VECTOR FUNCTION

VECTOR FUNCTION- GRADIENT OF A SCALAR, DIRECTION DERIVATIVE, DIVERGENCE OF A VECTOR, CURL OF A VECTOR, SCALAR POTENTIAL

1. VECTOR FUNCTION

2. CONTENT INTRODUCTION GRADIENT OF A SCALAR DIRECTION DERIVATIVE DIVERGENCE OF A VECTOR CURL OF A VECTOR SCALAR POTENTIAL

3. 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

4. VECTOR DIFFERENTIAL OPERATOR The vector differential Hamiltonian operator DEL (or nabla ) is denoted by ∇ and is defined as:

5. 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 = grad f is a vector quantity. grad f or ∇f , which is read “del f ”  

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

7. 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 .  

8. EXAMPLE 2 Find the directional derivative of the function f (x , y) = x 2 y 3 – 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) = 8j Directional derivative in the direction of the vector 2 i + 5 j = ∇f . = ( 8j ). = ans  

9. 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.

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

11. 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.

12. EXAMPLE 4 Show that A= (3 y 4 z 2 i+(4 x 3 z 2) j-(3 x 2 y 2 )k is solenoidal. Sol: here, A = (3 y 4 z 2 ) i +(4 x 3 z 2) j-(3 x 2 y 2 )k by definition, = ( ) = 0 Hence, it is a solenoidal.  

13. 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

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

15. = (2z-2x)i – (0-0)j + (2z-x)k At (1, 0, 2) = 2i - 0j + 3k ans

16. 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

17. EXAMPLE 6 Show 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,

18. = (3-3)i – (2-2)j + (3-3)k =0 hence, it is irrotational.

19. 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.

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

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

22. 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 DERIVATIVES FORMULA 1 The Del Operator 2 Gradient of a scalar function is a vector quantity. 3 Divergence of a vector is a scalar quantity. ∇.A 4 Curl of a vector is a vector quantity. ∇*A

23. 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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