Both conservative and solenoidal vector field

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SUMMARY

A vector field A that is both solenoidal and conservative can be expressed as A = -∇Φ, where Φ is a scalar function. The discussion clarifies that if A is solely a conservative vector field, it is represented as A = ∇φ, leading to confusion regarding the negative sign. The requirement for A to be solenoidal does not alter the fundamental representation of A as a gradient of a scalar function, but it emphasizes the divergence-free condition of the vector field.

PREREQUISITES
  • Understanding of vector calculus, specifically gradient and divergence operations.
  • Familiarity with the concepts of conservative and solenoidal vector fields.
  • Knowledge of scalar functions and their gradients.
  • Basic principles of mathematical physics related to vector fields.
NEXT STEPS
  • Study the properties of solenoidal vector fields in detail.
  • Learn about the implications of vector fields being conservative in physics.
  • Explore the mathematical derivation of the relationship between scalar functions and vector fields.
  • Investigate examples of solenoidal and conservative vector fields in applied contexts.
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Students of physics and mathematics, particularly those studying vector calculus and field theory, as well as educators seeking to clarify concepts related to vector fields.

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Homework Statement



if a vecor A is both solenoidal and conservative; is it correct that

A=-▼Φ

that is

A=- gradΦ

Φ is a scalar function

thanks
 
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If A is just a conservative vector field, then A= -\nabla \phi for some scalar function \phi. I'm not sure what requiring that it also be solenoidal adds.
 
HallsofIvy said:
If A is just a conservative vector field, then A= -\nabla \phi for some scalar function \phi. I'm not sure what requiring that it also be solenoidal adds.

well i thouhgt if A is only conservative
then

A= \nabla \phi (according to my textbook)

not

A= -\nabla \phi

so what is right?:confused:
 

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