Path Independence: My Book vs Reality

In summary, the conversation discusses the surface integral of a gradient field and its path independence under the condition of continuity. However, there is confusion about the definition of a gradient field and its relation to a conservative field. It is suggested that integrating the field equation can help determine if it is a conservative field.
  • #1
bmrick
44
0
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 dependant.

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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  • #2
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.
 

FAQ: Path Independence: My Book vs Reality

What is path independence?

Path independence is a concept in physics and mathematics which states that the value of a physical quantity does not depend on the path taken to reach it. In other words, the final result remains the same regardless of the route or process used to obtain it.

How does path independence differ in my book and reality?

In your book, you may have simplified or idealized scenarios to make them easier to understand. In reality, however, there are often external factors and variables that can affect the outcome, making it less predictable and not completely path independent.

Can you provide an example of path independence in physics?

A common example of path independence in physics is the work done by a conservative force, such as gravity. The work done by gravity only depends on the initial and final positions of an object, not the path it takes to get there.

Why is path independence important in physics?

Path independence is important in physics because it allows us to simplify complex systems and make accurate predictions. It also helps us understand the underlying principles and laws that govern the physical world.

Is path independence always applicable in real-world situations?

No, path independence is not always applicable in real-world situations. While it is a useful concept in many cases, there are often external factors and variables that can affect the outcome and make it non-path independent.

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