VECTOR FUNCTION

CONTENT

• INTRODUCTION

• GRADIENT OF A SCALAR

• DIRECTION DERIVATIVE

• DIVERGENCE OF A VECTOR

• CURL OF A VECTOR

• SCALAR POTENTIAL

INTRODUCTION

In this chapter, a vector field or a scalar field can be differentiated w.r.t. position in three ways to produce another vector field or scalar field. The chapter details the three derivatives, i.e.,

1. gradient of a scalar field

2. the divergence of a vector field

3. the curl of a vector field

VECTOR DIFFERENTIAL OPERATOR

* The vector differential Hamiltonian operator DEL(or nabla) is denoted by ∇ and is defined as:

=i +j +k

x y z

GRADIENT OF A SCALAR

* Let f(x,y,z) be a scalar point function of

position defined in some region of space. Then

gradient of f is denoted by grad f or ∇ f and is

defined as

grad f = ∇f =

ograd f is a vector quantity.ograd f or ∇f , which is read “del f ”

*EXAMPLE 1 Find gradient of F if F = y- at (1,1,1, ) Solution: by definition, ∇f = = -(6xy)i + (3-3)j – (2z)k = -6i+0j-2k ans

DIRECTIONAL DERIVATIVE* The directional derivative of a scalar point function f at a point f(x,y,z) in the direction of a vector a , is the component ∇f in the direction of a.*If a is the unit vector in the direction of a, then direction derivative of ∇f in the direction of a of is defined as = ∇f .

*EXAMPLE 2 Find the directional derivative of the function f (x, y) = x2y3 – 4y at the point (2, –1) in the direction of the vector v = 2 i + 5 j.Solution: by definition, ∇f = ∇f =at (2,-1) = 8jDirectional derivative in the direction of the vector 2 i + 5 j = ∇f . = (8j). = ans

DIVERGENCE OF A VECTOR

* Let f be any continuously differentiable vector

point function. Then divergence of f and is

written as div f and is defined as

which is a scalar quantity.

31 2fff

div f = •f = + +x y z

*EXAMPLE 3Find the divergence of a vector A= 2xi+3yj+5zk .Solution: by definition, = () = 2+3+5 = 10 ans

31 2fff

div f = •f = + +x y z

SOLENOIDAL VECTOR

* A vector point function f is said to be solenoidal vector if its divergent is equal to zero i.e., div f=0 at all points of the function. For such a vector, there is no loss or gain of fluid.

0321 fff zyxf

*EXAMPLE 4Show that A= (3y4z 2 i+(4 x3z 2)j-(3x2y2)k is solenoidal.Sol: here, A= (3y4z 2)i+(4 x3z 2)j-(3x2y2)k by definition, = () = 0 Hence, it is a solenoidal.

0321 fff zyxf

CURL OF A VECTOR

* Let f be any continuously differentiable vector

point function. Then the vector function curl of

f(x,y,z) is denoted by curl f and is defined as

zyxfzyxfzyxf

zyxzyx

,,,,,,

f,, curl

321

kji

f

*EXAMPLE 5Find the curl of A= (xy)i-(2 xz )j+(2yz)k at the point (1, 0, 2). Solution: here, A = (xy)i-(2xz )j+(2yz)k by definition,

yzxzxyzyx

22

kji

zyxfzyxfzyxf

zyxzyx

,,,,,,

f,, curl

321

kji

f

= (2z-2x)i – (0-0)j + (2z-x)kAt (1, 0, 2) = 2i - 0j + 3k ans

k2j2i22

xyy

xzx

xyz

yzx

xzz

yzy

IRROTATIONAL VECTOR

* Any motion in which curl of the velocity vector is a null vector i.e., curl v=0 is said to be irrotational.

*Otherwise it is rotational

0f zyx ,, curl

*EXAMPLE 6Show that F=(2x+3y+2z)i + (3x+2y+3z)j + (2x+3y+3z)k is irrotational. Solution: F=(2x+3y+2z)i+(3x+2y+3z)j+(2x+3y+3z)k by definition,

3z3y2x3z2y3x2z3y2x

f,, curl

zyx

zyx

kji

f

0f zyx ,, curl

= (3-3)i – (2-2)j + (3-3)k=0 hence, it is irrotational.

k2z3y2x3z2y3x

yx

j2z3y2x3z3y2x

zx

izy

3z2y3x3z3y2x

SCALAR POTENTIAL

*If f is irrotational, there will always exist a scalar function f(x,y,z) such that

f=grad g.

This g is called scalar potential of f.

*EXAMPLE 7A fluid motion is given by V=(ysinz-sinx)i + (xsinz+2yz)j + (xycosz+y2) . Find its velocity potential. Solution: V=(ysinz-sinx)i + (xsinz+2yz)j + (xycosz+y2) by definition,

= (ysinz-sinx)i+(xsinz+2yz)j+(xycosz+ y2)

by equating corresponding equation we get,* = ysinz-sinx integrating w r to x ; = *= xsinz+2yzintegrating w r to y ; = *= xycosz+ y2 integrating w r to z ; = Hence, = +C

*SUMMARY

DERIVATIVES FORMULA

1 The Del Operator ∇ =k

2 Gradient of a scalar function is a vector quantity.

grad f = ∇f =

3 Divergence of a vector is a scalar quantity.

∇.A

4 Curl of a vector is a vector quantity. ∇*A

0.

0

A

• So, any vector differential equation of the form B=0 can be solved identically by writing B=. • We say B is irrotational. • We refer to as the scalar potential.

• So, any vector differential equation of the form .B=0 can be solved identically by writing B=A. • We say B is solenoidal or incompressible. • We refer to A as the vector potential.

Scalar and vector potential

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