Question

Write the following using \(\epsilon_{i j k}\) notation and simplify if possible. a. \(\mathbf{C} \times(\mathbf{A} \times(\mathbf{A} \times \mathbf{C}))\) b. \(\nabla \times(\nabla \times \mathbf{A})\) c. \(\nabla \times \nabla \phi\).

Short answer

The simplified forms of the given expressions in \(\epsilon_{ijk}\) notation are: a. 0, b. \(\nabla(\nabla.\mathbf{A}) - \nabla^2\mathbf{A}\) (cannot be further simplified without knowledge about \(\mathbf{A}\)), c. 0.

Step-by-step solution

Convert \(\mathbf{C} \times(\mathbf{A} \times(\mathbf{A} \times \mathbf{C}))\) into \(\epsilon_{i j k}\) notation

The vector triple cross product can be converted to \(\epsilon_{i j k}\) notation as follows: \(\mathbf{C} \times(\mathbf{A} \times(\mathbf{A} \times \mathbf{C})) = \epsilon_{ijk}C_j(A_l \epsilon_{l m n} A_m C_n) = 0\). The reason this simplifies to zero is because the term \( A_l \epsilon_{l m n} A_m\) is identically zero, as two indices of epsilon cannot be same.

Convert \(\nabla \times(\nabla \times \mathbf{A})\) into \(\epsilon_{i j k}\) notation

This can be re-written using the vector identity \(\nabla \times(\nabla \times \mathbf{A}) = \nabla(\nabla.\mathbf{A}) - \nabla^2\mathbf{A}\), where \(\nabla.\) is the divergence operator, and \(\nabla^2\) is the Laplacian. This can't further be simplified in \(\epsilon_{ijk}\) notation without knowledge about A.

Convert \(\nabla \times \nabla \phi\) into \(\epsilon_{i j k}\) notation

Consider \(\phi\) as a scalar function. Taking both gradient (\(\nabla\)) and curl (\(\nabla \times\)) of a scalar function will yield the zero vector, hence \(\nabla \times \nabla \phi = 0\), regardless of what \(\phi\) is.

Key concepts

Vector Triple Product

The vector triple product involves three vectors in a specific order and often leads to simplifications using algebraic identities. If you have vectors \( \mathbf{A}, \mathbf{B}, \mathbf{C} \), the product \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) \) can be expanded using the vector triple product identity:

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

In the given problem, simplifying \( \mathbf{C} \times (\mathbf{A} \times (\mathbf{A} \times \mathbf{C})) \) requires understanding that any vector \( \mathbf{A} \times (\mathbf{A} \times \mathbf{C}) \) results in terms where identical indices contract to zero in \( \epsilon \) notation.
As a result, what's left becomes zero, showcasing how powerful and elegant these identities can be when dealing with complex vector calculations.

Epsilon Notation

Epsilon notation, symbolized as \( \epsilon_{ijk} \), is a useful tool in vector calculus for simplifying cross products.
It's a fully antisymmetric tensor, meaning it changes sign when any two indices are swapped and it becomes zero if any two indices are the same. For example:

• \( \epsilon_{123} = 1 \)
• If \( i = j \), then \( \epsilon_{ijk} = 0 \)

In vector calculus exercises, this notation often helps streamline computations.
In the given exercise steps, \( \mathbf{C} \times (\mathbf{A} \times (\mathbf{A} \times \mathbf{C})) \) illustrates how simplifications are derived immediately to zero using \( \epsilon \) notation.
By removing direct evaluations and human errors, calculations become more intuitive and less error-prone.

Curl and Divergence

Curl and divergence are fundamental vector calculus operators that offer insights into field behavior.
The curl of a vector field, represented as \( abla \times \mathbf{F} \), measures the field's rotation or circulation at a point. It uses the same epsilon notation for its computation, implying the changes in vector field orientation.
Divergence, denoted \( abla \cdot \mathbf{F} \), is a scalar quantity representing how much a field diverges from a point, capturing the source or sink nature of a field.In simpler terms:

• Curl shows how much something twists.
• Divergence shows how much something spreads or compresses.

In the exercise, the expression \( abla \times (abla \times \mathbf{A}) \) uses identities linking curl and divergence with the Laplacian, expanding it into understanding more about vector field properties.

Laplacian

The Laplacian, represented as \( abla^2 \), is a differential operator often found in physics and engineering.
It's used to assess the "spread" of a function and is essentially the divergence of the gradient of a field.
Mathematically, for a scalar function \( \phi \), it is given by:

• \( abla^2 \phi = abla \cdot abla \phi \)

It's a vital tool in analyzing functions related to heat, sound, and light because it accounts for how values diffuse over space.In the exercise, using the identity \( abla \times (abla \times \mathbf{A}) = abla(abla \cdot \mathbf{A}) - abla^2 \mathbf{A} \) shows its application.
While simplifying \( abla \times abla \phi \), we observed it resolves directly to zero.
Understanding the concepts behind these expressions makes analyzing and applying mathematical physics much more approachable and understandable.