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1. Noida Institute of Engineering and Technology, Greater Noida Vector Calculus B N Tripathi Department of Mathematics 2/14/2023 1 Unit: 4 Engineering mathematics II (AAS0203) Course Details (B. Tech. I st / II nd Sem ) NIET Greater Noida Engg Mathematics-II AAS0203

2. INTRODUCTION In this unit, vector differential calculus is considered, which extends the basic concepts of (ordinary) differential calculus to vector functions, by introducing derivative of a vector function and the new concepts of gradient, divergence and curl. Vector integral calculus: vector integral calculus extends the concept of (ordinary) integral calculus to vector functions. It has applications in fluid flow, design of underwater transmission cables, heat flow in stars, study of satellites. In this unit we consider three important integral theorem green’s theorem. Stokes's theorem states that the circulation of a vector field around the boundary of a surface in space equals the integral of the normal component of the curl of the field over the surface. Gauss divergence theorem ,which is important in electricity, magnetism and fluid flow. 2/14/2023 2 Content NIET Greater Noida Engg Mathematics-II AAS0203

3. Vector differentiation : Gradient Curl and divergence and their Physical interpretation Directional derivative Tangent and Normal planes Vector Integration: Line Integral Surface integral Volume integral Gauss’s Divergence theorem Green’s theorem Stokes's theorem and their applications 2/14/2023 3 Topics Covered NIET Greater Noida Engg Mathematics-II AAS0203

4. 2/14/2023 4 Course Objective The Objective of this course is to familiarize the engineering students with techniques of solving Ordinary Differential Equations, Fourier series expansion, Laplace Transform and vector calculus and its application in real world. It aims to equip the students with adequate knowledge of mathematics that will enable them in formulating problems and solving problems analytically. NIET Greater Noida Engg Mathematics-II AAS0203

5. 2/14/2023 5 Objective of Unit Calculate the gradient, curl and divergence, directional derivative, tangent and normal planes of a vector field. Set up and evaluate line integrals ,surface integrals, volume integrals of scalar functions or vector fields along curves …………the flux of a vector field through a surface. Set up and evaluate integrals over parametric surfaces. Set up and evaluate Gauss’s divergence theorem, Green’s theorem, Stokes's theorem. NIET Greater Noida Engg Mathematics-II AAS0203

6. 2/14/2023 6 Course Outcome Course Outcomes 2020-21 B.Tech . Ist Year -second semester Course Name: Mathematics-II CO1 Understand the concept of differentiation and apply for solving differential equations CO2 Understand the concept of convergence of sequence and series. Also evaluate Fourier series NIET Greater Noida Engg Mathematics-II AAS0203

7. 2/14/2023 7 Course Outcome CO3 Understand the basic idea of Laplace transform and apply for ordinary differential equations. CO4 Remember the concept of vector and apply for directional derivatives, tangent and normal planes. Also evaluate line, surface and volume integrals CO5 Understand the basic concept Proportion & Partnership, Problem of ages, Allegation & Mixture, Direction, Blood relation , Simple & Compound interest NIET Greater Noida Engg Mathematics-II AAS0203

8. 2/14/2023 8 CO-PO and PSO Mapping CO-PO and PSO Mapping 2020-21 B.Tech . Ist Year -Second semester Course Name: Mathematics-II CO PO-1 PO-2 PO-3 PO-4 PO-5 PO-6 PO-7 PO-8 PO-9 PO-10 PO-11 PO-12 CO1 3 3 3 3 3 1 2 - - 2 2 3 CO2 3 3 3 2 2 - - - - 2 2 1 CO3 3 2 3 2 3 1 - - - 2 2 2 CO4 3 2 3 2 3 1 1 - - 2 - 3 CO5 1 1 1 1 1 - - - - 2 - 2 Mean 2.6 2.2 2.6 2 2.4 1.0 1.5 - - 2 2 2.2 NIET Greater Noida Engg Mathematics-II AAS0203

9. 2/14/2023 9 Gradient Divergence Curl Physical interpretation Prerequisite and Recap NIET Greater Noida Engg Mathematics-II AAS0203

10. 2/14/2023 10 Objective of Topic Understand the parametric equations of curves and surfaces. Differentiate vector functions of a single variable. Undrstand and be able to find the unit tangent vector , the unit principal normal, Gradient, Curl, Divergence and Directional derivative. NIET Greater Noida Engg Mathematics-II AAS0203

11. 2/14/2023 11 Vector Differentiation NIET Greater Noida Engg Mathematics-II AAS0203

12. 2/14/2023 12 Point Function CO4 A variable quantity whose value at any point in a region of space depends upon the position of the point is called the point function. There are two types of point function Scalar point function Examples: The temperature distribution in a medium, Distribution of atmospheric pressure in space Vector point function Examples: The velocity of a moving fluid at any instant and gravitational force etc. NIET Greater Noida Engg Mathematics-II AAS0203

13. 2/14/2023 13 Gradient of a scalar field CO4 The gradient of a scalar function f ( x 1 , x 2 , x 3 , ..., x n ) is denoted by ∇ f or where ∇ (the nabla symbol) denotes the vector differential operator, del. The notation "grad(f)" is also commonly used for the gradient. In 3-dimensional cartesian coordinate system it is defined by: NIET Greater Noida Engg Mathematics-II AAS0203

14. 2/14/2023 14 Geometrical interpretation of gradient CO4 The gradient of scalar field f is a vector normal to the surface f(x, y, z)=c and has magnitude equal to the rate of change of f along this normal. NIET Greater Noida Engg Mathematics-II AAS0203

15. 2/14/2023 15 Gradient CO4 Q.1 Find gradφ when φ is given by ∅ = 3x 2 y-y 3 z 2 at the point (1,-2,1). [A.K.T.U 2019-2020] Solution:- gradφ = (3x 2 y-y 3 z 2 ) = ( 3x 2 y-y 3 z 2 ) + (3x 2 y-y 3 z 2 ) + ( 3x 2 y-y 3 z 2 ) = (6xy) + (3x 2 - 3y 2 z 2 ) + (-2y 3 z) = -12 -9 -16 at the point (1,-2,-1) NIET Greater Noida Engg Mathematics-II AAS0203 y 2

16. 2/14/2023 16 FAQ on Gradient CO4 Find grad f when f is given by f=3x 2 y-y 3 z at the point (1, -2, -1). [UPTU 2007] Ans : You Tube link: https://youtu.be/IwgqKjA6wko Show that is constant vector. [UPTU 2008] You Tube Link : https://youtu.be/E6FJrHFJ0bs NIET Greater Noida Engg Mathematics-II AAS0203

17. 2/14/2023 17 FAQ on Gradient CO4 I f u = x + y + z , v = x 2 + y 2 + z 2 and w = x y + y z + z x prove that the grad u, grad v, grad w are coplanar vectors. [UKTU 2010, UPTU 2015] You Tube Link : https://youtu.be/j36lJKSJMQk 4 . Find a unit vector normal to the surface x 2 y+2xz = 4 at the point (2, -2, 3). Ans :(1/√14)(-i+3j+2k) You Tube Link: https://youtu.be/DhwMOrl6Q9g NIET Greater Noida Engg Mathematics-II AAS0203

18. 2/14/2023 18 Find the angle between the surfaces x 2 + y 2 +z 2 = 9 and z = x 2 +y 2 – 3 at the point (2, -1, 2). Ans : You Tube Link: https://youtu.be/d4OyeuRTZNA What is the greatest rate of increase of u = xyz 2 at (1, 0, 3). Ans : 9 You Tube Link: https://youtu.be/U2-jCP-L0Lo FAQ on Gradient …CO4 NIET Greater Noida Engg Mathematics-II AAS0203

19. 2/14/2023 19 In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. Divergence CO4 NIET Greater Noida Engg Mathematics-II AAS0203

20. 2/14/2023 20 Divergence CO4 If F = P i + Q j +R k is a vector field and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, the divergence of F is the function of three variables defined by: NIET Greater Noida Engg Mathematics-II AAS0203

21. 2/14/2023 21 In terms of the gradient operator T he divergence of F can be written symbolically as the dot product of and F: Divergence CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

22. 2/14/2023 22 Divergence CO4 Q .1 Find the divergence of the vector =(xyz) +(3x 2 y) + ( xz 2 _ y 2 z ) at the point (2,-1,1). Solution: div = (xyz) + ( 3x 2 y) + (xz 2_ y 2 z) = yz + 3x 2 + 2xz - y 2 = -1+12+4-1 =14 at (2,-1, 1 ) NIET Greater Noida Engg Mathematics-II AAS0203

23. 2/14/2023 23 Curl CO4 Definition : 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

