# Exact and Inexact Differential

• FourierX
In summary, the question is asking if (x2 -y)dx + xdy = dF is an exact differential, where dF is path dependent with a line strikethrough it similar to "h" in Planck's constant. Exact differentials are those where the left hand side is exactly equal to d(something), while inexact differentials have an additional term. The condition for an exact differential is that the partial derivatives of M and N with respect to x and y must be exactly equal.
FourierX

## Homework Statement

Is (x2 -y)dx + xdy = dF an exact differential ?

dF = path dependant ( i.e. d has a line strikethrough it as h in Planck's constant.)

## Homework Equations

(x2 -y)dx + xdy = dF

## The Attempt at a Solution

Hi FourierX!

Exact differential means that the left hand side is exactly d(something), for example y2cosxdx + 2ysinxdy is an exact differential because it is d(y2sinx)

Mdx + Ndy = f(x) is exact only when this condition is met $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$ (It has to be EXACTLY equal...no minus signs allowed etc..)

## 1. What is the difference between exact and inexact differentials?

An exact differential is a differential that depends only on the current state of the system and not on the path taken to reach that state. In contrast, an inexact differential depends on the path taken and is not a function of state variables alone.

## 2. How do you determine if a differential is exact or inexact?

A differential is exact if it can be expressed as the total differential of a single-valued function, while an inexact differential cannot be expressed in this way.

## 3. What is the significance of exact and inexact differentials in thermodynamics?

In thermodynamics, exact differentials are used to describe reversible processes, while inexact differentials are used to describe irreversible processes. This allows for a more accurate and comprehensive understanding of energy transfer and changes in state.

## 4. Can an exact differential be converted to an inexact differential?

No, an exact differential cannot be converted to an inexact differential. However, it is possible for an inexact differential to be approximated by an exact differential under certain conditions.

## 5. How are exact and inexact differentials used in practical applications?

Exact and inexact differentials are used in a variety of scientific fields, including physics, chemistry, and engineering, to describe and analyze physical systems. They are particularly useful in thermodynamics, where they help to accurately model and predict energy transfer and changes in state.

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