Is Divergence Commutative in Vector Calculus?

[Pythagorean]

is (DEL dot A) the same as (A dot DEL)?

I know the dot product is commutative, but this involves an operator.

if the answer is YES, than why does one of the product rules read like this:

DEL X (A X B) = (B dot DELL)A - (A dot DEL)B + A(DEL dot B) - B(DEL dot A)

they commute the operator with its operand.

if the answer is NO, than what is the relationship between (DEL dot A) and (A dot DELL)?

this is not a homework question.

 

[Pythagorean]

Oh... it's called the convective operator for anyone interested... I tried so many different combinations in google

http://mathworld.wolfram.com/ConvectiveOperator.html

 

[tacman]

The answer is NO, divergence is not commutative. The commutative property states that the order of operands does not affect the result of an operation. However, in the case of divergence, the order of the operator and its operand does matter.

The expression (DEL dot A) means the divergence of the vector field A, while (A dot DEL) means the dot product of the vector A with the gradient operator. These two operations are not equivalent, and therefore, they do not commute.

The product rule you have mentioned involves the cross product, which is a different operation than the dot product. In this case, the gradient operator is being applied to the vector field A, and because the gradient operator is a differential operator, it does not commute with its operands.

In summary, there is no direct relationship between (DEL dot A) and (A dot DEL) as they are different operations. While the dot product is commutative, the divergence operator is not.