Zero curl and gradient of some scalar potential

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SUMMARY

A vector field with zero curl is necessarily the gradient of a scalar potential. This conclusion arises from the fundamental theorem of vector calculus, which states that a field exhibiting no rotation indicates that it can be represented as the gradient of a scalar function. The visual representation of such fields resembles gradient flows, reinforcing the relationship between zero curl and scalar potentials.

PREREQUISITES
  • Understanding of vector calculus
  • Familiarity with the concepts of curl and gradient
  • Knowledge of scalar fields
  • Basic principles of vector fields
NEXT STEPS
  • Study the fundamental theorem of vector calculus
  • Learn about the properties of curl and gradient in vector fields
  • Explore scalar potential functions and their applications
  • Investigate examples of fields with zero curl in physics
USEFUL FOR

This discussion is beneficial for students of mathematics, physicists, and engineers who seek to deepen their understanding of vector fields and their properties, particularly in relation to scalar potentials.

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Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential?

Thanks!
 
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redredred said:
Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential?

Thanks!

If the curl is zero the field does not rotate so the field lines look like gradient flows.
 

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