Differential Operators : Vector Calculus #Differential #Operators : #Vector #Calculus

                                     DIFFERENTIAL OPERATORS                                                                          

 Contents :

 1. Scalar  Point Function

 2. Vector Point Function          

 3. Delta Neighborhood

 4. Limit

 5. Continuity

 6. Directional Derivative at a point 

 7. Level Surface

1. Scalar Point Funtion :

             Let S be a domain in space. If to each point p 𝛜 S there corresponds a scalar  f( p ) then  is called  a scalar point function over the domain S.

            e.g. Let  f(p) be the density at any point P of a material body occupying a region R. Then f  is a scalar scalar point function defined for the region R.

             e.g. Let f(P) be the temperature at any point of a body occupying a certain region S. Then s is a

              scalar point function defined for the region S. 

2. Vector Point Function :

             Let S be a domain in space. If to each point p 𝛜 S there corresponds a vector f(p), then f is

            Called a vector point function over S.

             e.g. Let f(p) denote the velocity of a particle at a point P in a region R then f is a vector                   point function defined in the region R.

Note :

       1.  If oxyz be a frame of  reference in space then for  a point P = (x,y,z) , we can write  f(p)         = f(x,y,z) so that a scalar point function appears as a scalar function of three variables.

       2.     If r denotes the position vector of P with respect to the origin O then f(p)  may be written as            f(r).

       e.g. Let O be the origin in space. For each point P in the space there corresponds a unique real         number equal to OP. The function so defined on the space is called a distance function. It is             denoted by r.

        For P=(x,y,z) ,       r(P) = OP = √ ( x²+y²+z²).

e.g. Let O be the origin  in space S. For each P 𝞊 S there corresponds a unique vector OP. The            function so defined on the space S is called the position vector function. It is denoted              by r.

        For P= (x,y,z) , r(P) = OP = x i + y j + z k.

3. Delta Neighbourhood :

           Let P be a point in space and 𝝳 > 0 . The set of all the points Q such that PQ  < 𝝳 is  called 𝝳 – neighbouthood. If P is deleted from 𝝳 – nbd of P then it is called deleted 𝝳- nbd of P.

4. Limit :

            Let f  be a scalar point function defined on a deleted – nbd of P and l 𝞊 R. If for each 𝞊 > 0 ,there exist  𝝳 > 0 , such that 0< QP < 𝝳 => | f(Q) – l | < 𝞊 then we say that Lim f = l as Q  tends to P.

        5. Continuity :

           Let Let f  be a scalar point function defined on a deleted – nbd of P and l 𝞊 R. If for each 𝞊               > 0 , there exist  𝝳 > 0 , such that 0< QP < 𝝳 => | f(Q) – f(P) | < 𝞊 then we say that  f             is continuous at P.

        6.Directional Derivative at a point :

                   Let p be a point in space and L be a ray through p in the direction of the unit vector e. Let a                     vector point function f be defined in a nbd D of p. Let Q ≠ p and Q 𝞊 L ⋂ D.

If Lim ( f(Q) – f(P) ) / QP as Q tends to P  exists then we say that the limit is the directional  derivative of f at P in the direction of e. It is denoted by ∂f/∂e or ∂f/∂s where s= PQ.

         Note :

·         If e= i = unit vector along OX then ∂f/∂e = ∂f/∂i or ∂f/∂x

·         If e= j = unit vector along OY then ∂f/∂e = ∂f/∂j or ∂f/∂y

·         If e= i = unit vector along OZ then ∂f/∂e = ∂f/∂k or ∂f/∂z

        Theorem : If r is the position vector function and e is a unit vector , then ∂r/∂e = e.

        Proof :

                   Suppose r is the position vector function and e is a unit vector.

                   Let P be a point in the domain of r.

                   Let  Q ≠ P and Q 𝞊 L where L is the ray through P in the direction of e.

                  ∂r/∂e = Lim ( r(Q) – r(P) ) / QP                 ( as Q -> P )

                           = Lim  ( OQ – OP ) / QP               ( as Q -> P)

                           = Lim PQ/PQ                                    ( as Q -> P )

                           = Lim (PQ) e / PQ                           ( as Q -> P )

                          = Lim e                                              ( as Q -> P )

                 ∂r/∂e =e

                                  **    Hence the proof  **

           Note :

·         ∂r/∂x = ∂r/∂i = i

·         ∂r/∂y = ∂r/∂j = j

·          ∂r/∂e = ∂r/∂k = k

            7. Level Surface :

                   Let f be a scalar point function and c be a real number . Q is  a point in the                     domain S, such that Q 𝞊 S => f(Q) = c then  S is called the level surface.

                     For different value c , f constitutes a family of level surface  in three                                 dimensional space.    

                                                                    

           Note :

·         If f  is a scalar point function and P is any given point, then the surface S such that

Q 𝞊 S => f(Q) = c = f(P) defines  a level surface  through the point P.

·         If P,Q are two points on the level surface  f, then f(P ) = f(Q)

#Differential #Operators  :  #Vector #Calculus