Summary 2 vector fields by 1ezLyH5

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									CAL III                  Summary 2          VECTOR FIELDS        SANCHEZ

VECTOR FIELDS: a vector field in a plane is a function F that maps a point in a region of a
plane into a vector in the plane.
   F(x, y)=<P,Q> where P and Q are scalar functions.
   Ex: F(x, y) =2xyi + 3(x+y)j
   A vector field in space is a function F that maps a point in a region in space into a vector in
space.
  F(x, y, z)=<P, Q, R>
  Ex. every point in space is map into its gravity vector at that point

                         k(xi  yj  zk)
       F(x, y, z)  
                               r3
THE GRADIENT VECTOR FIELD: if f is a function, the gradient ,
          f    f  f
f  i        j k                 is a vector field.
          x    y  z
The gradient operator: the gradient is an operator. Other operators are the differential operator,
the integral operator.
The gradient is a linear operator.
                   
i        j     k
        x     y    z
Properties:
 The critical points of a function are the points where f(x,y,z)= o or does not exist.
 (af+bg) = af + bg
 (fg)=fg + gf

Problem:
Find the gradient vector fields of the following functions:
a) f ( x, y )  xy  f  F ( x, y )  yi  xj
b) f ( x, y )  2 x 2  y 2  f  F ( x, y )  4 xi  2 yj

                     1
                       2
                                                             1
                                                                               
                                                                                1
                                                                                          
c) f ( x, y )  sin y 2  x 2  f  F ( x, y )  2 x sin y 2  x 2 i  2 y sin y 2  x 2 j
                                                               2                2
                     xy                          y   x     xy
d ) f ( x, y , z )       f  F ( x , y , z )  i  j      k
                      z                          z   z     z2

The divergence of a vector field: if F is a vector field, then the divergence of F is the
                                                             P Q R
scalar defined by divF    F       , ,        P, Q, , R            
                                  x y z                      x y z
Pr operties :
1.   (aF  bG)  a  F  b  G
          
2.   ( fG )  ( f )(  G )  (f )  G
The Curl of a vector Field: the Curl of a vector Field, Curl F, is defined by
           i      j      k
                            R Q   P R   Q P 
Curl F=                        y  z   j z  x   k  x  y 
                           = i                                      
          x     y     z                                        
           P     Q       R

Theorem: If the function f(x, y, z) has continuous second-order partial derivatives, then the Curl
of the gradient of the function f is 0, that is Curl(gradf)=0
Proof.
1) Let f ( x, y, z ) be a function with gradf  f  Px, y, z i  Qx, y, z  j  Rx, y, z k
2)Curl gradf   Curl Px, y, z i  Qx, y, z  j  Rx, y, z k 
                      R Q   P R   Q P 
                 i  y  z   j  z  x   k  x  y 
                                                             
                                                          
                        f    f      f    f      f    f  
                  i         j        k       
                      y  z  z  y    z  x  x  z    x  y  y  x  
                                                                                  
                     2 f  2 f          2 f  2 f   2 f  2 f 
                 i                  j             k        
                     yz zy           zx xz   xy yx 
                                                                 
                                  
                  0i  oj  ok  0


Problem: verify the above theorem for the function f ( x, y, z)  x 2  2 yz  xz 2
The gradf  F  f  (2 x  z 2 )i  2 zj  (2 y  2 xz )k

                      i            j        k
                                          
 Curl ( gradf )                                   i 2  2  j (2 z  2 z )  0  0  0i  oj  ok
                     x           y        z
                   2x  z 2       2z    2 y  2 xz

Examples: Calculate the divergence and curl of the given vector field F
a) F(x, y, z) = xi +yj+zk
                                               P Q R
divF    F        , ,       xi  yj  zk             111  3
                   x y z                       x y z
         i       j    k
                    
curlF                    i 0  0  j (0  0)  k (0  0)  0
        x      y    z
         x       y     z
                        2         2        2
b) ) F(x,y,z) = xy i  yz j  x zk


divF    F 
                     
                     , ,
                   x y z
                                       
                             xy 2 i  yz 2 j  x 2 zk      
                                                         P Q R
                                                         x y z
                                                                   y2  z2  x2

          i       j          k
                           
curlF                           i 0  2 yz   j (2 xz  0)  k (0  2 xy )  2 yzi  2 xzj  2 xyk
         x      y         z
        xy 2     yz 2       x2z

								
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