Manipulating Equations with Del Operators

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SUMMARY

This discussion focuses on manipulating equations involving del operators, specifically the divergence operator. The participants analyze the equation div(A + B) = div(E) under the assumption that A, B, and E are twice differentiable vector fields. It is established that while div can cancel under certain conditions, one cannot simply conclude that E = A + B without further analysis. The discussion also highlights that if the divergence of a vector field is zero, it corresponds to the curl of another vector field.

PREREQUISITES
  • Understanding of vector calculus, specifically divergence and curl.
  • Familiarity with del operators and their applications in vector fields.
  • Knowledge of differential equations and their manipulation.
  • Experience with vector field analysis and properties of constant fields.
NEXT STEPS
  • Study the properties of divergence and curl in vector calculus.
  • Learn about the implications of zero divergence in vector fields.
  • Explore the relationship between divergence and the Laplacian operator.
  • Investigate examples of vector fields with zero divergence and their physical interpretations.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with vector calculus and require a deeper understanding of del operators and their applications in manipulating equations.

dm164
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I'm trying to understand how to manipulate equations with del operators.

If I have a equation like :

div( A + B ) = div(E)
and assume A,B,E are twice differential vectors

do div cancel?

can I say E = A + B?

If I write is like this
div( A + B - E ) = 0
div( A + B - (A + B)) = 0
div( 0 ) = 0.

but div( constant ) = 0 also
 
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Look at examples of vector fields that are constant. A+B might be one constant field with a zero divergence and E might be a different one.
 
dm164 said:
I'm trying to understand how to manipulate equations with del operators.

If I have a equation like :

div( A + B ) = div(E)
and assume A,B,E are twice differential vectors

do div cancel?

can I say E = A + B?

if the divergence of a vector field is zero then it is the curl of another vector field.
 

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