Chapter 11 Line, surface and volume integrals

1. Chapter 11 Line, surface and volume integrals 11.1 Line integrals • the forms of the line integral represents open curve integral represents closed curve (loop) integral (1) (2) (i) (ii)

2. Chapter 11 Line, surface and volume integrals along the following paths: Ex: (i) from (1,1) to (4,2) (ii) (iii) line followed line Sol: (i) (ii)

3. Chapter 11 Line, surface and volume integrals (iii) line y=1, dy=0 line x=4, dx=0 Ex: Evaluate the line integral , where C is the circle in the xy-plane defined by y x Hint: set

4. Chapter 11 Line, surface and volume integrals • physical examples of line integrals (1) (2) Φ: electrostatic potential energy (3) : induced magnetic field : carrying current (4) : external magnetic field : carrying current

5. Chapter 11 Line, surface and volume integrals • line integrals with respect to a scalar u : is a parameter S : the arc length along the curve C infinitesimal displacement infinitesimal arc length O

6. Chapter 11 Line, surface and volume integrals Ex: Evaluate the line integral , where C is the semicircle of radius a running from A=(a,0) to B=(-a,0), and for which Sol: • for three-dimensional orthogonal curvilinear coordinates

7. Chapter 11 Line, surface and volume integrals 11.3 Green’s theorem in a plane: express the line integral around a loop as a double integral over the enclosed region R function and its partial derivative are single-valued, finite, and continuous inside and on the boundary of the closed curve C Proof: (1) for the curve STU for the curve SVU

8. Chapter 11 Line, surface and volume integrals (2) for the curve TSV for the curve TUV

9. Chapter 11 Line, surface and volume integrals Ex: Show that the area of a region R enclosed by a simple closed curve C is given by Hence calculate the area of the ellipse (1) set (2) set

10. Chapter 11 Line, surface and volume integrals (3) set • using Green’s theorem to investigate the path independence of the line integrals * If the line integral is independent of the path: (1) for a closed loop is a sufficient condition for I=0

11. Chapter 11 Line, surface and volume integrals (2) for a open curve an exact differential of some function Φ for a closed loop is also a necessary condition for I=0

12. Chapter 11 Line, surface and volume integrals 11.4 Conservative fields and potentials If the line integral between two points is independent of the path, the vector field is called conservative. • A vector is conservative, if and only if , any of the following is true (1) is independent of the path, and (2) There exist a single-valued function (3) (4) is an exact differential

13. Chapter 11 Line, surface and volume integrals Ex: Evaluate the line integral , where , A is the point (c,c,h) and B is the point (2c,c/2,h), along the different paths. (i) , given by (ii) , given by Show that the vector is in fact conservative, and find such that Sol: (i) along

14. Chapter 11 Line, surface and volume integrals (ii) along is conservative.

15. Chapter 11 Line, surface and volume integrals 11.5 Surface integrals (1) (2) (3) a unit normal closed surface open surface * The surface is divided into N elements of area each with a unit normal

16. Chapter 11 Line, surface and volume integrals • Express any surface integral over S as a double integral over the region R in the xy-plane. surface S: unit normal: and are evaluated on the surface

17. Chapter 11 Line, surface and volume integrals Ex: Evaluate the surface integral , where and S is the surface of the hemisphere with method (1): method (2): surface

18. Chapter 11 Line, surface and volume integrals * using polar coordinate Hint:

19. Chapter 11 Line, surface and volume integrals • vector area of surfaces Ex: Find the vector area of the surface of the hemisphere Sol: The area element in spherical coordinates is * is the projected area of the hemisphere on the xy- plane, and not the surface area of the hemisphere. * For a closed surface, the vector area is always zero.

20. Chapter 11 Line, surface and volume integrals • For a open surface, the vector area depends only on its perimeter, or boundary curve C. • A surface is confined to the xy-plane.

21. Chapter 11 Line, surface and volume integrals Ex: Find the vector area of the surface of the hemisphere , by evaluating the line integral Sol: The perimeter C of the hemisphere is a circle

22. Chapter 11 Line, surface and volume integrals • physical examples of surface integrals: (1) total electric charge on a surface: (2) flux of vector field through surface: (3) net mass flux of fluid crossing surface: (4) the electromagnetic flux of energy out of a given volume V bounded by a surface S: (5) solid angle: • for a closed surface • if O is outside S: • if O is inside S:

23. Chapter 11 Line, surface and volume integrals • the manipulation of surface integrals: (1) take a surface S, by a parametric representation: (2) S has a normal vector: (3)

24. Chapter 11 Line, surface and volume integrals Ex 1: Compute the flux of fluid through the parabolic cylinder surface if the velocity vector is Sol: on the surface, Z Y X

