91Ó°ÊÓ

\(21-27=\) Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(\mathbf{F},\) \(\mathbf{G}\) are vector fields, then \(f \mathbf{F}, \mathbf{F} \cdot \mathbf{G},\) and \(\mathbf{F} \times \mathbf{G}\) are defined by \((f \mathbf{F})(x, y, z)=f(x, y, z) \mathbf{F}(x, y, z)\) \((\mathbf{F} \cdot \mathbf{G})(x, y, z)=\mathbf{F}(x, y, z) \cdot \mathbf{G}(x, y, z)\) \((\mathbf{F} \times \mathbf{G})(x, y, z)=\mathbf{F}(x, y, z) \times \mathbf{G}(x, y, z)\) $$\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl} \mathbf{F}-\mathbf{F} \cdot \operatorname{curl} \mathbf{G}$$

Step by step solution

01

Define Divergence and Curl

Understand that the divergence of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is given by \( \operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \). The curl of a vector field \( \mathbf{F} \) is defined as \( \operatorname{curl} \mathbf{F} = abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \).

02

Express Cross Product

Knowing \( \mathbf{F} \times \mathbf{G} \), where \( \mathbf{F} = (F_1, F_2, F_3) \) and \( \mathbf{G} = (G_1, G_2, G_3) \), express it as \( \mathbf{F} \times \mathbf{G} = \left( F_2G_3 - F_3G_2, F_3G_1 - F_1G_3, F_1G_2 - F_2G_1 \right) \).

03

Calculate Divergence of Cross Product

Take the divergence of \( \mathbf{F} \times \mathbf{G} \). It involves computing \[ \operatorname{div}(\mathbf{F} \times \mathbf{G}) = \frac{\partial}{\partial x}(F_2G_3 - F_3G_2) + \frac{\partial}{\partial y}(F_3G_1 - F_1G_3) + \frac{\partial}{\partial z}(F_1G_2 - F_2G_1) \].

04

Apply Curl

Calculate \( \operatorname{curl} \mathbf{F} \) and \( \operatorname{curl} \mathbf{G} \) using their definitions. Substitute these curls into the formula \( \mathbf{G} \cdot \operatorname{curl} \mathbf{F} - \mathbf{F} \cdot \operatorname{curl} \mathbf{G} \).

05

Verify Identity

Show that the expressions from the divergence of the cross product and the dot products with the curls are equal. Compare both expansions and confirm the equality as stated. This involves careful term-by-term comparison of the expanded expressions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divergence

Divergence is a fundamental concept in vector calculus that measures how much a vector field spreads out from a given point. Imagine water flowing from a sprinkler - the way it spreads out or converges back gives us an intuitive sense of divergence.
For a vector field \( \mathbf{F} = (F_1, F_2, F_3) \), its divergence is calculated using the formula:

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

A positive divergence at a point implies the field is acting as a source there, while a negative divergence suggests a sink.
This concept is heavily used in physics, especially in analyzing fluid dynamics, where it helps describe the flow and conservation of mass.

Curl

Curl represents the rotation or swirling motion present within a vector field. Imagine a swirling tornado or eddies in a stream; these are natural occurrences of what curl describes.
The curl of a vector field \( \mathbf{F} \) is defined using the cross product of the del operator (\( abla \)) and \( \mathbf{F} \):

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

The curl gives us a vector that points in the direction of the axis around which the field rotates, with its magnitude indicating the strength of rotation.
In electromagnetic theory, for example, the curl is crucial for understanding magnetic fields and their behavior.

Cross Product

The cross product is a mathematical operation performed on two vectors in three-dimensional space. It results in a third vector that is perpendicular to the plane formed by the initial pair. Think of it as a way to "twist" one vector around another.
Given vectors \( \mathbf{F} = (F_1, F_2, F_3) \) and \( \mathbf{G} = (G_1, G_2, G_3) \), their cross product is calculated as:

• \( \mathbf{F} \times \mathbf{G} = (F_2G_3 - F_3G_2, F_3G_1 - F_1G_3, F_1G_2 - F_2G_1) \)

This results in a vector orthogonal to both \( \mathbf{F} \) and \( \mathbf{G} \).
The cross product is used extensively in physics and engineering, such as in determining torque, where force and lever arm directions are involved.