24. 2/14/2023 24 Curl CO4 Q .1 Find the Curl of the vector = (xyz) + (3x 2 y) + ( xz 2 _ y 2 z ) at the point (2,-1,1). Solution : Curl = = (-2yz-0) + (xy-z 2 ) + (6xy-xz) = 2 -3 -14 at (2,-1, 1). NIET Greater Noida Engg Mathematics-II AAS0203

25. 2/14/2023 25 FAQ on divergence and curl CO4 If F = xz 3 i - 2x 2 yz j +2yz 4 k. Find the divergence and curl of the vector field F. [UPTU 2007] . Ans : div F = z 3 -2x 2 z+8yz 3, curl F = 2(x 2 y+z 4 ) i +3xz 2 j - 4 xyz k . You Tube Link: https://youtu.be/yq5olnzDCGc If F = x 2 y i - 2xy j +(y 2 -xy) k. Find the divergence and curl of the vector field F. [UPTU 2007] Ans : div F = z 3 -2x 2 z+8yz 3, curl F = 2(x 2 y+z 4 ) i +3xz 2 j - 4 xyz k . 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

26. 2/14/2023 26 Solenoidal and Irrotational Fields CO4 If curl F=0 then F is called an irrotational vector. If F is irrotational, then there exists a scalar point function ɸ such that F=∇ɸ where ɸ is called the scalar potential of F. The work done in moving an object from point A to B in an irrotational field is = ɸ(B)- ɸ(A). The curl signifies the angular velocity at any point is equal to the half the curl of the linear velocity at that point of the body. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

27. 2/14/2023 27 Solenoidal and Irrotational Fields CO4 The field with null divergence is called solenoidal , and the field with null-curl is called irrotational field. The divergence of the curl of any vector field A must be zero, i.e. ∇· (∇×A)=0 This shows that a solenoidal field can be expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

28. 2/14/2023 28 Solenoidal and Irrotational Fields CO4 Q.1 Show that vector =(x+3y ) i + (y-3z) j +(x-2z) k is solinoidal . Solution:- div f = (x+3y) + (y-3z) + (x-2z) = 1+1-2 =0 so given vector is solinoidal . 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

29. 2/14/2023 29 Solenoidal and Irrotational Fields CO4 Q.2 Show that vector = ( y+z ) i + ( z+x ) j +( x+y ) k is irrotational . Solution: v ecto r is irrotational if curl =0 curl = = i ( x+y ) - ( z+x ) -j ( x+y ) - ( y+z ) +k ( z+x ) - ( y+z ) = i (0) –j(0) +k(0) =0 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

30. 2/14/2023 30 Prove that (y 2 -z 2 +3yz-2x) i +(3xz+2xy) j +(3xy-2xz +2z) k is both solenoidal and irrotational . You Tube Link: https://youtu.be/nBYS24WTRcI A vector field is given by (x 2 +xy 2 ) i +(y 2 +x 2 y) j. Show that field is irrotational and find the scalar potential. Ans : x 3 /3 +y 3 /3 +x 2 y 2 /2 +c. You Tube Link: https://youtu.be/fsMouTxce_A FAQ on Solenoidal & Irrotational field CO4 NIET Greater Noida Engg Mathematics-II AAS0203

31. 2/14/2023 31 Directional derivative CO4 The directional derivative  of a scalar field f at a point P(x, y, z) in the direction of the unit vector a is given by NIET Greater Noida Engg Mathematics-II AAS0203

32. 2/14/2023 32 Directional derivative CO4 Q.1 Find the directional derivative of in the direction of the line Solution:- grad = i + j + k = (10xy + z 2 ) i + (5 x 2 -10yz)j + (-5 y 2 + 5zx) k NIET Greater Noida Engg Mathematics-II AAS0203

33. 2/14/2023 33 Directional derivative CO4 = i - 5 j at (1,1,1) = Directional derivative = grad =( i - 5j ) . ( ) = NIET Greater Noida Engg Mathematics-II AAS0203

34. 2/14/2023 34 Q.1 Find the directional derivative of φ at P(3, 1, 2) in the direction of the vector You Tube link : https://youtu.be/1wJwS5XdA-4 Ans: -9/49√14 FAQ Of Directional Derivative CO4 NIET Greater Noida Engg Mathematics-II AAS0203

35. 2/14/2023 35 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

36. 2/14/2023 36 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

37. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 37 MCQ s (4.1) CO4 Q.1 Find the directional derivative of in the direction of at the point(1,1). [M.T.U.(SUM) 2011] Ans : 8 Q.2 Find the tangent to the curve at the point t = 1. at the point t = 1. Ans : Q.3 Show that the vector is solinoidal .

38. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 38 MCQ s Continued…CO4 Q.4 If ,find . [M.T.U.2012] Ans : 0 Q.5 If and are irrotational , prove that will be solinoidal . [M.T.U.2013; 2011]

39. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 39 Weekly Assignment 4.1 CO4 at t he point . Q.2 Determine the value of constants a, b, c if is irrotational . Q.1 Find the directional derivative of P(1,1,1) in the direction of the line . [GBTU 2010] Ans : 35/3 [AKTU. 2017]

40. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 40 Weekly Assignment 4.1 Continued..CO4 Q.3 Show that the vector field where = is irrotational and find a scalar such that [MTU 2013; GBTU 2010] Ans : Q.4 Find the angle between the normal to the surfaces at the point(-1, -1, 2) and (4, 1, -1). Ans : Q.5 Show that the vector field F = is irrotational as well as solenoidal . Find the scalar potential. [UPTU 2006] Ans : .

41. 2/14/2023 41 Vector Integration NIET Greater Noida Engg Mathematics-II AAS0203

42. 2/14/2023 42 Objective of Topic After studying this topic ,student will be able to caiculate the Line integral , Surface integral, Volume integral and Gauss’s divergence theorem ,Green’s theorem,Stoke’s theorem . NIET Greater Noida Engg Mathematics-II AAS0203

43. 2/14/2023 43 Line Integral of a Vector Field CO4 NIET Greater Noida Engg Mathematics-II AAS0203 Any integral which to be evaluated along a curve is called a line integral. The line integral along of vector F along C and is denoted by

44. 2/14/2023 44 Work done by a force CO4 Let F represent the force acting on a particle moving along an arc AB. The work done during a small displacement Therefore the total work done by F during the small displacement from A to B is given by NIET Greater Noida Engg Mathematics-II AAS0203

45. 2/14/2023 45 Work done by a force CO4 Q.1 Find the total work done by a force moving a point from (0, 0) to (a, b) by the lines x = 0, x = a, y = 0, y = b. [ UPTU 2006 ] Solution: = dx i + dy j + dz k . = ( + ) dx -2xy dy Total work done = = + NIET Greater Noida Engg Mathematics-II AAS0203

46. 2/14/2023 46 Work done by a force CO4 = dx = = = -ab 2 So Total work done = + -ab 2 NIET Greater Noida Engg Mathematics-II AAS0203

47. 2/14/2023 47 FAQ Of Work done by a force CO4 Q.1 Find the total work done by a force moving a point along t h e line from (1,2,3) to (3,5,7) . You tube link: https://youtu.be/xH4lJhmwJPk Q.2 Find the total work done by a force moving a point from (0, 0) to (a, b) by the lines x = 0, x = a, y = 0, y = b. [UPTU 2006] You tube link: https://youtu.be/2SB3IVCwW1w NIET Greater Noida Engg Mathematics-II AAS0203

48. 2/14/2023 48 Surface Integral CO4 Any integral which to be evaluated over a surface is called a surface integral. The surface integral over a surface is denoted by In order to evaluate the surface integral, it is convenient to express them as a double integrals taken over the orthogonal projection of S on one of the coordinates planes. Hence – NIET Greater Noida Engg Mathematics-II AAS0203

49. 2/14/2023 49 Example 1 surface integral CO4 Evaluate , where S is the portion of y = 2 x 2 + 1 in the first octant bounded by x = 0, x = 2, z = 4 and z = 8 . Solution The projection graph on the xz -plane is shown in Fig 1. NIET Greater Noida Engg Mathematics-II AAS0203

50. 2/14/2023 50 Example 1 (2) CO4 Let y = g ( x, z ) = 2 x 2 + 1. Since g x ( x , z ) = 4 x and g z ( x , z ) = 0, then NIET Greater Noida Engg Mathematics-II AAS0203

51. 2/14/2023 51 FAQ CO4 Q.1 Evaluate and S is the surface of the plane 2x + y +2z =6 in the first octant. Ans : 36. Q.2 Evaluate and S is the surface of the cylinder x 2 + y 2 = 16 included in the first octant between z = 0 to z = 5. Ans : 90 NIET Greater Noida Engg Mathematics-II AAS0203

