17.05 Curl and Divergence The operations in this section have applications in the study of fluid flow, electricity and magnetism. We will mostly be looking at techniques of calculation today. If

F  P i Q j R k :

 R Q   P R   Q P  curl F      i   k  j   y z   z x   x y  To help remember this, we introduce the vector operator  (del):

i

    j k x y z

We used this symbol previously as part of the symbol for gradient vector, but now we treat it as an “operator”, or something that can be applied to a function. When “del” operates on a scalar function it produces the gradient vector:

f  i

f f f f f f  j k  i  j k x y z x y z

If we think of  as a vector with components

   , , we can calculate the cross x y z

product:

i  F  x P

j  y Q

k  R Q   R P   Q P        i  k  j  z  y z   x z   x y  R

 R Q   P R   Q P       i   k  j  y z   z x   x y 

curl F

curl F   F

Hanford High School Calculus III

17.05 Curl and Divergence – G. Kelly

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Ex.1:

F  x, y, z   xz i  xyz j  y 2 k i  curl F    F  x xz

Find curl F

j k   y z xyz  y 2

             y 2    xyz   i     y 2    xz   j    xyz    xz   k z z y   x  y   x 

  2 y  xy  i  0  x j   yz  0 k   y  2  x  i  x j  yz k

i  curl  f     f  x f x

j  y f y

k   2 f 2 f   2 f 2 f   2 f 2 f    i   j       k z  yz zy   zx xz   xy yx  f z  0 i  0 j 0 k  0

A conservative vector field is one where F  f , so if F is conservative, curl F  0 . From example 1, curl F   y  2  x  i  x j  yz k Since curl F  0 , F is not conservative.

The inverse is true only if F is defined everywhere (no holes): If curl F  0 , then F is a conservative vector field.

The name curl refers to the tendency of an object to rotate in the vector field. With fluid flow, if curl equals zero at a point, then an object in the fluid at that point would move but not rotate.

Hanford High School Calculus III

17.05 Curl and Divergence – G. Kelly

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div F 

Divergence:

P Q R   x y z

This gives us a scalar field. Curl is a vector field. Since  

Ex. 4:

   i  j  k , we can write div F   F x y z

dot product

F  x, y, z   xz i  xyz j  y 2 k div F    F 

    xz    xyz     y 2  x y z

 z  xz Note: div curl F  0 (page 1101)

In fluid flow, divergence relates to the net rate of change of the mass per unit volume (density) at a point. If div F  0 , then F is incompressible (like water).

div  f      f  

2 f 2 f 2 f  2  2  2 f 2 x y z

2    is called the Laplace Operator because it occurs in Laplace’s equation (page 922). Laplace’s equation (in three dimensions):

2 f 

2 f 2 f 2 f    0 x 2 y 2 z 2

Solutions to this equation are harmonic functions.

Hanford High School Calculus III

17.05 Curl and Divergence – G. Kelly

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