# Conservative Vector Fields and the Intersect Rule

## DOI:

https://doi.org/10.58445/rars.359## Keywords:

Multivariable Calculus, Vector Calculus, Vector Physics, Newtonian Mechanics, Vector Fields, Gradient Fields## Abstract

Conservative vector fields are defined as the gradient of a scalar-valued potential function. Gradient fields are irrotational, as in, the curl in all conservative vector fields is zero. Additionally, line integrals in conservative vector fields are path independent, and line integrals over closed paths are always equal to zero. Gradient fields represent conservative forces, and the associated potential function is analogous to potential energy associated with said conservative forces. The Intersect Rule provides a new, unique shortcut for determining if a vector field is conservative and deriving potential functions, by treating the indefinite integral as a set.

## References

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Khan, Salman Amin, and Grant Sanderson. “Fundamental Theorem of Line Integrals (Article).” Khan Academy, Khan Academy, 2011, www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals-in-vector-fields-articles/a/fundamental-theorem-of-line-integrals.

Khan, Salman Amin, and Grant Sanderson. “Conservative Vector Fields (Article).” Khan Academy, Khan Academy, 2011, www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals-in-vector-fields-articles/a/conservative-fields.

Physics, Libretexts. “3.5: Conservative Forces, Potential Energy, and Conservation of Energy.” Physics LibreTexts, Libretexts, 20 Feb. 2022, phys.libretexts.org/Bookshelves/Conceptual_Physics/Introduction_to_Physics_(Park)/03%3A_Unit_2-_Mechanics_II_-_Energy_and_Momentum_Oscillations_and_Waves_Rotation_and_Fluids/03%3A_Work_and_Energy/3.05%3A_Conservative_Forces_Potential_Energy_and_Conservation_of_Energy#:~:text=If%20only%20conservative%20forces%2C%20such,c%3D%E2%88%92%CE%94PE.

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