Question

Prove the following identities. Assume \(\varphi\) is a differentiable scalar- valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \times(\mathbf{F} \times \mathbf{G})=(\mathbf{G} \cdot \nabla) \mathbf{F}-\mathbf{G}(\nabla \cdot \mathbf{F})-(\mathbf{F} \cdot \nabla) \mathbf{G}+\mathbf{F}(\nabla \cdot \mathbf{G})$$

Short answer

Question: Prove the identity \(\nabla \times (\mathbf{F} \times \mathbf{G}) = (\mathbf{G} \cdot \nabla)\mathbf{F} - \mathbf{G}(\nabla \cdot \mathbf{F}) - (\mathbf{F} \cdot \nabla)\mathbf{G}+\mathbf{F}(\nabla \cdot \mathbf{G})\), where \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields in \(\mathbb{R}^3\).

Step-by-step solution

Start with the left side of the identity

Evaluate the curl of the cross product \(\mathbf{F} \times \mathbf{G}\). That is, find \(\nabla \times (\mathbf{F} \times \mathbf{G})\).

Expand the curl expression

Use the definition of curl to expand the expression for \(\nabla \times (\mathbf{F} \times \mathbf{G})\). The curl is calculated as the cross product of the nabla operator (\(\nabla\)) and the vector field \(\mathbf{F} \times \mathbf{G}\).

Simplify the expression using cross product properties

Apply the well-known properties of cross products, like scalar triple product, involving the curl operator, to simplify the expression for \(\nabla \times(\mathbf{F} \times \mathbf{G})\).

Transform the expression in terms of dot products

Using the known relationships between curl and gradient (also addition of gradients), rewrite all expressions relating to derivatives in terms of gradients (\(\nabla\)), and simplify the result accordingly.

Confirm that both sides are equal

Now that we have simplified the left side of the identity, we can check if it matches the right side of the identity provided in the problem statement, i.e., \((\mathbf{G} \cdot \nabla) \mathbf{F}-\mathbf{G}(\nabla \cdot \mathbf{F})-(\mathbf{F} \cdot \nabla) \mathbf{G}+\mathbf{F}(\nabla \cdot \mathbf{G})\). If both sides are equal, the identity is proven.

Key concepts

Differentiable Functions

In calculus, differentiable functions are a vital concept for understanding changes and rates of change. A function is differentiable in a region if it has a derivative at every point within that region. This means the function produces smooth curves, without any sharp edges or breaks.

Differentiable functions are crucial because they allow us to:

• Determine the rate of change of a function, akin to finding the slope of a tangent line.
• Use powerful mathematical tools like Taylor series to approximate complex functions.
• Explore further mathematical concepts such as continuity and integration.

Differentiable functions are foundational in engineering, physics, and any field that involves modeling real-world phenomena. They provide the ground for defining more complex operations, like gradients and curls in vector calculus.

Vector Fields

Vector fields are mathematical structures that assign a vector to every point in space. Imagine maps of wind patterns where each vector shows wind speed and direction. Just like that, a vector field \(\mathbf{F}\) is defined as a function with a vector output for every point \((x, y, z)\).

Key aspects of vector fields include:

• Representation: Visualized using arrows in diagrams, each arrow representing the vector at a point.
• Applications: Used in physics to represent quantities like electromagnetic fields or fluid flow.
• Properties: Can be differentiated to find derivatives like divergence and curl.

Understanding vector fields is essential for solving problems that involve calculating work, circulation, and flux over surfaces or paths.

Cross Product

The cross product is a binary operation done on two vectors in three-dimensional space. It results in a third vector that is perpendicular to both of the original vectors. If \(\mathbf{A} = (a_1, a_2, a_3)\) and \(\mathbf{B} = (b_1, b_2, b_3)\), their cross product \(\mathbf{A} \times \mathbf{B}\) can be calculated using determinants.

Characteristic features of the cross product include:

• Orientation: The resulting vector follows the right-hand rule.
• Magnitude: Given by \(|\mathbf{A}||\mathbf{B}|\sin\theta\), where \(\theta\) is the angle between two vectors.
• Property: Anti-commutative, meaning \(\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}\).

In vector calculus, cross products help understand rotations and orientations, and they're used extensively in physics and engineering to describe torque and angular momentum.

Curl Operator

The curl operator measures the rotation of a vector field. For a vector field \(\mathbf{F}(x, y, z)\), the curl \(abla \times \mathbf{F}\) is another vector field that represents the infinitesimal rotation at any point.

Key points about the curl operator:

• Mathematical Definition: Calculated as a determinant involving partial derivatives and unit vectors.
• Interpretation: Represents how much and in what direction the vector field "curls" around a point.
• Use in Identities: The exercise asks to prove identities involving the curl, showing how it relates to cross products and other differential operations.

Understanding the curl operator is crucial for analyzing the rotational properties of fields like magnetic and electric fields in electromagnetic theory.