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Chapter 3

Chapter 3. 1. Line Integral Volume Integral Surface Integral Green’s Theorem Divergence Theorem (Gauss’ Theorem) Stokes’ Theorem. z. 4 . . . y. 3. 3. x. Example (Volume Integral). Solution.

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Chapter 3

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  1. Chapter 3 1. Line Integral • Volume Integral • Surface Integral • Green’s Theorem • Divergence Theorem (Gauss’ Theorem) • Stokes’ Theorem

  2. z 4    y 3 3 x Example (Volume Integral)

  3. Solution Since it is about a cylinder, it is easier if we use cylindrical polar coordinates, where

  4. A B O Line Integral Ordinary integral f (x) dx, we integrate along the x-axis. But for line integral, the integration is along a curve. f (s) ds = f (x, y, z) ds

  5. Scalar Field, V Integral If there exists a scalar field V along a curve C, then the line integral of Valong C is defined by

  6. Example

  7. Solution

  8. Exercise 2.6

  9. 2.8.2 Vector Field, Integral Let a vector field and The scalar product is written as

  10. Example 2.15

  11. Solution

  12. Exercise 2.7

  13. * Double Integral *

  14. 2.9 Volume Integral 2.9.1 Scalar Field, F Integral If V is a closed region and F is a scalar field in region V, volume integral F of V is

  15. z 2 y O 3 1 x Example 2.20 Scalar function F= 2 x defeated in one cubic that has been built by planes x= 0, x= 1, y= 0, y= 3, z= 0 and z= 2. Evaluate volume integral F of the cubic.

  16. Solution

  17. 2.9.2 Vector Field, Integral If V is a closed region and , vector field in region V, Volume integral of V is

  18. Example 2.21 Evaluate , where V is a region bounded by x= 0, y= 0, z= 0 and 2x+y+z= 2, and also given

  19. Solution If x =y= 0, plane 2x + y + z = 2 intersects z-axis at z= 2. (0,0,2) If x=z= 0, plane 2x +y+z= 2 intersects y-axis at y= 2. (0,2,0) If y=z = 0, plane 2x +y+z= 2 intersects x-axis at x = 1. (1,0,0)

  20. z 2 2x + y + z = 2 O y 2 y = 2 (1 x) 1 x • We can generate this integral in 3 steps : • Line Integral from x=0 to x=1. • Surface Integral from line y= 0 to line y= 2(1-x). • Volume Integral from surface z= 0 to surface 2x + y + z= 2 that is z= 2 (1-x) -y

  21. Therefore,

  22. z 4    y 3 3 x Example 2.22 Evaluate where and V is region bounded by z = 0, z = 4 and x2 + y2 = 9

  23. Using polar coordinate of cylinder, ; ; ; where

  24. Therefore,

  25. Exercise 2.8

  26. 2.10 Surface Integral 2.10.1 Scalar Field, V Integral If scalar field V exists on surface S, surface integral Vof S is defined by where

  27. Example 2.23 Scalar field V=x y z defeated on the surface S : x2+y2= 4 between z= 0 and z= 3 in the first octant. Evaluate Solution Given S : x2+y2= 4 , so grad S is

  28. Also, Therefore, Then,

  29. Surface S : x2+y2= 4 is bounded by z= 0 and z= 3 that is a cylinder with z-axis as a cylinder axes and radius, So, we will use polar coordinate of cylinder to find the surface integral. z 3 O y 2  2 x

  30. Polar Coordinate for Cylinder where (1st octant) and

  31. Using polar coordinate of cylinder, From

  32. Therefore,

  33. Exercise 2.9

  34. 2.10.2 Vector Field, Integral If vector field defeated on surface S, surface integral of S is defined as

  35. Example 2.24

  36. z 3 O y 3 3 x Solution

  37. Using polar coordinate of sphere,

  38. Exercise 2.9

  39. 2.11 Green’s Theorem If c is a closed curve in counter-clockwise on plane-xy, and given two functions P(x, y) and Q(x, y), where S is the area of c.

  40. y x2 + y2 = 22 2 C2  C3 x O 2 C1 Example 2.25 Solution

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