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1 Chapter 2 Vector Calculus 1.Elementary 2.Vector Product 3.Differentiation of Vectors 4.Integration of Vectors 5.Del Operator or Nabla (Symbol ) 6.Polar Coordinates
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2 Chapter 2 Continued 7.Line Integral 8.Volume Integral 9.Surface Integral 10.Green’s Theorem 11.Divergence Theorem (Gauss’ Theorem) 12.Stokes’ Theorem
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3 2.1 Elementary Vector Analysis Definition 2.1 (Scalar and vector) Vector is a directed quantity, one with both magnitude and direction. For instance acceleration, velocity, force Scalar is a quantity that has magnitude but not direction. For instance mass, volume, distance
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4 We represent a vector as an arrow from the origin O to a point A. The length of the arrow is the magnitude of the vector written as or. O A or O A
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5 2.1.1 Basic Vector System Unit vectors,, Perpendicular to each other In the positive directions of the axes have magnitude (length) 1
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6 Define a basic vector system and form a right-handed set, i.e
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7 2.1.2 Magnitude of vectors Let P = ( x, y, z ). Vector is defined by with magnitude (length)
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8 2.1.3 Calculation of Vectors 1. Vector Equation Two vectors are equal if and only if the corresponding components are equals
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9 2. Addition and Subtraction of Vectors 3. Multiplication of Vectors by Scalars
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10 Example 2.1
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11 2.2Vector Products 1) Scalar Product (Dot product) 2) Vector Product (Cross product)
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12 3) Application of Multiplication of Vectors a)Given 2 vectors and, projection onto is defined by b) The area of triangle comp b a
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13 c) The area of parallelogram d) The volume of tetrahedrone e) The volume of parallelepiped A A
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14 Example 2.3
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15 2.4 Vector Differential Calculus Let A be a vector depending on parameter u, The derivative of A(u) is obtained by differentiating each component separately,
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16 The n th derivative of vector is given by The magnitude of is
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17 Example 2.4
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18 Example 2.5 The position of a moving particle at time t is given by x 4t + 3, y t 2 + 3t, z t 3 + 5t 2. Obtain The velocity and acceleration of the particle. The magnitude of both velocity and acceleration at t 1.
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19 Solution The parameter is t, and the position vector is The velocity is given by The acceleration is
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20 At t 1, the velocity of the particle is and the magnitude of the velocity is
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21 At t 1, the acceleration of the particle is and the magnitude of the acceleration is
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22 2.4.1 Differentiation of Two Vectors If both and are vectors, then
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23 2.4.2 Partial Derivatives of a Vector If vector depends on more than one parameter, i.e
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24 Partial derivative of with respect to is given by e.t.c.
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25 Example 2.6
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26 Exercise 2.1
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27 2.5 Vector Integral Calculus The concept of vector integral is the same as the integral of real-valued functions except that the result of vector integral is a vector.
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28 Example 2.7
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29 Exercise 2.2
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30 2.6 Del Operator Or Nabla (Symbol ) Operator is called vector differential operator, defined as
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31 2.6.1 Grad (Gradient of Scalar Functions) If x, y, z is a scalar function of three variables and is differentiable, the gradient of is defined as
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32 Example 2.8
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34 Exercise 2.3
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35 Solution
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36 2.6.1.1 Grad Properties If A and B are two scalars, then
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37 2.6.2 Directional Derivative
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38 Example 2.9
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39 Solution Directional derivative of in the direction of
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42 2.6.3 Unit Normal Vector Equation (x, y, z) constant is a surface equation. Since (x, y, z) constant, the derivative of is zero; i.e.
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43 This shows that when (x, y, z) constant, Vector grad is called normal vector to the surface (x, y, z) constant y ds grad z x
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44 Unit normal vector is denoted by Example 2.10 Calculate the unit normal vector at (-1,1,1) for 2 yz xz xy 0.
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45 Solution Given 2 yz xz xy 0. Thus
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46 2.6.4 Divergence of a Vector
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47 Example 2.11
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48 Exercise 2.4
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49 Remarks
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50 2.6.5 Curl of a Vector
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51 Example 2.12
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52 Solution
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53 Exercise 2.5
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54 Answer Remark
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55 2.7 Polar Coordinates Polar coordinate is used in calculus to calculate an area and volume of small elements in easy way. Lets look at 3 situations where des Cartes Coordinate can be rewritten in the form of Polar coordinate.
