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You have to introduce a new symbol for this function, although the physical meaning can be the same. Relationships between partial derivatives. Reminder to the chain rule. composite function:. Example :. Internal energy of an ideal gas. Let’s calculate. with the help of the chain rule. - PowerPoint PPT Presentation
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Relationships between partial derivatives
Reminder to the chain rule
,...),...)y,x(v,...),y,x(u(F,...)y,x(F
composite function: ....),v,u(F
,...),y,x(u ,...,...)y,x(v
You have to introduce a new symbol for this function, although the physicalmeaning can be the same
Example: Internal energy of an ideal gas Tcnun)T(U V0
nR
PV)V,P(T
R
PVcun)V,P(U V0
,...),...)y,x(v,...),y,x(u(F,...)y,x(F
Let’s calculate x
,...)y,x(F
with the help of the chain rule
x
,...)y,x(v
v
,...)v,u(F
x
,...)y,x(u
u
,...)v,u(F
x
,...)y,x(F
Example: xysinyx,...)y,x(F2/322
xycosyyxxysinx2yx2
3
x
,...)y,x(F 2/32222
explicit:
Now let us build a composite function with: 22 yx)y,x(u and xy)y,x(v
vsinu)v,u(F 2/3 vsinu2
3
u
)v,u(F
x2
x
)y,x(u
vcosuv
)v,u(F 2/3
yx
)y,x(v
x
)y,x(v
v
)v,u(F
x
)y,x(u
u
)v,u(F
x
)y,x(F
u
)v,u(F
x
)y,x(u
v
)v,u(Fy
x
)y,x(v
vsinu
2
3 x2 vcosu 2/3 y xycosyyxxysinx2yx2
3
x
)y,x(F 2/32222
Composite functions are important in thermodynamics
-Advantage of thermodynamic notation:
Example: ))Z,X(Y,X(F)Z,X(F
If you don’t care about new Symbol for F(X,Y(X,Z))
wrong conclusion from X
Y
Y
F
X
F
X
F
0X
Y
Y
F
-Thermodynamic notation: ZXYZ X
Y
Y
F
X
F
X
F
can be well distinguished
Apart from phase transitions thermodynamic functions are analytic
y
)y,x(F
xx
)y,x(F
y yx
)y,x(F
xy
)y,x(F 22
See later consequences for physics
(Maxwell’s relations, e.g.)
Inverse functions and their derivatives
Reminder: )x(yfunction )y(xinverse function defined according to
y))y(x(y
Example:1x
1)x(y
function 1y)1x( 1yxy xyy1
y
y1)y(x
y
y)y1(
y
1y
y1
1
1)y(x
1))y(x(y
X
Y0 2 4 6 8 100
2
4
6
8
10
Y
X
y=y(x,z=const.)
What to do in case of functions of two independent variables y(x,z)
keep one variable fixed (z, for instance)
)z,x(y )z,y(xis inverse to if y)z),z,y(x(y
y)z),z,y(x(y Let’s apply the chain rule to
1dy
dx
dx
)z),y(x(dy
Result from intuitive relation: 1y
x
x
y
Thermodynamic notation:
1Y
X
X
Y
ZZ
0 1 2 30
1
2
3
dY/d
X*d
X/d
Y
X
0 2 4 6 8 100
2
4
6
8
10
Y
X
0 2 4 6 80
2
4
6
dY/d
X
X
0 2 4 6 8 100
2
4
6
8
10
x
Y0 2 4 6 8
0
2
4
6
dX
/dY
Y
Numerical example
