# Similarity between exact equations and potential functions

• leehufford
An "exact differential" is a special case of a "differential": a differential is exact if it can be expressed as the gradient of a function.In summary, the conversation discusses the concept of exact differential equations and potential functions. The speaker recognizes the similarities between this concept and the use of exact equations in finding potential functions in vector calculus. They ask for clarification on the end goal and interchangeability of the two processes. The expert summarizer explains that potentials exist if there is an "exact differential" and provides resources for further understanding of the theory. The conversation concludes with the speaker gaining a better understanding of the concept.

#### leehufford

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

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/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-b-vector-fields-and-line-integrals/session-63-potential-functions/MIT18_02SC_MNotes_v2.2to3.pdf

UltrafastPED said:
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/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-b-vector-fields-and-line-integrals/session-63-potential-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

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.

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 $\dfrac{\partial f}{\partial x}= u(x,y)$ and $\dfrac{\partial f}{\partial y}= v(x,y)$
(and that $\dfrac{\partial f}{\partial z}= w(x,y,z)$). That is the concept introduced in "multi-variable Calculus", usually in connection with path integrals, and is exactly the idea behind "exact differential equations".