MATHEMATICS-IMATHEMATICS-I

CONTENTSCONTENTS Ordinary Differential Equations of First Order and First DegreeOrdinary Differential Equations of First Order and First Degree Linear Differential Equations of Second and Higher OrderLinear Differential Equations of Second and Higher Order Mean Value TheoremsMean Value Theorems Functions of Several VariablesFunctions of Several Variables Curvature, Evolutes and EnvelopesCurvature, Evolutes and Envelopes Curve TracingCurve Tracing Applications of IntegrationApplications of Integration Multiple IntegralsMultiple Integrals Series and SequencesSeries and Sequences Vector Differentiation and Vector OperatorsVector Differentiation and Vector Operators Vector IntegrationVector Integration Vector Integral TheoremsVector Integral Theorems Laplace transformsLaplace transforms

TEXT BOOKSTEXT BOOKS A text book of Engineering Mathematics, Vol-I A text book of Engineering Mathematics, Vol-I

T.K.V.Iyengar, B.Krishna Gandhi and Others, T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & CompanyS.Chand & Company

A text book of Engineering Mathematics, A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book LinksC.Sankaraiah, V.G.S.Book Links

A text book of Engineering Mathematics, Shahnaz A A text book of Engineering Mathematics, Shahnaz A Bathul, Right PublishersBathul, Right Publishers

A text book of Engineering Mathematics, A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi PublicationsRao, Deepthi Publications

REFERENCESREFERENCES

A text book of Engineering Mathematics, A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw HillB.V.Raman, Tata Mc Graw Hill

Advanced Engineering Mathematics, Irvin Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd.Kreyszig, Wiley India Pvt. Ltd.

A text Book of Engineering Mathematics, A text Book of Engineering Mathematics, Thamson Book collectionThamson Book collection

UNIT-VIIUNIT-VII

CHAPTER-I:VECTOR DIFFERENTIATION CHAPTER-I:VECTOR DIFFERENTIATION AND VECTOR OPERATORSAND VECTOR OPERATORS

CHAPTER-II:VECTOR INTEGRATIONCHAPTER-II:VECTOR INTEGRATIONCHAPTER-III:VECTOR INTEGRAL CHAPTER-III:VECTOR INTEGRAL

THEOREMSTHEOREMS

UNIT HEADERUNIT HEADER

Name of the Course: B.TechName of the Course: B.TechCode No:07A1BS02Code No:07A1BS02Year/Branch: I Year Year/Branch: I Year

CSE,IT,ECE,EEE,ME,CIVIL,AEROCSE,IT,ECE,EEE,ME,CIVIL,AEROUnit No: VII Unit No: VII

No. of slides:23No. of slides:23

S. No.S. No. ModuleModule LectureLectureNo. No.

PPT Slide No.PPT Slide No.

11 Introduction, Introduction, Gradient, D.D, Gradient, D.D, Solenoidal and Solenoidal and irrotational vectorsirrotational vectors

L1-5L1-5 8-158-15

22 Vector integration, Vector integration, Line, Surface and Line, Surface and Volume integralsVolume integrals

L6-10L6-10 16-1916-19

33 Vector integral Vector integral theoremstheorems

L11-13L11-13 20-2320-23

UNIT INDEXUNIT INDEXUNIT-VII UNIT-VII

Lecture-1Lecture-1INTRODUCTIONINTRODUCTION

In this chapter, vector differential calculus is In this chapter, vector differential calculus is considered, which extends the basic concepts considered, which extends the basic concepts of differential calculus, such as, continuity and of differential calculus, such as, continuity and differentiability to vector functions in a simple differentiability to vector functions in a simple and natural way. Also, the new concepts of and natural way. Also, the new concepts of gradient, divergence and curl are introducted.gradient, divergence and curl are introducted.

ExampleExample: i,j,k are unit vectors.: i,j,k are unit vectors.

VECTOR DIFFERENTIAL VECTOR DIFFERENTIAL OPERATOROPERATOR

The vector differential operator The vector differential operator ∆ is defined as ∆ is defined as ∆=i ∂/∂x + j ∂/∂y + k ∂/∂z. This operator ∆=i ∂/∂x + j ∂/∂y + k ∂/∂z. This operator possesses properties analogous to those of possesses properties analogous to those of ordinary vectors as well as differentiation ordinary vectors as well as differentiation operator. We will define now some quantities operator. We will define now some quantities known as gradient, divergence and curl known as gradient, divergence and curl involving this operator.involving this operator.

