Problem 37 - Calculus: Early Transcendental Functions

Question

If \(\mathbf{F}\) and \(\mathbf{G}\) are vector fields, prove that $$ \nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}) $$

Step-by-Step Solution

Answer

The given identity is proven by using the BAC-CAB rule, which yields \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{F} \cdot(\nabla \times \mathbf{G}) - \mathbf{G} \cdot(\nabla \times \mathbf{F})\). This can be rearranged to \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot(\nabla \times \mathbf{F}) - \mathbf{F} \cdot(\nabla \times \mathbf{G})\), thereby proving the given identity.

Step 1: Write down known Vectors and Operators

We have two vector fields \(\mathbf{F}\) and \(\mathbf{G}\), and we have to work with the divergence operator \(\nabla \cdot\) and the curl operator \(\nabla \times\). We can generally express this as \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot(\nabla \times \mathbf{F}) - \mathbf{F} \cdot(\nabla \times \mathbf{G})\)

Step 2: Applying the BAC-CAB rule

The BAC-CAB rule is a mnemonic for a result in vector calculus. The result is expressed in the following way: For three vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{A} \times \mathbf{C}) - \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})\). In this problem, we can identify the vectors as \(\mathbf{A} = \nabla\), \(\mathbf{B} = \mathbf{F}\), and \(\mathbf{C} = \mathbf{G}\). So, we get that \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{F} \cdot(\nabla \times \mathbf{G}) - \mathbf{G} \cdot(\nabla \times \mathbf{F})\)

Step 3: Rearranging

From step 2 we have the following equation \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{F} \cdot(\nabla \times \mathbf{G}) - \mathbf{G} \cdot(\nabla \times \mathbf{F})\). It can be rearranged to the following \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot(\nabla \times \mathbf{F}) - \mathbf{F} \cdot(\nabla \times \mathbf{G})\), proving the required identity.

Key Concepts

Divergence Operator

In vector calculus, the divergence operator is a vital concept often denoted as \( abla \cdot \). It is used to measure a vector field's tendency to originate from or converge at a point. In essence, divergence captures the "source" or "sink" magnitude of a vector field at a given point. It tells us how much a vector field like wind or fluid flow is expanding or compressing at a particular location.

The divergence of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is calculated using the formula:

• \( abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \)

This equation shows us how to compute divergence in a three-dimensional Cartesian coordinate system, but the concept is that in any dimension, divergence sums the rates of change of each vector component in its respective direction.

Divergence is particularly valuable in physics, especially in fluid dynamics and electromagnetism, where it's used to describe the behavior of fields. It's integral to understanding phenomena like

• Expansion of gases
• Current flow in electrical circuits
• Behavior of magnetic and electric fields

Curl Operator

The curl operator is another fundamental tool in vector calculus, often symbolized by \( abla \times \). It helps us understand the rotational attributes of a vector field. Whereas divergence focuses on points spreading out or converging, curl analyzes how a vector field loops or swirls around a point.

The curl of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is given as:

• \( abla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right) \)

Curl results in a vector that shows the axis of rotation and the magnitude of rotation. Think of it like analyzing the swirls in a whirlpool or the eddy currents in water.

This concept is crucial in various physical fields, particularly where rotation and circulation are significant, such as:

• Fluid motion
• Electromagnetic fields
• Mechanics and dynamic systems

Understanding curl helps in visualizing how vector fields will behave under certain conditions, giving insights into rotational forces and impacts.

BAC-CAB Rule

The BAC-CAB rule is a handy mnemonic for remembering a specific identity in vector calculus concerning the cross product, often involving three vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \). This rule is particularly useful for simplifying expressions that involve multiple cross products by using known relationships.

Mathematically, this rule states:

• \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C}) \mathbf{B} - (\mathbf{A} \cdot \mathbf{B}) \mathbf{C} \)

In the given problem, it is used to prove an identity related to the divergence and curl of the cross product of two vector fields. By applying this rule:

• Vector \( \mathbf{A} \) corresponds to \( abla \)
• Vector \( \mathbf{B} \) corresponds to \( \mathbf{F} \)
• Vector \( \mathbf{C} \) corresponds to \( \mathbf{G} \)

The BAC-CAB rule allows for rearranging and simplifying complex components of vector calculus. It is invaluable for resolving calculations related to cross products, particularly in physics and engineering.