Is this differential exact or not?

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Homework Help Overview

The discussion revolves around determining whether the differential expression 3x²y³dx + 3x³y²dy is exact, with references to Euler's criterion for exactness in differential forms.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the conditions under which a differential is considered exact, specifically referencing the equality of certain partial derivatives. Some participants express a need for clarification on the application of Euler's criterion.

Discussion Status

There is an indication that some participants believe the differential may be exact, and a suggestion has been made to find a potential function. However, the discussion does not reach a consensus, as others emphasize the need for mathematical rigor in confirming exactness.

Contextual Notes

Participants are exploring the definitions and implications of exact differentials, with some confusion regarding terminology such as "potential function" versus "anti-derivative." There may be assumptions about the familiarity with differential forms and calculus concepts among participants.

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Homework Statement


Is the differential 3x^2y^3dx+3x^3y^2dy exact or not?
Please show me how to do this problem using Euler's criterion.

Homework Equations





The Attempt at a Solution


Euler says if partial deriv of first term = partial deriv of 2nd term then it must be exact right?
 
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jenzao said:

Homework Statement


Is the differential 3x^2y^3dx+3x^3y^2dy exact or not?
Please show me how to do this problem using Euler's criterion.

Homework Equations





The Attempt at a Solution


Euler says if partial deriv of first term = partial deriv of 2nd term then it must be exact right?

To be exact; the partial derivative with respect to y of the first term (the one proportional to dx) must be equal to the partial derivative with respect to x of the second term.
 
Yes it appears to be exact. Now all you need to do is to find the potential function.
 
Or, more mathematically, find the anti-derivative. "Potential function" is physics terminology.
 

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