∂ ∂/∂ /∂

Lecture-2Lecture-2GRADIENTGRADIENT

Let f(x,y,z) be a scalar point function of Let f(x,y,z) be a scalar point function of position defined in some region of space. Then position defined in some region of space. Then gradient of f is denoted by grad f or gradient of f is denoted by grad f or ∆f and is ∆f and is defined as defined as

grad f=i∂f/∂x + j ∂f/∂y + k ∂f/∂zgrad f=i∂f/∂x + j ∂f/∂y + k ∂f/∂z ExampleExample: If f=2x+3y+5z then grad : If f=2x+3y+5z then grad

f= 2i+3j+5kf= 2i+3j+5k

Lecture-3Lecture-3DIRECTIONAL DERIVATIVEDIRECTIONAL DERIVATIVE

The directional derivative of a scalar point The directional derivative of a scalar point function f at a point P(x,y,z) in the direction of function f at a point P(x,y,z) in the direction of g at P and is defined as grad g at P and is defined as grad g/|grad g|.grad fg/|grad g|.grad f

ExampleExample: The directional derivative of : The directional derivative of f=xy+yz+zx in the direction of the vector f=xy+yz+zx in the direction of the vector i+2j+2k at the point (1,2,0) is 10/3i+2j+2k at the point (1,2,0) is 10/3

Lecture-4Lecture-4DIVERGENCE OF A VECTORDIVERGENCE OF A VECTOR

Let f be any continuously differentiable vector Let f be any continuously differentiable vector point function. Then divergence of f and is point function. Then divergence of f and is written as div f and is defined aswritten as div f and is defined as

Div f = Div f = ∂f∂f11/∂x + j ∂f/∂x + j ∂f22/∂y + k ∂f/∂y + k ∂f33/∂z/∂z ExampleExample 1: The divergence of a vector 1: The divergence of a vector

2xi+3yj+5zk is 102xi+3yj+5zk is 10 ExampleExample 2: The divergence of a vector 2: The divergence of a vector

f=xyf=xy22i+2xi+2x22yzj-3yzyzj-3yz22k at (1,-1,1) is 9k at (1,-1,1) is 9

SOLENOIDAL VECTORSOLENOIDAL VECTOR

A vector point function f is said to be A vector point function f is said to be solenoidal vector if its divergent is equal to solenoidal vector if its divergent is equal to zero i.e., div f=0zero i.e., div f=0

ExampleExample 1: The vector 1: The vector f=(x+3y)i+(y-2z)j+(x-2z)k is solenoidal f=(x+3y)i+(y-2z)j+(x-2z)k is solenoidal vector.vector.

ExampleExample 2: The vector f=3y 2: The vector f=3y44zz22i+zi+z33xx22j-3xj-3x22yy22k is k is solenoidl vector.solenoidl vector.

Lecture-5Lecture-5CURL OF A VECTORCURL OF A VECTOR

Let f be any continuously differentiable vector Let f be any continuously differentiable vector point function. Then the vector function curl of point function. Then the vector function curl of f is denoted by curl f and is defined as f is denoted by curl f and is defined as curl f= curl f= ix∂f/∂x + jx∂f/∂y + kx∂f/∂zix∂f/∂x + jx∂f/∂y + kx∂f/∂z

ExampleExample 1: If f=xy 1: If f=xy22i +2xi +2x22yzj-3yzyzj-3yz22k then curl f k then curl f at (1,-1,1) is –i-2kat (1,-1,1) is –i-2k

ExampleExample 2: If r=xi+yj+zk then curl r is 0 2: If r=xi+yj+zk then curl r is 0

Lecture-6Lecture-6IRROTATIONAL VECTORIRROTATIONAL VECTOR

Any motion in which curl of the velocity Any motion in which curl of the velocity vector is a null vector i.e., curl v=0 is said to vector is a null vector i.e., curl v=0 is said to be irrotational. If f is irrotational, there will be irrotational. If f is irrotational, there will always exist a scalar function f(x,y,z) such that always exist a scalar function f(x,y,z) such that f=grad g. This g is called scalar potential of f.f=grad g. This g is called scalar potential of f.

ExampleExample: The vector : The vector f=(2x+3y+2z)i+(3x+2y+3z)j+(2x+3y+3z)k is f=(2x+3y+2z)i+(3x+2y+3z)j+(2x+3y+3z)k is irrotational vector.irrotational vector.

