Path Independence: My Book vs Reality

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SUMMARY

The discussion centers on the concept of path independence in gradient fields, specifically addressing the assertion that surface integrals of continuous gradient fields are path independent. The user challenges this notion by presenting a continuous gradient field, exemplified by the equation grad f = y^2 + y + 5, which appears path dependent. The consensus reached is that for a line integral between two points to be path independent, the gradient field must be conservative. The discussion emphasizes that continuity alone does not guarantee path independence; rather, the field must be confirmed as conservative through integration.

PREREQUISITES
  • Understanding of gradient fields and their properties
  • Knowledge of conservative vector fields
  • Familiarity with line integrals and surface integrals
  • Ability to perform integration of vector fields
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  • Study the properties of conservative vector fields in depth
  • Learn how to determine if a vector field is conservative using integration
  • Explore the implications of path independence in physics and engineering
  • Investigate the relationship between continuity and path independence in vector calculus
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Mathematicians, physics students, and engineers interested in vector calculus, particularly those exploring the concepts of gradient fields and their applications in real-world scenarios.

bmrick
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my book says that the surface integral of a gradient field is path independent so long as the gradient field is continuous. This seems fishy to me. I'm envisioning a continuous gradient filed where z=grad f(x,y) and the object traced out looks like a mountain range. The equation for such a field might look like grad f = y^2 +y+5. Such a field is clearly continuous, and yet the path integral is definitely path dependent.

What does make sense to me is that if a gradient field of a conservative field exist, THEN the line integral between two points on the gradient field is path independent. And an intuitive test is to integrate the field equation you're working with and test it for conservation.
Is this right?
 
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Perhaps this is a question of "wording". If you know that you have a "gradient" field (not just a given vector field that might be a gradient) then, yes, as long as the field is continuous, it is path independent. I don't know what you mean by "the object traced out". Are you referring to a given path and considering that it might be non-differentiable at points? No, that would NOT be "definitely path dependent". I don't know where you got that idea. The integral is a "smoothing" process and "corners" in the path will not be important.
 

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