1. (MTH 250) Lecture 26 Calculus

2. Previous Lecture’s Summary Recalls Introduction to double integrals Iterated integrals Theorem of Fubini Properties of double integrals Integrals over non-rectangular regions Reversing the order of integration

3. Today’s Lecture Recalls Polar Coordinates Rectangular Coordinates. Cylindrical Coordinates Spherical Coordinates Equations of Surfaces Conversion of Coordinate Systems Directional Derivatives Gradients.

4. Recalls

5. Recalls

6. Definition : Recalls

7. Theorem: Recalls

8. Definition: Recalls

9. Properties of double integrals Theorem:

10. Polar coordinates

13. Rectangular coordinates

14. Cylindrical coordinates

17. Spherical coordinates

21. Equations of surfaces

22. Conversion of Coordinate systems

23. Directional Derivatives

25. Theorem: Directional Derivatives

28. Gradient Definition: Remark:

29. Gradient Properties:

30. Gradient

31. Gradient

32. Gradient

33. Gradient Theorem: Remark:

34. Gradient

35. Gradient

36. Lecture Summary Polar Coordinates Rectangular Coordinates. Cylindrical Coordinates Spherical Coordinates Equations of Surfaces Conversion of Coordinate Systems Directional Derivatives Gradients.