathbf{G} = 0 \), it means that \( \mathbf{F} \) and \( \mathbf{G} \) are divergence-free fields. However, this information alone does not directly tell us about their curl, \( abla \times \mathbf{F} \) and \( abla \times \mathbf{G} \).

Step 4: Analyze the Divergence with Identity

From the vector identity for the divergence of a cross product, examine:\[abla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (abla \times \mathbf{F}) - \mathbf{F} \cdot (abla \times \mathbf{G})\]. There's no general guarantee that \( abla \times \mathbf{F} \) or \( abla \times \mathbf{G} \) are zero. Therefore, unless both curl terms somehow cancel out, the divergence of \( \mathbf{F} \times \mathbf{G} \) may not be zero.

Step 5: Draw the Conclusion

Since there is no specific information about the curls \( abla \times \mathbf{F} \) and \( abla \times \mathbf{G} \), and the expression for \( abla \cdot (\mathbf{F} \times \mathbf{G}) \) depends on them, it cannot be concluded that \( \mathbf{F} \times \mathbf{G} \) has zero divergence in general. Thus, \( \mathbf{F} \times \mathbf{G} \) does not necessarily have zero divergence.

Key Concepts

Cross Product

The cross product is a crucial operation in vector calculus that involves two vector fields. This operation results in a new vector that is perpendicular to both of the original vectors. It's a way to combine vectors in three dimensions that helps us understand how they interact in space.

For vectors \( \mathbf{F} \) and \( \mathbf{G} \), the cross product is formally defined as:\[\mathbf{F} \times \mathbf{G} = \begin{vmatrix}i \mathbf{i} & \mathbf{j} & \mathbf{k} \i F_1 & F_2 & F_3 \i G_1 & G_2 & G_3 i\end{vmatrix}\]Where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors along the x, y, and z axes. This determinant gives us another vector. Let's simplify this explanation:

• The vector \( \mathbf{F} \) has components \( F_1, F_2, F_3 \).
• The vector \( \mathbf{G} \) has components \( G_1, G_2, G_3 \).

A key property of the cross product is that it is zero if the vectors are parallel. This means it is hugely helpful in determining orthogonal relationships in spatial analyses.

Curl of a Vector Field

The curl of a vector field is a vector operator that describes the infinitesimal rotation of the field at every point. In simpler words, it gives us a measure of how much and in which direction a vector field rotates or swirls.

Mathematically, the curl of a vector field \( \mathbf{F} \) is defined by:\[abla \times \mathbf{F} = \begin{vmatrix}i \mathbf{i} & \mathbf{j} & \mathbf{k} \i \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \i F_1 & F_2 & F_3 i\end{vmatrix}\]Here, \( abla \) (del) is the vector differential operator. This operation translates the components of the vector field into a new vector that holds the information about the rotational aspects of the original field.

The curl is particularly useful in physics when dealing with rotational dynamics and electromagnetic fields, where such rotational behavior is often crucial. It's important to know that the curl can be zero even when the divergence is not, and vice versa, indicating different types of behaviors for field flows.

Divergence-free Vector Fields

A divergence-free vector field is one where the vectors do not spread out or come together, meaning the field is incompressible. This property is expressed mathematically when the divergence of the field is zero.

For a vector field \( \mathbf{F} \), having zero divergence means:\[abla \cdot \mathbf{F} = 0\]This condition suggests that what goes into any region of space in the vector field comes out entirely, maintaining an invariant flow.

Divergence-free fields have crucial applications:

• In fluid dynamics, where they represent an incompressible fluid flow.
• In electromagnetism, with magnetic fields, which are also divergence-free.

In the context of considering cross products and their properties, knowing that each vector field \( \mathbf{F} \) and \( \mathbf{G} \) is divergence-free doesn't automatically imply anything about the divergence of their cross product. Understanding the interactions in these fields can help in predicting physical behaviors in complex systems.