1. Chapter 13-Vector Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

2. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

3. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

4. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

5. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let v denote the velocity of a wind that blows through a wind tunnel parallel to the sides of the tunnel. Suppose that at each point P = (x, y, z), the magnitude of v (P) is proportional to the height of P above the ground. Assume that the ground is at height z = 0 and that the vector i points down the tunnel in the direction that the wind is blowing. Describe the velocity as a vector field. Vector Fields in Physics

6. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Determine the field lines of the vector field F(x, y) = i + y j that is shown below. Integral Curves (Streamlines)

7. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let  be a positive constant. The differential equation is called the harmonic oscillator equation. It is known that u (t) is a solution of this equation if and only if u (t) = Acos (  t) + B sin (  t) for some constants A and B. Use this fact to determine the integral curves of the vector field F(x, y) = −  yi +  xj. Integral Curves (Streamlines)

8. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let the temperature at a point in the plane be given by T(x, y) = x 2 +6y 2. Calculate the gradient of T and give a physical interpretation. Gradient Vector Fields and Potential Functions

9. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Gradient Vector Fields and Potential Functions

10. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Gradient Vector Fields and Potential Functions

11. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: The vector field F(x, y)=(y − 3x 2 ) i + (x + sin (y)) j is known to be conservative. Find a scalar valued function u for which F =  u. Finding Potential Functions EXAMPLE: The vector field F(x, y, z) = y 2 i + (2xy + z) j + (y + 3) k is known to be conservative. Find a scalar valued function u for which F =  u.

12. Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 1. Is {(x, y, z) : 0 < x, 0 < y, 0 < z} a region? If the answer is No, then why not? 2. Is {(x, y, z) : 0 < xyz} a region? If the answer is No, then why not? 3. True or false: If a particle is in motion due to a conservative force field, then the kinetic energy of the particle is conserved. 4. The vector field F(x, y) = y i+(x + 3) j is known to be conservative. Find a scalar-valued function u for which F =  u. Quick Quiz

13. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals DEFINITION: Let C be a directed curve (in the plane or in space) with parameterization t  r(t), a≤ t ≤ b. Suppose that F is a continuous vector field defined on C. Then the line integral (or path integral) of F over C is denoted by the symbol  C F· dr and defined to be

14. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals EXAMPLE: Let C be the planar directed curve parameterized by r(t) =, 0 ≤ t ≤ 1. Calculate  R F· dr for F(x, y) = e x i + xyj.

15. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals THEOREM: Let t  r(t), a ≤ t ≤ b be a continuously differentiable parameterization for a directed curve C in the plane or in space. Suppose that F is a continuous vector field whose domain contains C. Let s  p(s), 0 ≤ s ≤ L be the arc length parameterization of C with p (0) = r (a) and p (L) = r (b). Let T(s) = p’ (s) denote the unit tangent vector to C at the point p(s). Then the line integral of F over C is given by the equation

16. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals

17. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Dependence on Path EXAMPLE: Let F(x, y) = −y i + x j. Set P 0 = (1, 0) and P 1 = (−1, 0). Consider two paths from P 0 to P 1 : C parameterized by r(t) = cos (t) i + sin (t) j, 0 ≤ t ≤  and C * parameterized by r * (t) = cos (t) i − sin (t) j, 0 ≤ t ≤ . Is the line integral of F over C equal to the line integral of F over C * ?

18. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Closed Curves EXAMPLE: Calculate where C comprises the line segment from P 0 = (1, 0, 0) to P 1 = (0, 1, 0), the line segment from P 1 to P 2 = (0, 0, 2), and the line segment from P 2 to P 0.

19. Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

20. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

21. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

22. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

23. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Consider the vector field F(x, y, z) = yzi + xzj + xyk. Show that F is path independent. Calculate the line integral of F from P 0 = (0,−1, 2) to P 1 = (2, 1, 4).

24. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Characterization of Path-Independent Vector Fields

25. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Characterization of Path-Independent Vector Fields

26. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Closed Vector Fields

27. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Is the vector field F(x, y) = (x 2 − sin (y))i +(y 3 + cos (x))j conservative? Closed Vector Fields

28. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Closed Vector Fields

29. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let M (x, y) = sin (y) + y sin (x) and N (x, y) = x cos (y) − cos (x) + 1. Is the vector field F(x, y) = M (x, y) i + N (x, y) j conservative on the rectangle G = {(x, y) : −1 < x < 1,−1 < y < 1}? If it is, then find a function u such that F =  u. Closed Vector Fields

30. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vector Fields in Space

31. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Is the vector field in space given by F(x, y, z) = xz i + yz j +(y 2 /2) k a conservative vector field? Vector Fields in Space

32. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Suppose that F(x, y, z) = M(x, y, z)i+ N(x, y, z)j + R(x, y, z) k is a continuously differentiable vector field on a simply connected region G in space. If F satisfies all three equations then there is a twice continuously differentiable function u on G such that  u = F. In short, a closed vector field on a simply connected region in space is conservative. Vector Fields in Space

33. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Is the vector field F(x, y, z) = yz 2 i + (xz2−z)j + (2xyz−y) k conservative on the box G = {(x, y, z) : 0 < x < 2, 0 < y < 2, 0 < z < 2}? If it is, find a function u for which F =  u. Vector Fields in Space

34. Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

35. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence of a Vector Field

36. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence of a Vector Field EXAMPLE: Define F(x, y) = xi + yj, G(x, y) = −xi − y 3 j, and H(x, y) = x 2 i − y 2 j. Calculate the divergence of each vector field at the origin and relate your answer to the physical properties of the flow that the vector field represents.

37. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Curl of a Vector Field EXAMPLE: Define F(x, y) = xi + yj, G(x, y) = −xi − y 3 j, and H(x, y) = x 2 i − y 2 j. Calculate the divergence of each vector field at the origin and relate your answer to the physical properties of the flow that the vector field represents.

38. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Curl of a Vector Field EXAMPLE: Define F(x, y, z) = y 2 z i − x 3 j + xy k. Calculate curl(F). EXAMPLE: Sketch the vector field F(x, y, z) = −yi + xj + 0k and its curl.

39. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Identities Involving div, curl, grad, and  THEOREM: Let u be a twice continuously differentiable scalar-valued function on a region G in the plane or in space. Let F be a twice continuously differentiable vector field on G. Then i) div( grad (u)) =  u, ii) div(curl (F)) = 0, iii) curl(grad (u)) = 0, iv) curl(curl (F)) =grad(div (F)) −  F, v) div(uF) = u div(F) +grad (u) · F.

40. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Identities Involving div, curl, grad, and  EXAMPLE: For F(x, y, z) =, verify the identity curl(curl (F)) =grad(div (F)) −  F.

41. Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Calculate the divergence of x 2 yzi + 7yj − xyk at the point (1, 2,−1). 2. Calculate the curl of x 2 i + (12/z) j − y 3 k at the point (−4, 3, 2). 3. Calculate  × (  u) for u (x, y, z) = x 2 y 3 + z 4. 4. True or false: F is conservative implies curl(F) = 0.

42. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

43. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

44. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

45. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate the area A inside the ellipse x 2 /a 2 + y 2 /b 2 = 1.

46. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Vector Form of Green’s Theorem EXAMPLE: Let F(x, y) = x i + y j and R = {(x, y) : x 2 + y 2 < 1}. Assume that F represents the velocity of the flow of a fluid in the region. Explain what Green’s Theorem tells us about this fluid flow.

47. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Green’s Theorem for More General Regions

48. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Green’s Theorem for More General Regions

49. Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. True or false: If a simple closed curve C is part of the boundary of a region, then the positive orientation of C is always counterclockwise. 2. If C is a positively oriented simple closed curve that encloses a region of area 3 and if F(x, y) = 5y i − 2x j, then what is the value of 3. If C is a positively oriented simple closed curve that encloses a region of area 3, if F(x, y) = 7x i − 5y j, and if n is the outward unit normal along C, then what is the value of

50. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Integral for Surface Area DEFINITION: Let S be the graph of a continuously differentiable function (x, y)  f (x, y) defined on a region R in the xy-plane. We refer to as the element of surface area on S. The surface area of the graph of f over R is defined by

51. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Integral for Surface Area EXAMPLE: Find the area of the portion of the plane 3x + 2y + z = 6 that lies over the interior of the circle x 2 + y 2 = 1 in the xy-plane.

52. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Integrating a Function over a Surface

53. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Integrating a Function over a Surface

54. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application EXAMPLE: Assuming that it has uniform mass distribution , determine the center of mass of the upper half of the sphere x 2 + y 2 + z 2 = a 2.

55. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Element of Area for a Surface that is Given Parametrically

56. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Surface Integrals Over Parameterized Surfaces

57. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Surface Integrals Over Parameterized Surfaces EXAMPLE: Integrate the function  (x, y, z) = z − xy over the surface S that is parameterized by r(u, v) =, 0 ≤u ≤ 1, 0 ≤ v ≤ 2.

58. Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

59. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Orientable Surfaces and Their Boundaries EXAMPLE: Suppose that a > 0. What are the two orientations of the sphere x 2 + y 2 + z 2 = a 2 ?

60. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Component of Curl in the Normal Direction

61. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stoke’s Theorem

62. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stoke’s Theorem

63. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stokes’s Theorem on a Region with Piecewise Smooth Boundary

64. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stokes’s Theorem on a Region with Piecewise Smooth Boundary

65. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application

66. Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

67. Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence Theorem

68. Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence Theorem

69. Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Applications

70. Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz