Similarity between exact equations and potential functions

  • Thread starter leehufford
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  • #1
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Hello,

I'm currently in a differential equations course, and we are learning how to solve exact equations in the form of Mdx + Ndy = 0.

I immediately recognized this from my vector calculus class as it was used to find a potential function of a vector field (assuming the vector field was conservative). We integrated both M dx and N dy, combined them on a term by term basis and came up with a potential function.

In differential equations we start to do the same thing but then we set the resulting function from integration equal to N (or M, whichever one we didn't integrate).

I was just wondering if both of these processes have the same end goal and/or are interchangeable. If they aren't interchangeable why are they different? Seems like we get a function as an answer to each problem. The problems seem to slightly deviate from each other but I was wondering if this is just my professor's style. Thanks in advance,

Lee
 

Answers and Replies

  • #3
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Potentials exist if there is an "exact differential"; that is, the expression must be integrable.

For a concise statement see http://mathworld.wolfram.com/ExactDifferential.html

Here is a treatment of the theory: http://ocw.mit.edu/courses/mathemat...ntial-functions/MIT18_02SC_MNotes_v2.2to3.pdf
Thanks for the reply. After reading the MIT document it seems like a potential function is the integral of a solution to an exact differential equation. Is that a correct assessment? Does that make the gradient of the function the original differential equation? Sorry it's still a little hazy.

Lee
 
  • #4
UltrafastPED
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You have it just about right.

The potential is the integral of the exact differential; when you take the gradient you obtain the integrated function.

In physics the potential is path independent; the gradient provides the negative of the force.
 
  • #5
HallsofIvy
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I get annoyed at physics notation for mathematics concepts! (A pet peeve of mine.)

In mathematics notation, a differential in two (or three) variables, u(x,y)dx+ v(x,y)dy (or df= u(x,y,z)dx+ v(x,y,z)dy+ w(x,y,z)dz) is an exact differential (if there is a function, f such that [itex]\dfrac{\partial f}{\partial x}= u(x,y)[/itex] and [itex]\dfrac{\partial f}{\partial y}= v(x,y)[/itex]
(and that [itex]\dfrac{\partial f}{\partial z}= w(x,y,z)[/itex]). That is the concept introduced in "multi-variable Calculus", usually in connection with path integrals, and is exactly the idea behind "exact differential equations".
 

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