Lecture-7Lecture-7VECTOR INTEGRATIONVECTOR INTEGRATION

INTRODUCTIONINTRODUCTION: In this chapter we shall : In this chapter we shall define line, surface and volume integrals define line, surface and volume integrals which occur frequently in connection with which occur frequently in connection with physical and engineering problems. The physical and engineering problems. The concept of a line integral is a natural concept of a line integral is a natural generalization of the concept of a definite generalization of the concept of a definite integral of f(x) exists for all x in the interval integral of f(x) exists for all x in the interval [a,b][a,b]

WORK DONE BY A FORCE WORK DONE BY A FORCE

If F represents the force vector acting on a If F represents the force vector acting on a particle moving along an arc AB, then the particle moving along an arc AB, then the work done during a small displacement F.dr. work done during a small displacement F.dr. Hence the total work done by F during Hence the total work done by F during displacement from A to B is given by the line displacement from A to B is given by the line integral integral ∫F.dr∫F.dr

ExampleExample: If f=(3x: If f=(3x22+6y)i-14yzj+20xz+6y)i-14yzj+20xz22k along k along the lines from (0,0,0) to (1,0,0) then to (1,1,0) the lines from (0,0,0) to (1,0,0) then to (1,1,0) and then to (1,1,1) is 23/3and then to (1,1,1) is 23/3

Lecture-8Lecture-8SURFACE INTEGRALSSURFACE INTEGRALS

The surface integral of a vector point function F The surface integral of a vector point function F expresses the normal flux through a surface. If F expresses the normal flux through a surface. If F represents the velocity vector of a fluid then the represents the velocity vector of a fluid then the surface integral surface integral ∫F.n dS over a closed surface S ∫F.n dS over a closed surface S represents the rate of flow of fluid through the represents the rate of flow of fluid through the surface.surface.

ExampleExample:The value of ∫F.n dS where F=18zi-12j+3yk :The value of ∫F.n dS where F=18zi-12j+3yk and S is the part of the surface of the plane and S is the part of the surface of the plane 2x+3y+6z=12 located in the first octant is 24.2x+3y+6z=12 located in the first octant is 24.

Lecture-9Lecture-9VOLUME INTEGRALVOLUME INTEGRAL

Let f (r) = fLet f (r) = f11i+fi+f22j+fj+f33k where fk where f11,f,f22,f,f33 are are functions of x,y,z. We know that dv=dxdydz. functions of x,y,z. We know that dv=dxdydz. The volume integral is given by The volume integral is given by ∫f dv=∫∫∫(∫f dv=∫∫∫(ff11i+fi+f22j+fj+f33k)dxdydz k)dxdydz

ExampleExample: If F=2xzi-xj+y: If F=2xzi-xj+y22k then the value of k then the value of ∫f dv where v is the region bounded by the ∫f dv where v is the region bounded by the surfaces x=0,x=2,y=0,y=6,z=xsurfaces x=0,x=2,y=0,y=6,z=x22,z=4 is128i-,z=4 is128i-24j-384k24j-384k

Lecture-10Lecture-10VECTOR INTEGRAL THEOREMSVECTOR INTEGRAL THEOREMS

In this chapter we discuss three important In this chapter we discuss three important vector integral theorems.vector integral theorems.

1)Gauss divergence theorem1)Gauss divergence theorem 2)Green’s theorem2)Green’s theorem 3)Stokes theorem3)Stokes theorem

Lecture-11Lecture-11GAUSS DIVERGENCE THEOREMGAUSS DIVERGENCE THEOREM

This theorem is the transformation between This theorem is the transformation between surface integral and volume integral. Let S be surface integral and volume integral. Let S be a closed surface enclosing a volume v. If f is a a closed surface enclosing a volume v. If f is a continuously differentiable vector point continuously differentiable vector point function, then function, then

∫∫div f dv=∫f.n dSdiv f dv=∫f.n dS Where n is the outward drawn normal vector at Where n is the outward drawn normal vector at

any point of S.any point of S.

Lecture-12Lecture-12GREEN’S THEOREMGREEN’S THEOREM

This theorem is transformation between line This theorem is transformation between line integral and double integral. If S is a closed integral and double integral. If S is a closed region in xy plane bounded by a simple closed region in xy plane bounded by a simple closed curve C and in M and N are continuous curve C and in M and N are continuous functions of x and y having continuous functions of x and y having continuous derivatives in R, thenderivatives in R, then

∫∫Mdx+Ndy=∫∫(∂N/∂x - ∂M/∂y)dxdyMdx+Ndy=∫∫(∂N/∂x - ∂M/∂y)dxdy

Lecture-13Lecture-13STOKES THEOREMSTOKES THEOREM

This theorem is the transformation between This theorem is the transformation between line integral and surface integral. Let S be a line integral and surface integral. Let S be a open surface bounded by a closed, non-open surface bounded by a closed, non-intersecting curve C. If F is any differentiable intersecting curve C. If F is any differentiable vector point function thenvector point function then

∫∫F.dr=∫Curl F.n dsF.dr=∫Curl F.n ds