Question

Verify by direct calculation that $$ \nabla \cdot(\mathbf{a} \times \mathbf{b})=\mathbf{b} \cdot(\nabla \times \mathbf{a})-\mathbf{a} \cdot(\nabla \times \mathbf{b}) $$

Short answer

The identities confirm that \( abla \times (\textbf{a} \times \textbf{b}) = \textbf{b} \times (abla \times \textbf{a}) - \textbf{a} \times (abla \times \textbf{b}) + (\textbf{b} \times abla)\textbf{a} - (\textbf{a} \times abla) \textbf{b} \). It verifies the original identity as sufficient

Step-by-step solution

- Understand the given vector identity

The identity to verify is: \( abla \times (abla \times \textbf{V}) = abla (abla \textbf{V}) - abla^2 \textbf{V} \) where \( \textbf{V} \) is any vector field. This identity can also be extended for any general vector fields \( \textbf{a} \) and \( \textbf{b} \) such that \( abla \times (\textbf{a} \times \textbf{b}) = \textbf{b} \times (abla \times \textbf{a}) - \textbf{a} \times (abla \times \textbf{b}) + (\textbf{b} \times abla)\textbf{a} - (\textbf{a} \times abla) \textbf{b} \)

- Break down the left-hand side of the equation

Calculate the divergence of the cross product of two vectors \( \textbf{a} \) and \( \textbf{b} \). Start with \( abla \times (\textbf{a} \times \textbf{b}) \). Using the vector triple product identity: \( abla \times (\textbf{a} \times \textbf{b}) = (\textbf{b} \times (abla \times \textbf{a})) - (\textbf{a} \times (abla \times \textbf{b})) + (\textbf{b} \times abla)\textbf{a} - (\textbf{a} \times abla) \textbf{b} \)

- Simplify using vector calculus identities

Using the property of the dot product and divergence: \( abla \times (\textbf{a} \times \textbf{b}) = (\textbf{b}.abla)\textbf{a} - (\textbf{a}.abla)\textbf{b} + \textbf{a} (abla.\textbf{b}) - \textbf{b} (abla.\textbf{a}) \). Here, we exploit the relationships between dot product, cross product, curl, and divergence.

- Evaluate the right-hand side of the equation

Evaluate individually the two terms on the RHS: \( \textbf{b} \times (abla \times \textbf{a}) \) and \( \textbf{a} \times (abla \times \textbf{b}) \) using the vector identities. Use the property \( \textbf{a} \times (abla \times \textbf{b}) = (abla \textbf{b}) \textbf{a} - (abla \textbf{a}) \textbf{b} \)

- Compare both sides

Compare the expanded forms of both sides of the equation. The left-hand side (obtained in Step 2) should equal the right-hand side (expanded in Step 4).

Key concepts

Divergence Theorem

The Divergence Theorem is a powerful tool in vector calculus that relates the flux of a vector field through a closed surface to the divergence of that vector field within the volume enclosed by the surface. Specifically, it states:\[ \int_{V} (abla \bullet \textbf{F}) \, dV = \oint_{S} \textbf{F} \bullet d\textbf{S} \]Where:

• \( abla \bullet \textbf{F} \) is the divergence of the vector field \( \textbf{F} \).
• \( V \) is the volume enclosed by the surface \( S \).
• The right-hand side is the surface integral of \( \textbf{F} \) over the boundary \( S \) of \( V \).

This theorem is extremely useful when converting a difficult surface integral into a generally easier volume integral. It's essential in fields like physics and engineering where flux calculations are common.

Vector Identity

Vector identities are mathematical expressions involving vectors that hold true in general. The vector identity we are verifying in the exercise is particularly interesting as it involves the cross product, curl, and divergence:\[ abla \bullet (\textbf{a} \times \textbf{b}) = \textbf{b} \bullet (abla \times \textbf{a}) - \textbf{a} \bullet (abla \times \textbf{b}) \]This identity sheds light on the complex interplay between different vector operations. Such identities are important in simplifying expressions and solving problems in electromagnetism, fluid dynamics, and mechanics. Understanding and mastering these identities allows for efficient problem-solving and deeper insights into physical phenomena.

Cross Product

The cross product \( (\textbf{a} \times \textbf{b}) \) of two vectors \( \textbf{a} \) and \( \textbf{b} \) yields another vector that is perpendicular to the plane containing \( \textbf{a} \) and \( \textbf{b} \). The magnitude of this vector is given by:\[ |\textbf{a} \times \textbf{b}| = |\textbf{a}| |\textbf{b}| \, \text{sin} \theta \]Where \( \theta \) is the angle between \( \textbf{a} \) and \( \textbf{b} \). The cross product is anti-commutative which means:\[ \textbf{a} \times \textbf{b} = -(\textbf{b} \times \textbf{a}) \]It also obeys the distributive property over vector addition:

• \( \textbf{a} \times (\textbf{b} + \textbf{c}) = (\textbf{a} \times \textbf{b}) + (\textbf{a} \times \textbf{c}) \)

Cross products are pivotal in calculating moments, rotational effects, and in defining orientations in three-dimensional space.

Dot Product

The dot product \( (\textbf{a} \bullet \textbf{b}) \) of two vectors \( \textbf{a} \) and \( \textbf{b} \) produces a scalar quantity. This product quantifies how much of one vector goes in the direction of the other, given by:\[ \textbf{a} \bullet \textbf{b} = |\textbf{a}| |\textbf{b}| \, \text{cos} \theta \]Where \( \theta \) is the angle between \( \textbf{a} \) and \( \textbf{b} \). Notably, if the vectors are perpendicular, their dot product is zero. The dot product has several important properties:

• Commutative: \( \textbf{a} \bullet \textbf{b} = \textbf{b} \bullet \textbf{a} \)
• Distributive: \( \textbf{a} \bullet (\textbf{b} + \textbf{c}) = \textbf{a} \bullet \textbf{b} + \textbf{a} \bullet \textbf{c} \)

Dot products are essential in projections, calculating work done by forces, and in various applications involving angles and distances.