It seems I am not so familiar with definite integral being changed to differentials.Can anyone justify me how the T1,P1 values and the limits are being changed to simply T and P as given in the attachment.
Dear Rajorshi Koyal,
If z=f(x,y) is a smooth function where
z is the dependent variable, x and y are the independent variables.
we have
dz=(∂f/∂x)dx +(∂f/∂y)dy, it is called the total differentiation.
where(∂f/∂x) is the partial derivative of f(x,y) with respect to x
(∂f/∂y) is the partial derivative of f(x,y) with respect to y.
In your question,
We have V = V(P, T), V depends on the variables P and T
and then
dV=(∂V/∂P)Tdp+(∂V/∂T)PdT ....(1`)
Assuming that P varies between P₁ and P₂ and T varies between T₁ and T₂, Integrating both sides with your initial conditions, the required forms yields.
V=delta= ∫(∂V/∂P)Tdp+∫(∂V/∂T)PdT..... (2)
the boundaries of p from P₁ to P₂
he boundaries of T from T₁ to T₂
where (∂V/∂P)T computed at the value T (T₁ or T₂ )
and (∂V/∂T)P computed at the value P (P₁ or P₂ )
Integrating (1) we obtain (2) and differentiating (2) we obtain(1).
The choice based on your initial conditions of the mathematical model.
Best regards
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