52. 2/14/2023 52 Volume integral CO4 Let F be a vector point function and volume V enclosed by a closed surface Then the volume integral is given by NIET Greater Noida Engg Mathematics-II AAS0203

53. 2/14/2023 53 Volume integral CO4 Q.1 If = (2x 2 – 3z) i -2xy j -4xk, then evaluate dV , where V is bounded by the planes x=0 ,y=0, z=0, and 2x+2y+z=4. Solution:- = (2x 2 – 3z) + (-2xy) + (-4xk) = 4x -2x =2x so dV = 2x dx dy dz = 2x dz dy dx NIET Greater Noida Engg Mathematics-II AAS0203

54. 2/14/2023 54 Volume integral CO4 = 2x dy dx = 2x (4-2x-2y) dy dx = dx = dx = 2x (2-x) 2 dx = 2 (4x -4x 2 + x 3 ) dx =2 dx = 2(8 -32/3 +4) =8/3 NIET Greater Noida Engg Mathematics-II AAS0203

55. 2/14/2023 55 FAQ On Volume integralCO4 Q.1 If F = 2z i – x j + y k evaluate where V is the region bounded by the surfaces x = 0, y = 0, x = 2, y = 4, z = x 2 , z = 2. Ans : (32/15) (3 i + 5 k ) Q.2 Evaluate dV , w h ere = 45 x 2 y and V is t h e closed region bounded by the planes 4x+2y+z=8, x=0, y=0,z=0. NIET Greater Noida Engg Mathematics-II AAS0203

56. 2/14/2023 56 IF P , Q ,  P / y ,  Q / x are continuous on R , which is bounded by a simply closed curve C , then THEOREM Green’s Theorem in the Plane NIET Greater Noida Engg Mathematics-II AAS0203

57. 2/14/2023 57 Example 1 CO4 2/14/2023 Evaluate where C is shown in Fig 2. NIET Greater Noida Engg Mathematics-II AAS0203

58. 2/14/2023 58 Example 1 (2) CO4 2/14/2023 Solution If P ( x , y ) = x 2 – y 2 , Q ( x , y ) = 2 y – x , then and Thus NIET Greater Noida Engg Mathematics-II AAS0203

59. 2/14/2023 59 Example 2 (2) CO4 2/14/2023 Solution We have P ( x , y ) = x 5 + 3 y and then Hence Since the area of this circle is 4 , we have NIET Greater Noida Engg Mathematics-II AAS0203

60. 2/14/2023 60 Example 2 CO4 2/14/2023 Evaluate where C is the circle ( x – 1) 2 + ( y – 5) 2 = 4 shown in Fig 3. NIET Greater Noida Engg Mathematics-II AAS0203

61. 2/14/2023 61 Example 2 (2) CO4 2/14/2023 Solution We have P ( x , y ) = x 5 + 3 y and then Hence Since the area of this circle is 4 , we have NIET Greater Noida Engg Mathematics-II AAS0203

62. 2/14/2023 62 Example 3 CO4 2/14/2023 Find the work done by F = (– 16 y + sin x 2 ) i + (4 e y + 3 x 2 ) j along C shown in Fig 4. NIET Greater Noida Engg Mathematics-II AAS0203

63. 2/14/2023 63 Example 3 (2) CO4 2/14/2023 Solution We have Hence from Green’s theorem In view of R, it is better handled in polar coordinates, since R : NIET Greater Noida Engg Mathematics-II AAS0203

64. 2/14/2023 64 Example 3 (3) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

65. 2/14/2023 65 FAQ on Green’s theorem CO4 2/14/2023 Use Green’s theorem evaluate the integral where C is the square formed by the lines. Ans : 0 [MTU 2011]. Using Green’s theorem find the area of the region in the first quadrant bounded by the curves Ans log 2 [UPTU 2009] NIET Greater Noida Engg Mathematics-II AAS0203

66. 2/14/2023 66 THEOREM CO4 2/14/2023 Let D be a closed and bounded region on 3-space with a piecewise smooth boundary S that is oriented outward. Let F ( x, y, z ) = P ( x, y, z ) i + Q ( x, y, z ) j + R ( x, y, z ) k be a vector field for which P, Q , and R are continuous and have continuous first partial derivatives in a region of 3-space containing D . Then THEOREM Divergence Theorem NIET Greater Noida Engg Mathematics-II AAS0203