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56 2.7.1 Polar Coordinate for Plane (r, θ) x ds y d d
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57 2.7.2 Polar Coordinate for Cylinder ( , , z ) x y z dv z ds
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58 2.7.3 Polar Coordinate for Sphere (r, y x r z
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59 Example 2.13 (Volume Integral) x z y 4 3 3
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60 Solution Since it is about a cylinder, it is easier if we use cylindrical polar coordinates, where
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61 2.8 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 A O B
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62 2.8.1 Scalar Field, V Integral If there exists a scalar field V along a curve C, then the line integral of V along C is defined by
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63 Example 2.14
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64 Solution
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66 Exercise 2.6
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67 2.8.2 Vector Field, Integral Let a vector field and The scalar product is written as
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69 Example 2.15
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70 Solution
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73 Exercise 2.7
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74 * Double Integral *
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78 2.9Volume 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
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79 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. z x y 3 O 2 1
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80 Solution
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81 2.9.2 Vector Field, Integral If V is a closed region and, vector field in region V, Volume integral of V is
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82 Evaluate, where V is a region bounded by x 0, y 0, z 0 and 2 x y z 2, and also given Example 2.21
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83 If x y 0, plane 2 x y z 2 intersects z -axis at z 2. (0,0,2) If x z 0, plane 2 x y z 2 intersects y -axis at y 2. (0,2,0) If y z 0, plane 2 x y z 2 intersects x -axis at x = 1. (1,0,0) Solution
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84 We can generate this integral in 3 steps : 1.Line Integral from x 0 to x 1. 2.Surface Integral from line y 0 to line y 2(1 x ). 3.Volume Integral from surface z 0 to surface 2 x y z 2 that is z 2 (1 x ) y z x y 2 O 2 1 2x + y + z = 2 y = 2 (1 x)
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85 Therefore,
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86 Example 2.22 Evaluate where and V is region bounded by z = 0, z = 4 and x 2 + y 2 = 9 x z y 4 3 3
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87 ;; ; where Using polar coordinate of cylinder,
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88 Therefore,
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89 Exercise 2.8
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90 2.10 Surface Integral 2.10.1 Scalar Field, V Integral If scalar field V exists on surface S, surface integral V of S is defined by where
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91 Example 2.23 Scalar field V x y z defeated on the surface S : x 2 y 2 4 between z 0 and z 3 in the first octant. Evaluate Solution Given S : x 2 y 2 4, so grad S is
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92 Also, Therefore, Then,
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93 Surface S : x 2 y 2 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. x z y 2 2 3 O
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94 Polar Coordinate for Cylinder where(1 st octant) and
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95 Using polar coordinate of cylinder, From
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96 Therefore,
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97 Exercise 2.9
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98 2.10.2 Vector Field, Integral If vector field defeated on surface S, surface integral of S is defined as
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99 Example 2.24
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100 Solution x z y 3 3 3 O
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103 Using polar coordinate of sphere,
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105 Exercise 2.9
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106 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.
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107 Example 2.25 y 2 x 2 C3C3 C2C2 C1C1 O x 2 + y 2 = 2 2 Solution
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115 2.12 Divergence Theorem (Gauss’ Theorem) If S is a closed surface including region V in vector field
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116 Example 2.26
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117 Solution x z y 2 2 4 O S3S3 S4S4 S2S2 S1S1 S5S5
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128 2.13Stokes’ Theorem If is a vector field on an open surface S and boundary of surface S is a closed curve c, therefore
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129 Example 2.27 Surface S is the combination of
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130 Solution z y x 3 4 O S3S3 C2C2 S2S2 C1C1 S1S1 3
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131 We can also mark the pieces of curve C as C 1 :Perimeter of a half circle with radius 3. C 2 :Straight line from (-3,0,0) to (3,0,0). Let say, we choose to evaluate first. Given
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132 So,
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133 By integrating each part of the surface,
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134 and Then,
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135 By using polar coordinate of cylinder ( because is a part of the cylinder),
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136 Therefore, Also,
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138 (ii) For surface, normal vector unit to the surface is By using polar coordinate of plane,
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140 (iii) For surface S 3 : y = 0, normal vector unit to the surface is dS = dxdz The integration limits : So,
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