Line Integrals / Conservative Vector Fields

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SUMMARY

The discussion centers on evaluating the line integral of the vector field F = < z^2/x, z^2/y, 2zlog(xy)> along a path from P = (1/2, 4, 2) to Q = (2, 3, 3) using the potential function f = z^2log(xy). The integral is computed as f(2,3,3) - f(1/2,4,2), resulting in 9*log(6) - 4*log(2). The necessity of specifying that the path lies in the region where x, y, z are positive is crucial due to the domain restrictions of the logarithmic function and the behavior of the vector field components.

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  • Understanding of vector calculus, specifically line integrals.
  • Familiarity with conservative vector fields and potential functions.
  • Knowledge of logarithmic functions and their domains.
  • Ability to evaluate multivariable functions and their gradients.
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  • Study the properties of conservative vector fields and their implications for line integrals.
  • Learn about the evaluation of line integrals using potential functions in vector calculus.
  • Explore the domain restrictions of logarithmic functions and their impact on vector fields.
  • Investigate the concept of path independence in line integrals for conservative fields.
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Students and professionals in mathematics, physics, and engineering who are working with vector calculus, particularly those focusing on line integrals and conservative vector fields.

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Homework Statement



[tex]F = < z^2/x, z^2/y, 2zlog(xy)>[/tex]
[tex]F = \nabla f[/tex], where [tex]f = z^2log(xy)[/tex]

Homework Equations



Evaluate [tex]\int F \cdot ds[/tex] for any path c from [tex]P = (1/2, 4, 2)[/tex] to [tex]Q = (2, 3, 3)[/tex] contained in the region [tex]x > 0, y > 0, z > 0[/tex]

Why is it necessary to specify that the path lie in the region where [tex]x, y, z[/tex] are positive?

The Attempt at a Solution



I did [tex]f(2,3,3) - f(1/2,4,2)[/tex] to get [tex]9*log(6) - 4*log(2)[/tex]

I don't really have an idea of how to answer the second question. Does it have to do with closed paths?
 
Last edited:
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think about the domain of the log function.
 
or the first 2 elements of F
 

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