67. 2/14/2023 67 Example 1 CO4 2/14/2023 Let D be the region bounded by the hemisphere Solution The closed region is shown in Fig 5. NIET Greater Noida Engg Mathematics-II AAS0203

68. 2/14/2023 68 Example1 (2) CO4 2/14/2023 Fig 5. NIET Greater Noida Engg Mathematics-II AAS0203

69. 2/14/2023 69 Example 1 (2) CO4 2/14/2023 Triple Integral: Since F = x i + y j + z k , we see div F = 3. Hence Surface Integral: We write  S =  S1 +  S2 , where S 1 is the hemisphere and S 2 is the plane z = 1. If S 1 is a level surfaces of g ( x, y ) = x 2 + y 2 + ( z – 1) 2 , then a unit outer normal is NIET Greater Noida Engg Mathematics-II AAS0203

70. 2/14/2023 70 Example 1 (3) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

71. 2/14/2023 71 Example 1 (4) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

72. 2/14/2023 72 Example 2 CO4 2/14/2023 IF F = xy i + y 2 z j + z 3 k , evaluate  S (F  n ) dS , where S is the unit cube defined by 0  x  1, 0  y  1, 0  z  1. Solution We see div F =   F = x + 2 yz + 3 z 2 . Then NIET Greater Noida Engg Mathematics-II AAS0203

73. 2/14/2023 73 Example 2 (2) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

74. 2/14/2023 74 FAQ on Gauss Divergence Theorem CO4 2/14/2023 Verify the divergence theorem for taken over the cube bounded by the planes x = 0, x = 1. y = 0, y = 1, z = 0, z = 1. [UPTU 2009] Use divergence theorem to evaluate the surface integral where S is the portion of the plane x + 2y +3 z = 6 which lies in the first quadrant. NIET Greater Noida Engg Mathematics-II AAS0203

75. 2/14/2023 75 Stokes’s Theorem CO4 2/14/2023 Vector Form of Green’s Theorem If F ( x , y ) = P ( x , y ) i + Q ( x , y ) j , then Thus, Green’s Theorem can be written as NIET Greater Noida Engg Mathematics-II AAS0203

76. 2/14/2023 76 THEOREM CO4 2/14/2023 Let S be a piecewise smooth orientable surface bounded by a piecewise smooth simple closed curve C . Let F ( x , y , z ) = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k be a vector field for which P , Q , R , are continuous and have continuous first partial derivatives in a region of 3-space containing S . If C is traversed in the positive direction, then where n is a unit normal to S in the direction of the orientation of S . THEOREM NIET Greater Noida Engg Mathematics-II AAS0203

77. 2/14/2023 77 Example 1 CO4 2/14/2023 Let S be the part of the cylinder z = 1 – x 2 for 0  x  1, −2  y  2. Verify Stokes’ theorem if F = xy i + yz j + xz k . Fig 6 NIET Greater Noida Engg Mathematics-II AAS0203

78. 2/14/2023 78 Example 1 (2) CO4 2/14/2023 Solution See Fig 6. Surface Integral: From F = xy i + yz j + xz k we find NIET Greater Noida Engg Mathematics-II AAS0203

79. 2/14/2023 79 Example 1 (3) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

80. 2/14/2023 80 Example 1 (4) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

81. 2/14/2023 81 Example 1 (5) CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203

82. 2/14/2023 82 Example 2 CO4 2/14/2023 Evaluate where C is the trace of the cylinder x 2 + y 2 = 1 in the plane y + z = 2. Orient C counterclockwise as viewed from above. See Fig 6.. NIET Greater Noida Engg Mathematics-II AAS0203

83. 2/14/2023 83 Example 2 (1) CO4 2/14/2023 Fig 7. NIET Greater Noida Engg Mathematics-II AAS0203

84. 2/14/2023 84 Example 2 (2) CO4 2/14/2023 Solution The given orientation of C corresponding to an upward orientation of the surface S. NIET Greater Noida Engg Mathematics-II AAS0203

85. 2/14/2023 85 Example 2 (3) CO4 2/14/2023 Thus if g ( x, y, z ) = y + z – 2 = 0 defines the plane, then the upper normal is NIET Greater Noida Engg Mathematics-II AAS0203

