MA 242.003

1. MA 242.003 • Day 51 – March 26, 2013 • Section 13.1: (finish) Vector Fields • Section 13.2: Line Integrals

2. Chapter 13: Vector Calculus

3. Chapter 13: Vector Calculus “In this chapter we study the calculus of vector fields, …and line integrals of vector fields (work), …and the theorems of Stokes and Gauss, …and more”

4. Section 13.1: Vector Fields

5. Section 13.1: Vector Fields Wind velocity vector field 2/20/2007

6. Section 13.1: Vector Fields Wind velocity vector field 2/21/2007 Wind velocity vector field 2/20/2007

7. Section 13.1: Vector Fields Ocean currents off Nova Scotia

8. Section 13.1: Vector Fields Airflow over an inclined airfoil.

9. General form of a 2-dimensional vector field

10. General form of a 2-dimensional vector field

11. General form of a 2-dimensional vector field Examples:

12. General form of a 2-dimensional vector field Examples: QUESTION: How can we visualize 2-dimensional vector fields?

13. General form of a 2-dimensional vector field Examples: Question: How can we visualize 2-dimensional vector fields? Answer: Draw a few representative vectors.

14. Example:

15. Example:

16. Example:

18. We will turn over sketching vector fields in 3-space to MAPLE.

19. Gradient, or conservative, vector fields

20. Gradient, or conservative, vector fields

21. Gradient, or conservative, vector fields EXAMPLES:

22. Gradient, or conservative, vector fields EXAMPLES:

23. QUESTION: Why are conservative vector fields important?

24. QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)

25. QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3) Sections 13.2 and 13.3 are concerned with the following questions:

26. QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3) Sections 13.2 and 13.3 are concerned with the following questions: Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative.

27. QUESTION: Why are conservative vector fields important? ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3) Sections 13.2 and 13.3 are concerned with the following questions: Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative. Once you know you have a conservative vector field, “Integrate it” to find its potential functions.

28. Format of chapter 13: Sections 13.2, 13.3 - conservative vector fields Sections 13.4 – 13.8 – general vector fields

29. Section 13.2: Line integrals

30. Section 13.2: Line integrals GOAL: To generalize the Riemann Integral of f(x) along a line to an integral of f(x,y,z) along a curve in space.

32. We partition the curve into n pieces:

33. Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

34. Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

35. Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum: which is similar to a Riemann sum.

36. Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum: which is similar to a Riemann sum.

38. EXAMPLE:

39. Extension to 3-dimensional space

40. Extension to 3-dimensional space

41. Extension to 3-dimensional space Shorthand notation

42. Extension to 3-dimensional space Shorthand notation

43. Extension to 3-dimensional space Shorthand notation

44. Extension to 3-dimensional space Shorthand notation 3. Then

47. Line Integrals along piecewise differentiable curves