Proving the Curl Identity for a Simple Curl Equation

In summary, the conversation discusses the final term in the equation $\nabla \times (a\nabla b) = \epsilon_{ijk}\frac{\partial}{\partial x_j}(a\frac{\partial b}{\partial x_k})\hat e_i$, which should be the zero vector but it is not immediately obvious. It is shown that this term is indeed zero by using the properties of the Levi-Civita symbol and the fact that the curl of a gradient is always zero. This is a clever and elegant way to prove the final term's value.
  • #1
Hiero
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Homework Statement
Simplify ##\nabla \times ( a\nabla b)##
Relevant Equations
##\nabla\times \vec V = \epsilon_{ijk}\frac{\partial V_k}{\partial x_j}\hat e_i##
##\nabla s = \frac{\partial s}{\partial x_i}\hat e_i##
Attempt:

$$\nabla \times ( a\nabla b) = \epsilon_{ijk}\frac{\partial}{\partial x_j}(a\frac{\partial b}{\partial x_k})\hat e_i$$ $$ = \epsilon_{ijk}\big(\frac{\partial a}{\partial x_j}\frac{\partial b}{\partial x_k}+a\frac{\partial b}{\partial x_j\partial x_k}\big)\hat e_i$$ $$= \nabla a \times \nabla b + \text{(final term)}$$

That “final term” (a triple sum) should be the zero vector, but I cannot see how. Maybe I messed up elsewhere.

Thanks.
 
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  • #2
Well, [itex]\epsilon_{ijk} = -\epsilon_{ikj}[/itex] but [itex]\partial_j\partial_kb = \partial_k\partial_jb[/itex] so [tex]
\epsilon_{ijk}\partial_j\partial_kb = -\epsilon_{ikj}\partial_k\partial_jb.[/tex] But [itex]j[/itex] and [itex]k[/itex] are dummy indices, so we can relabel them on the right hand side ...
 
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  • #3
pasmith said:
Well, [itex]\epsilon_{ijk} = -\epsilon_{ikj}[/itex] but [itex]\partial_j\partial_kb = \partial_k\partial_jb[/itex] so [tex]
\epsilon_{ijk}\partial_j\partial_kb = -\epsilon_{ikj}\partial_k\partial_jb.[/tex] But [itex]j[/itex] and [itex]k[/itex] are dummy indices, so we can relabel them on the right hand side ...
That’s a very nice way to show it is zero. The last step (after relabeling) is that ##[x=-x] \implies [x=0]##
I have to admit it seems magical to use this property ignoring the invisible nested summation.

Basically though, each component cancels out in pairs by virtue of the two properties you mentioned:
pasmith said:
Well, [itex]\epsilon_{ijk} = -\epsilon_{ikj}[/itex] but [itex]\partial_j\partial_kb = \partial_k\partial_jb[/itex]
 
  • #4
Alternatively, [tex]\nabla \times (a\mathbf{v}) = (\nabla a) \times \mathbf{v} + a \nabla \times \mathbf{v}[/tex] and if [itex]\mathbf{v}[/itex] is a gradient then its curl is zero (which follows from the observation in my earlier post).
 

1. What is the Curl Identity for a Simple Curl Equation?

The Curl Identity for a Simple Curl Equation is a mathematical equation that states the curl of the curl of a vector field is equal to the negative of the divergence of that vector field. In other words, it is a relationship between the curl and divergence of a vector field.

2. Why is proving the Curl Identity important?

Proving the Curl Identity is important because it helps us understand the relationship between the curl and divergence of a vector field. It also allows us to simplify complicated vector equations and make calculations easier.

3. How is the Curl Identity proven?

The Curl Identity can be proven using vector calculus and the properties of the curl and divergence operators. It involves manipulating the equations and using mathematical principles to show that the two sides of the equation are equal.

4. What are some real-world applications of the Curl Identity?

The Curl Identity has many real-world applications, such as in fluid dynamics, electromagnetism, and heat transfer. It is used to analyze and understand the behavior of vector fields in these fields of study.

5. Are there any limitations to the Curl Identity?

The Curl Identity is limited to vector fields in three-dimensional space. It also assumes that the vector field is continuously differentiable, which may not always be the case in real-world situations. Additionally, it only applies to simple curl equations and may not hold true for more complex equations.

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