1. 5.1 Definition of the partial derivative the partial derivative of f(x,y) with respect to x and y are Chapter 5 Partial differentiation for general n-variable second partial derivatives of two-variable function f(x,y)

2. Chapter 5 Partial differentiation 5.2 The total differential and total derivative

3. Ex: Find the total derivative of with respect to, given that Chapter 5 Partial differentiation 5.3 Exact and inexact differentials If a function can be obtained by directly integrating its total differential, the differential of function f is called exact differential, whereas those that do not are inexact differential.

4.  Inexact differential can be made exact by multiplying a suitable function called an integrating factor Chapter 5 Partial differentiation Ex: Show that the differential xdy+3ydx is inexact. Properties of exact differentials:

5. for n variables Chapter 5 Partial differentiation Ex: Show that (y+z)dx+xdy+xdz is an exact differential 5.4 Useful theorems of partial differentiation

6. Chapter 5 Partial differentiation 5.5 The chain rule

7. Chapter 5 Partial differentiation 5.6 Change of variables Ex: Polar coordinates ρ and ψ, Cartesian coordinates x and y, x=ρcosφ, y=ρsinφ, transform into one in ρ and φ

8. Chapter 5 Partial differentiation 5.7 Taylor’s theorem for many-variables functions Ex: The Taylor’s expansion of f(x,y)=yexp(xy) about x=2, y=3.

9. 5.8 Stationary points of many-variables functions Chapter 5 Partial differentiation two-variable function about point

10. Ex: has a maximum at the point, a minimum at and a stationary point at the origin whose nature cannot be determined by the above procedures. Chapter 5 Partial differentiation Sol:

11. for a n-variable function at all stationary points Chapter 5 Partial differentiation

12. Ex: Derivative the conditions for maxima, minima and saddle points for a function of two variables, using the above analysis.

13. Find the maximum value of the differentiable function subject to the constraint Lagrange undetermined multipliers method Chapter 5 Partial differentiation 5.9 Stationary values under constraints

14. Ex: The temperature of a point (x,y) on a circle is given by T(x,y)=1+xy. Find the temperature of the two hottest points on the circle. Chapter 5 Partial differentiation  the stationary points of f(x,y,z) subject to the constraints g(x,y,z)=c1, h(x,y,z)=c2.

15. Ex: Find the stationary points of subject to the following constraints: Chapter 5 Partial differentiation

18. 5.11 Thermodynamic relations Maxwell’s thermodynamic relations: P: pressure V: volume T: temperature S: entropy U: internal energy Ex: Show that

19. Chapter 5 Partial differentiation Ex: Show that

20. Chapter 5 Partial differentiation 5.12 Differentiation of integrals indefinite integral definite integral (1) The integral’s limits are constant:

21. Chapter 5 Partial differentiation (2) The integral’s limits are function of x Ex: Find the derivative with respect to x of the integral