In 2-D infinite paths possible
The fundamental theorem of calculus: f = f(x)
f(x)
x2
x2
 df 
∫x  dx  dx = x∫ df = f ( x2 ) − f ( x1 )
1
1
y
z = z(x,y)
A
In 1-D only one way to integrate this.
C
a
b
x
For an exact differential
b
a
b
b
a
b
a
a
∫ dz = ∫ dz + ∫ dz = ∫ dz − ∫ dz = 0
C
Green’s theorem
In general, the differential dz = Mdx + Ndy is ‘exact’ if:
y
1.
∂M ∂N
=
∂y
∂x
2.∫ dz = 0
A
C
C
a
b
x
 ∂N ∂M 
Mdx
+
Ndy
=
∫C
∫∫A  ∂x − ∂y dxdy
Inexact differentials do not satisfy these conditions
e.g. dz = ydx - xdy
b
3.∫ dz
Is path independent
a
e.g. state variables, P = P(V,T)
Heat: Spontaneous flow of energy caused by
the temperature difference between two
objects
Work: Any other kind of energy transfer
Internal Energy: Total energy in a system
Sum of two inexact differentials can be exact.
First law of thermodynamics (in words)
Increase in internal energy = Heat added + work
done on the system
∆U = ∆Q + ∆W
(∆W is the work done ON the system)
1