25. Chapter 11 Line, surface and volume integrals Ex 2: Surface integral for a vector on a plane Sol: on the surface, set and Z Y X

26. Chapter 11 Line, surface and volume integrals 11.6 Volume integrals (1) total mass (2) total linear momentum Ex: Find an express for the angular momentum of a solid body rotating which angular velocity about an axis through the origin. Sol: angular momentum total angular momentum putting

27. Chapter 11 Line, surface and volume integrals • volumes of three-dimensional region the volume of the small circular cone: Ex: Find the volume enclosed between a sphere of radius a centered on the region and a circular cone of half-angle α with the vertex at the origin. Sol: on the surface

28. Chapter 11 Line, surface and volume integrals 11.7 Integral forms for grad, div, and curl at any point P V is small volume enclosing P and S is its bounding surface. C is a plane contour of area A enclosing the point P and is the normal to the enclosed planar area.

29. Chapter 11 Line, surface and volume integrals Z Proof: for Cartesian coordinates H G D C E F Y A B X

30. Chapter 11 Line, surface and volume integrals

31. Chapter 11 Line, surface and volume integrals • consider the plane ABDC, its vector area is

32. Chapter 11 Line, surface and volume integrals Ex: Show that the geometrical definition of grad leads to the usual expression for in Cartesian coordinates. Sol: consider a small rectangular volume two faces with x=constant are also consider another two faces y=constant and z=constant

33. Chapter 11 Line, surface and volume integrals Ex: By considering the infinitesimal volume element show the usual expression for in orthogonal curvilinear coordinates. Z R S T Q Sol: for face P X Y on and its opposite face, the net contribution is: , similarly, for faces

34. Chapter 11 Line, surface and volume integrals Ex: By considering the infinitesimal planar surface element PQRS, show the usual expression for in orthogonal curvilinear coordinates. Sol: for plane PQRS define by two vector and the unit normal is S→R P→S R→Q Q→P Hint: for S→R

35. Chapter 11 Line, surface and volume integrals at point P in the direction Similarly, we can obtain • set and using

36. Chapter 11 Line, surface and volume integrals 11.8 Divergence theorem and related theorems from the above for each volume divergence theorem Ex: if , use the surface integral represents its volume. Sol:

37. Chapter 11 Line, surface and volume integrals Ex: Evaluate the surface integral , where and S is the open surface of the hemisphere Sol: the total hemisphere surface is divided into two surfaces: for

38. Chapter 11 Line, surface and volume integrals • two-dimensional divergence theorem: Y tangent vector normal vector if the vector field is continuous and differentiable in R. R 2D 3D C X

39. Chapter 11 Line, surface and volume integrals • Green’s theorems: Two scalar functions and that are continuous and differentiable in some volume V bounded by a surface S for vector fields and (1) (2) (1)-(2)

40. Chapter 11 Line, surface and volume integrals • other related integral theorems: (1) Proof: is a constant vector and (2) Proof: set is a constant vector and

41. Chapter 11 Line, surface and volume integrals • physical application of divergence theorem: Ex: For a compressible fluid with time-varying position-dependent density and velocity , in which fluid is neither being created nor destroyed, show that Sol: for an arbitrary volume V, the conservation of mass is total mass continuity equation

42. Chapter 11 Line, surface and volume integrals • If a single source is located at the origin, the fluid radially at a rate , the velocity is • the flux across a sphere (center at origin) of radius is undefined at the origin ( has a singularity) except the origin • for any closed surface , that does not enclose the origin From the above, we conclude S enclose source S disclose source

43. Chapter 11 Line, surface and volume integrals *the divergence theorem is valid, by redefining a delta function three-dimensional Dirac delta function properties: if lies in V otherwise Ex: for any volume V containing the source at the origin Ex: if a source at and a sink at enclose source enclose sink enclose none, or enclose them both

44. Chapter 11 Line, surface and volume integrals 11.9 Stoke’s theorem and related theorems • for a previous theorem divide the surface S into small area with boundary and unit normal Stoke’s theorem Note:Stoke’s theorem involves a open surface. Divergence theorem involves a closed surface.

45. Chapter 11 Line, surface and volume integrals Ex: Given the vector field verify Stoke’s theorem for the hemispherical surface Sol: (1) for surface integral: (2) for line integral: The perimeter C is the circle

46. Chapter 11 Line, surface and volume integrals • two-dimensional Stoke’s theorem let and recover Green’s theorem • related integral theorems: (1) is a constant vector, by Stoke’s theorem

47. Chapter 11 Line, surface and volume integrals (2) is a constant vector, by Stoke’s theorem

48. Chapter 11 Line, surface and volume integrals Ex:From Ampere’s law derive Maxwell’s equation in the case where the currents are steady, i.e. Sol: Ampere’s law: Stoke’s theorem: S C