1. MA 242.003 Day 52 – April 1, 2013 Section 13.2: Line Integrals – Review line integrals of f(x,y,z) – Line integrals of vector fields

2. 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.

4. We partition the curve into n pieces:

5. 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:

7. which is similar to a Riemann sum.

8. 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.

11. Extension to 3-dimensional space

13. Shorthand notation

14. Extension to 3-dimensional space Shorthand notation

15. Extension to 3-dimensional space Shorthand notation

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

18. What is the geometrical interpretation of the line integral?

25. (continuation of example)

26. A major application: Line integral of a vector field along C

28. We generalize to a variable force acting on a particle following a curve C in 3-space.

29. Principle: Only the component of force in the direction of motion contributes to the motion.

30. Direction of motion

31. Principle: Only the component of force in the direction of motion contributes to the motion. Direction of motion

32. Principle: Only the component of force in the direction of motion contributes to the motion. Direction of motion

35. Partition C into n parts, and choose sample points in each sub – arc.

36. Notice that the unit tangent vector T gives the instantaneous direction of motion.

37. Partition C into n parts, and choose sample points in each sub – arc. Notice that the unit tangent vector T gives the instantaneous direction of motion. Remembering the work done formula

38. Partition C into n parts, and choose sample points in each sub – arc. Notice that the unit tangent vector T gives the instantaneous direction of motion.

40. which is a Riemann sum!

41. which is a Riemann sum! We define the work as the limit as.

45. Change in notation for line integrals of vector fields.