Question About Del: Why Does Formula Fail?

[AlonsoMcLaren]

Question about "del"

We know that A x (BxC)= (A·C)B-(A·B)C (*)

In the following example, we can treat ∇ as a vector and apply the formula (*) above to get the correct answer
∇x(∇xV)= ∇(∇·V)-∇^2 V

But in this example, the formula (*) seems to fail
∇x(UxV)≠U(∇·V)-V(∇·U)

Why?

 

[Char. Limit]

Because ∇ is NOT a vector, no matter how much we want it to act like one. For one, ∇ isn't commutative. (Vectors are)

 

[AlonsoMcLaren]

Because ∇ is NOT a vector, no matter how much we want it to act like one. For one, ∇ isn't commutative. (Vectors are)

Then why the textbook simply uses the formula (*) when deriving ∇x(∇xV)= ∇(∇·V)-∇^2 V ? Is it just a coincidence that the formula (*) works for ∇x(∇xV)= ∇(∇·V)-∇^2 V ?

 

[dextercioby]

It's just a coincidence. Nabla is a differential operator. You can't simply switch it from one term of a an equation to another without changing the result.

Vector identities are always easier to express in tensor notation, using delta Kronecker and epsilon Levi Civita.