If the absolute temperature ($T$) and the pressure ($p$) are kept constant, the variation of the internal energy ($dU$), which depends on the entropy variation ($dS$) and the volume Variation ($dV$), is expressed as:

Integrating this results in the following expression in terms of the internal energy ($U$), the entropy ($S$), and the volume ($V$):

As the internal energy differential ($dU$) depends on the differential inexact Heat ($\delta Q$), the pressure ($p$), and the volume Variation ($dV$) according to the equation:

and the expression for the second law of thermodynamics with the absolute temperature ($T$) and the entropy variation ($dS$) as:

we can conclude that:

Given that the internal energy ($U$) depends on the entropy ($S$) and the volume ($V$), the internal energy differential ($dU$) can be calculated as follows:

$dU = \left(\displaystyle\frac{\partial U}{\partial S}\right)_V dS + \left(\displaystyle\frac{\partial U}{\partial V}\right)_S dV$

To simplify the notation of this expression, we introduce the derivative of the internal energy ($U$) with respect to the entropy ($S$) while keeping the volume ($V$) constant as:

$DU_{S,V} \equiv \left(\displaystyle\frac{\partial U}{\partial S}\right)_V$

and the derivative of the internal energy ($U$) with respect to the volume ($V$) while keeping the entropy ($S$) constant as:

$DU_{V,S} \equiv \left(\displaystyle\frac{\partial U}{\partial V}\right)_S$

therefore, we can write:

The internal energy differential ($dU$) is a function of the variations in the entropy ($S$) and the volume ($V$), as well as the slopes the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) and the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$), which is expressed as:

When compared with the equation for the internal energy differential ($dU$):

it results in the slope of the internal energy ($U$) with respect to the variation in the volume ($V$):

The internal energy differential ($dU$) is a function of the variations in the entropy ($S$) and the volume ($V$), as well as the slopes the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) and the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$), which is expressed as:

When compared with the equation for the internal energy differential ($dU$):

it results in the slope of the internal energy ($U$) with respect to the variation in the entropy ($S$):

Since the internal energy differential ($dU$) is an exact differential, we should note that the internal energy ($U$) with respect to the entropy ($S$) and the volume ($V$) must be independent of the order in which the function is derived:

$D(DU_{S,V}){V,S}=D(DU{V,S})_{S,V}$

Using the relationship between the slope the partial derivative of internal energy with respect to entropy at constant volume ($DU_{S,V}$) and the absolute temperature ($T$)

,

and the relationship between the slope the partial derivative of internal energy with respect to volume at constant entropy ($DU_{V,S}$) and the pressure ($p$)

,

we can conclude that: