Is Divergence Commutative in Vector Calculus?

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SUMMARY

The discussion centers on the commutativity of the divergence operator (DEL dot A) and the operator acting on a vector field (A dot DEL) in vector calculus. It is established that these two expressions are not equivalent due to the nature of differential operators. The relationship between them is clarified through the product rule for vector calculus, specifically the formula for the curl of a cross product. The term "convective operator" is introduced as a relevant concept for further exploration.

PREREQUISITES
  • Understanding of vector calculus concepts, including divergence and curl.
  • Familiarity with differential operators and their properties.
  • Knowledge of the product rule in vector calculus.
  • Basic understanding of vector fields and their operations.
NEXT STEPS
  • Research the properties of differential operators in vector calculus.
  • Study the product rule for vector calculus in detail.
  • Explore the concept of the convective operator and its applications.
  • Learn about the implications of non-commutativity in vector calculus operations.
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, physics, and engineering, particularly those focusing on vector calculus and fluid dynamics.

Pythagorean
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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.
 
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