Understanding Perfect Differentials

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Homework Help Overview

The discussion revolves around understanding the concept of perfect differentials in the context of a specific mathematical expression involving variables x and y. The original poster is attempting to determine whether the given expression represents a perfect differential and is struggling with the differentiation process.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster differentiates terms with respect to different variables and questions why the results do not match. Some participants express similar confusion regarding the equality of the differentiated terms. Others suggest that canceling the denominator leads to equal results, raising questions about the implications of this approach.

Discussion Status

Participants are actively engaging with the problem, sharing their observations about the differentiation process and the implications of canceling terms. There is no explicit consensus on the nature of the problem or the correct approach, and one participant suggests consulting a professor for further clarification.

Contextual Notes

There is uncertainty regarding the interpretation of the term "perfect differential" and whether a unique perfect differential exists for the given expression. Participants are exploring various interpretations and potential manipulations of the expression.

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Homework Statement


Hi, I have an exam tomorrow and I'm trying to do the following question which I've almost solved:

Find the perfect differential of the following:

[(x² + 2xy) / (x + y)² ]dx - [x² / (x + y)² ]dy

I differentiated the first term with respect to 'y' and then the second term with respect to x, equated them but they weren't the same, what could I be doing wrong?

The problem is I can't seem to get both terms to equal each other unless i cancel the denominator..but why can't i get them to equal each other if i differentiate with the denominators too? Should be the same...?


Homework Equations





The Attempt at a Solution

 
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I get that they are not equal either.
 
quasar987 said:
I get that they are not equal either.

If I were to cancel the denominator and then differentiate they actually equal each other. I'm guessing this question was meant to trick me into cancelling it then?
 
I don't think so. Go see your prof about this.
 
What does "Find the perfect differential of the following" mean?

I know what determining whether a differential is a perfect differential or not means but this seems to imply that there is some perfect differential associated with this differential. My first guess my be to change it in someway so that this becomes a perfect differential- for example, multiplying the entire expression by (x+ y)^2 makes it a perfect differential- but that is certainly not unique so "the" perfect differential would not apply.
 

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