5.1 Definition of the partial derivative

1. Chapter 5 Partial differentiation 5.1 Definition of the partial derivative • the partial derivative of f(x,y) with respect to x and y are • 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. Chapter 5 Partial differentiation Ex: Find the total derivative of with respect to , given that 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. Chapter 5 Partial differentiation Ex: Show that the differential xdy+3ydx is inexact. • Inexact differential can be made exact by multiplying a suitable function called an integrating factor Properties of exact differentials:

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

10. Chapter 5 Partial differentiation 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. Sol:

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

12. Chapter 5 Partial differentiation Ex: Derivative the conditions for maxima, minima and saddle points for a function of two variables, using the above analysis.

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

14. Chapter 5 Partial differentiation 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. • the stationary points of f(x,y,z) subject to the constraints g(x,y,z)=c1, h(x,y,z)=c2.

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

16. Chapter 5 Partial differentiation

17. Chapter 5 Partial differentiation

18. Chapter 5 Partial differentiation 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 (1) The integral’s limits are constant: • indefinite integral • definite integral

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