86. 2/14/2023 86 FAQ on Stokes’s Theorem CO4 2/14/2023 Use Stoke’s theorem to evaluate where C is boundary of the triangle with vertices (2, 0 ,0), (0, 3, 0) and (0, 0, 6) oriented in anticlockwise direction. Ans : 15 . [UPTU 2009]. 2. Verify Stoke’ s theorem for function over the unclosed surface of cylinder bounded by the plane z = h and open at the end z = 0. [MTU 2011]. NIET Greater Noida Engg Mathematics-II AAS0203

87. 2/14/2023 87 Faculty Video Links, YouTube & NPTEL Video Links and Online Courses Details CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 Self Made Video Link: Vector Calculus Serial No. Topic You Tube video lecture link 1 Gradient of a scalar function Lecture 1 https://youtu.be/IwgqKjA6wko 2 Gradient of a scalar function Lecture 2 https://youtu.be/d4OyeuRTZNA 3 Gradient of a scalar function Lecture 3 https://youtu.be/j36lJKSJMQk 4 Gradient of a scalar function Lecture 4 https://youtu.be/E6FJrHFJ0bs 5 Greatest rate of increase https://youtu.be/U2-jCP-L0Lo

88. 2/14/2023 88 Faculty Video Links, YouTube & NPTEL Video Links and Online Courses Details CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 Self Made Video Link: Vector Calculus Serial No. Topic You Tube video lecture link 6 Unit vector normal to surface https://youtu.be/DhwMOrl6Q9g 7 Divergence and curl Lecture 1 https://youtu.be/DhwMOrl6Q9g 8 Solinoidal and Irrotational Lecture 1 https://youtu.be/fsMouTxce_A 9 Solinoidal and Irrotational Lecture 2 https://youtu.be/nBYS24WTRcI

89. 2/14/2023 89 Faculty Video Links, YouTube & NPTEL Video Links and Online Courses Details Continued…CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 Self Made Video Link: Vector Calculus Serial No. Topic You Tube video lecture link 10 Directional derivative Lecture1 https://youtu.be/yq5olnzDCGc 11 Directional derivative Lecture2 https://youtu.be/1wJwS5XdA-4 12 Work done by a force Lecture 1 https://youtu.be/2SB3IVCwW1w 13 Work done by a force Lecture 2 https://youtu.be/xH4lJhmwJPk

90. 2/14/2023 90 Faculty Video Links, YouTube & NPTEL Video Links and Online Courses Details Continued…CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 Youtube /other Video Links : Vector Calculus Serial No. Topic You Tube video lecture link 10 Line Integral https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals-vectors/v/line-integrals-and-vector-fields 11 Surface Integral   https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/3d-flux/v/vector-representation-of-a-surface-integral

91. 2/14/2023 91 Faculty Video Links, YouTube & NPTEL Video Links and Online Courses Details Continued…CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 Youtube /other Video Links : Vector Calculus Serial No. Topic You Tube video lecture link Volume integral http://nucinkis-lab.cc.ic.ac.uk/HELM/workbooks/workbook_29/29_2_surface_vol_vec_ints.pdf Applications of Green’s Theorem & Stoke's Theorem https://www.youtube.com/watch?v=Mb6Yb-SGqio https://www.khanacademy.org/math/multivariable-calculus/greens-theorem-and-stokes-theorem/stokes-theorem/v/stokes-theorem-intuition

92. 2/14/2023 92 Faculty Video Links, YouTube & NPTEL Video Links and Online Courses Details Continued…CO4 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 You tube/other Video Links : Vector Calculus Serial No. Topic You Tube video lecture link 1 Gauss divergence Theorem https://www.youtube.com/watch?v=eSqznPrtzS4

93. 2/14/2023 93 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 1. Statement of Green’s Theorem is……… 2. Statement of Stoke’s Theorem is……. 3. Statement of Gauss’s Divergence Theorem is…….

94. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 94 MCQ s (4.3 )CO4 . Q.1 If is the velocity of a fluid particle, what does represent ? [GBTU 2010] Ans : Circulation Q.2 State Green's theorem for a plane region. [M.T.U.2012] Q.3 State Stokes's theorem (Relation between line and surface integrals). [UPTU 2007, 2008, 2009, UKTU 2012] Q.4 Write the statement of divergence theorem for a given vector field [G.B.T.U.2012]

95. 2/14/2023 95 Prerequisite and Recap NIET Greater Noida Engg Mathematics-II AAS0203 Gauss divergence theorem Green’s theorem Stokes’s theorem

96. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 96 Weekly Assignment 4.2 CO4 Q.1 Find total work done the by a forc -2xy in moving a point from (0, 0) to (a, b) along the rectangle bounded by the lines x = 0, x = a, y = 0 and y = b. [UPTU 2006] Ans : Q.2 Find the work done in moving a particle in the force field along the curve defined by from x = 0 to x =2. [MTU 2012] Ans : 16 Q.3 Evaluate , where and S is the surface of the plane in the first octant. [MTU 2012] Ans : 24

97. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 97 Weekly Assignment 4.2 Continued……CO4 Q.4 Evaluate by Green’s theorem where C is the rectangle with vertices (0, 0), (π, 0), (π, π/2), (0, π/2) and hence verify Green’s theorem. [UPTU (SUM) 2007] Ans : 2( Q.5 Verify Green's theorem in the plane for where C is closed curve of the region bounded by y = x and y = x 2 . [G.B.T.U.(SUM) 2010; U.P.T.U.(SUM) 2008]

98. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 98 Weekly Assignment 4.2 Continued …CO4 Q.6 Use Stokes's theorem to evaluate where C is boundary of the triangle with vertices (2, 0 ,0), (0, 3, 0) and (0, 0, 6) oriented in anticlockwise direction. [MTU 2013] Ans : 21 Q.7 By using divergence theorem find where S is the surface of the ellipsoid [MTU 2013] Ans :

99. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 99 Weekly Assignment 4.2 CO4 Q.8 Verify divergence theorem for taken over the region bounded by the cylinder [UPTU 2005] Ans : 84π Q.9 Verify the divergence theorem for taken over the rectangular parallelepiped [MTU 2013] Ans : abc (a + b + c)

100. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 100 Old Question Papers CO4 Q.1 Show that vectors (1, 6, 4), (0, 2, 3) and (0, 1, 2) are linearly independent. [A.K.T.U 2019-2020] Q.2 Show that the vector field F = is irrotational as well as solenoidal . Find the scalar potential. . [A.K.T.U 2019-2020] Q.3 Find the volume of the region bounded by the surface y= x 2 ,x=y 2 and the planes z = 0, z = 3. [A.K.T.U 2019-2020] .

101. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 101 Old Question Papers CO4 Q .4 Find the directional derivative of ∅ ( x,y,z )= x 2 yz +4zx 2 at 1, -2, 1) in the direction of 2ı̂− ȷ̂− 2k. Find also the greatest rate of increase of ∅. [A.K.T.U 2019-2020] Q.5 Find gradφ when φ is given by ∅ = 3x 2 y-y 3 z 2 at the point (1,-2,1). . [A.K.T.U 2019-2020] Q.6 Determine the value of constants a, b, c if is irrotational . [A.K.T.U2017] .

102. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 102 Expected Questions for University exam CO4 Q.1 If u = x + y + z, , w = yz + zx + xy , prove that grad u, grad v & grad w. Q.2 If then show that ( a) [G.B.T.U 2011] (b ) [G.B.T.U 2011] (c) [G.B.T.U 2011; U.P.T.U 2008]

103. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 103 Expected Questions For University Exam..CO4 Q.3 If , evaluate , where c is the arc of the from (0, 0) to (1, 2). Ans : -7/6. Q.4 Find ,where and S is the having centre at (3, -1, 2) and radius 3. [UPTU 2006] Ans : 108π

104. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 104 Expected Questions For University Exam..CO4 Q.5 If , then evaluate where V is bounded by the planes x = 0, y = 0, z = 0 and 2x + 2y + z = 4. Ans : 8/3 Q.6 Use Green’s theorem evaluate the integral where C is the square formed by the lines [MTU 2011; GBTU 2010]. Ans :

105. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 105 Expected Questions For University Exam..CO4 Q.7 Verify Stoke’ s theorem for function the surface S is the surface S is the surface of the region bounded by x = 0, y = 0, z = 0, 2x + y + 2z = 8 which is not included on xz – plane. [UPTU 2007] Q.8 Verify the divergence theorem for taken over the cube bounded by the planes [UPTU 2009; GBTU 2010] Ans : 3/2

106. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 106 Summary CO4 The grad, div, and curl in Cartesian coordinates are

107. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 107 Summary CO4 Three important integral identities are

108. 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 108 References CO4 (1) B V RAMANA (2) KREYSZIG (3) B S GREWAL (4) PETER O’NEIL

109. 2/14/2023 109 2/14/2023 NIET Greater Noida Engg Mathematics-II AAS0203 Thank You !