Partial Derivatives

1
Partial Derivatives

• Partial Derivative
• The slope of one particular tangent to the curve
  at a given point

Example What are the partial derivatives of the
ideal gas equation of state?
2
Extrema of Multivariate Functions

• Total Derivative
• The total differential for a multivariate
  function is given by
• All xi must be independent variables
• For example, consider a function with two
  variables x and y

3
Extrema of Multivariate Functions

• Extrema
• An extremum point exists where all partial
  derivatives are zero
• For example, with the two variables x and y

4
Extrema of Multivariate Functions
Example Find the minimum of the paraboloid
5
Extrema of Multivariate Functions

• Saddle Points
• A point where all first partial derivatives are
  zero, but some second partial derivatives are
  positive and some are negative
• For a two variable function, the Hessian must be
  positive to ensure an extremum point

6
Extrema of Multivariate Functions Subject to
Constraints

• Constraints
• Later in the semester we will be interested in
  finding equilibrium states of a system subject to
  a given set of constraints (e.g. constant
  temperature, pressure, and mole number)
• This process requires optimizing a function
  subject to a constraint
• Mathematically, we express this optimization
  process as follows
• The constraint function g(x,y) provides a
  relationship between the variables x and y
• As a result, x and y are no longer independent
  variables
• For simple cases, we can solve the optimization
  problem in a straightforward manner for more
  complicated scenarios, we use the method of
  Lagrange multipliers

7
Extrema of Multivariate Functions Subject to
Constraints
Example Find the minimum of the
paraboloid subject to the constraint
8
Extrema of Multivariate Functions Subject to
Constraints

• The Method of Lagrange Multipliers
• Consider a function f (x,y) subject to the
  constraint g(x,y) const.
• Take the value hmax to be the maximum value of f
  along the locus of points defined by the
  intersection of f and g
• Now draw the plane f hmax through the f and g
  surfaces
• Call the locus of points lf defined by the
  intersection of f with f hmax the level curve
  of f (x,y)
• Similarly, call the locus of points lg defined by
  the intersection of g with f hmax the level
  curve of g(x,y)

9
Extrema of Multivariate Functions Subject to
Constraints

• The Method of Lagrange Multipliers
• At the point h hmax, the total differentials
  of both f and g must both equal zero
• Rearrangement of the above equations gives
• This implies the derivatives of f and g need only
  be the same within an arbitrary constant l.
  Optimization requires solving the following

and
10
Extrema of Multivariate Functions Subject to
Constraints
Example Again, find the minimum of the
paraboloid subject to the constraint
11
Extrema of Multivariate Functions Subject to
Constraints

• The Method of Lagrange Multipliers
• In general, for a multivariate function f (x1,
  x2, ,xt) with t variables and lets say two
  constraints g(x1, x2, ,xt) c1 and h(x1, x2,
  ,xt) c2, one solves the set of equations

12
Extrema of Multivariate Functions Subject to
Constraints

• The Method of Lagrange Multipliers
• Also, one can use the condition that the total
  differentials of the function to be optimized and
  the constraint functions must be zero to develop
  the following expression

13
Integrating Multivariate Functions

• Path of Integration
• Consider the multivariate integral
• Question Does it matter whether we integrate
  over x first then y or y first and then x?
• Answer It depends on the nature of the functions
  s and t

14
Integrating Multivariate Functions

• Path of Integration
• State Function the value of the integral is
  independent of the path
• Path Function the value of the integral is
  dependent on the path
• How do we know which case we have? If you can
  express sdx tdy as the differential of a
  function f (x,y), that isdf sdx tdy, then f
  is a state function
• The value of the integral depends only on the
  value of f at the final and initial states

15
Integrating Multivariate Functions

• Euler reciprocal relationship
• In general, one can use the Euler reciprocal
  relationship to check if an expression sdx tdy
  is an exact differential
• If this relationship holds, sdx tdy is an exact
  differential since

and
Example Is 6xy3dx 9x2y2dy an exact
differential?
16
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Chain rule
• -1 rule
• Change of variables rule

17
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Change of height of cylinder
• with radius at constant volume

18
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Change of surface area of cylinder
• with radius at constant volume

19
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Relation between constant-volume and
  constant-pressure heat capacities

20
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Relation between constant-volume and
  constant-pressure heat capacities

21
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Relation between constant-volume and
  constant-pressure heat capacities

change of variables
22
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Relation between constant-volume and
  constant-pressure heat capacities

change of variables
Maxwell relation
23
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Relation between constant-volume and
  constant-pressure heat capacities

change of variables
Maxwell relation
Ideal-gas model
24
Partial-derivative Relations

• Several standard relations involving partial
  derivatives are important in thermodynamics
• Example
• Relation between constant-volume and
  constant-pressure heat capacities

change of variables
Maxwell relation
Ideal-gas model
That famous result