1. 13
VECTOR CALCULUS

2. Here, we define two operations that:
VECTOR CALCULUS
Here, we define two operations that:
Can be performed on vector fields.
Play a basic role in the applications of vector calculus to fluid flow, electricity, and magnetism.

3. Each operation resembles differentiation.
VECTOR CALCULUS
Each operation resembles differentiation.
However, one produces a vector field whereas the other produces a scalar field.

4. 13.5 Curl and Divergence VECTOR CALCULUS
In this section, we will learn about:
The operations of curl and divergence
and how they can be used to obtain
vector forms of Green’s Theorem.

5. Suppose: F = P i + Q j + R k is a vector field on .
CURL
Suppose:
F = P i + Q j + R k is a vector field on
The partial derivatives of P, Q, and R all exist

6. Then, the curl of F is the vector field on defined by:
Equation 1
Then, the curl of F is the vector field on defined by:

7. As a memory aid, let’s rewrite Equation 1 using operator notation.
CURL
As a memory aid, let’s rewrite Equation 1 using operator notation.
We introduce the vector differential operator (“del”) as:

8. CURL
It has meaning when it operates on a scalar function to produce the gradient of f :

9. CURL
If we think of as a vector with components ∂/∂x, ∂/∂y, and ∂/∂z, we can also consider the formal cross product of with the vector field F as follows.

10. CURL

11. CURL
Equation 2
Thus, the easiest way to remember Definition 1 is by means of the symbolic expression

12. If F(x, y, z) = xz i + xyz j – y2 k find curl F.
Example 1
If F(x, y, z) = xz i + xyz j – y2 k
find curl F.
Using Equation 2, we have the following result.

13. CURL
Example 1

14. CURL
Most computer algebra systems (CAS) have commands that compute the curl and divergence of vector fields.
If you have access to a CAS, use these commands to check the answers to the examples and exercises in this section.

15. So, we can compute its curl.
Recall that the gradient of a function f of three variables is a vector field on
So, we can compute its curl.
The following theorem says that the curl of a gradient vector field is 0.

16. GRADIENT VECTOR FIELDS
Theorem 3
If f is a function of three variables that has continuous second-order partial derivatives, then

17. GRADIENT VECTOR FIELDS
Proof
By Clairaut’s Theorem,

18. GRADIENT VECTOR FIELDS
Notice the similarity to what we know from Section 12.4:
a x a = 0 for every three-dimensional (3-D) vector a.

19. CONSERVATIVE VECTOR FIELDS
A conservative vector field is one for which
So, Theorem 3 can be rephrased as: If F is conservative, then curl F = 0.
This gives us a way of verifying that a vector field is not conservative.

20. CONSERVATIVE VECTOR FIELDS
Example 2
Show that the vector field F(x, y, z) = xz i + xyz j – y2 k is not conservative.
In Example 1, we showed that: curl F = –y(2 + x) i + x j + yz k
This shows that curl F ≠ 0.
So, by Theorem 3, F is not conservative.

21. CONSERVATIVE VECTOR FIELDS
The converse of Theorem 3 is not true in general.
The following theorem, though, says that it is true if F is defined everywhere.
More generally, it is true if the domain is simply-connected—that is, “has no hole.”

22. CONSERVATIVE VECTOR FIELDS
Theorem 4 is the 3-D version of Theorem 6 in Section 12.3
Its proof requires Stokes’ Theorem and is sketched at the end of Section 12.8

23. CONSERVATIVE VECTOR FIELDS
Theorem 4
If F is a vector field defined on all of whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field.

24. CONSERVATIVE VECTOR FIELDS
Example 3
Show that F(x, y, z) = y2z3 i + 2xyz3 j + 3xy2z2 k is a conservative vector field.
Find a function f such that

25. CONSERVATIVE VECTOR FIELDS
Example 3 a
As curl F = 0 and the domain of F is , F is a conservative vector field by Theorem 4.

26. CONSERVATIVE VECTOR FIELDS
E. g. 3 b—Eqns. 5-7
The technique for finding f was given in Section 12.3
We have:
fx(x, y, z) = y2z3
fy(x, y, z) = 2xyz3
fz(x, y, z) = 3xy2z2

27. CONSERVATIVE VECTOR FIELDS
E. g. 3 b—Eqn. 8
Integrating Equation 5 with respect to x, we obtain: f(x, y, z) = xy2z3 + g(y, z)

28. CONSERVATIVE VECTOR FIELDS
Example 3 b
Differentiating Equation 8 with respect to y, we get: fy(x, y, z) = 2xyz3 + gy(y, z)
So, comparison with Equation 6 gives: gy(y, z) = 0
Thus, g(y, z) = h(z) and fz(x, y, z) = 3xy2z2 + h’(z)

29. CONSERVATIVE VECTOR FIELDS
Example 3 b
Then, Equation 7 gives: h’(z) = 0
Therefore, f(x, y, z) = xy2z3 + K

30. CURL
The reason for the name curl is that the curl vector is associated with rotations.
One connection is explained in Exercise 37.
Another occurs when F represents the velocity field in fluid flow (Example 3 in Section 12.1).

31. CURL
Particles near (x, y, z) in the fluid tend to rotate about the axis that points in the direction of curl F(x, y, z).
The length of this curl vector is a measure of how quickly the particles move around the axis.

32. F = 0 (IRROTATIONAL CURL)
If curl F = 0 at a point P, the fluid is free from rotations at P.
F is called irrotational at P.
That is, there is no whirlpool or eddy at P.

33. If curl F ≠ 0, the paddle wheel rotates about its axis.
F = 0 & F ≠ 0
If curl F = 0, a tiny paddle wheel moves with the fluid but doesn’t rotate about its axis.
If curl F ≠ 0, the paddle wheel rotates about its axis.
We give a more detailed explanation in Section as a consequence of Stokes’ Theorem.

34. DIVERGENCE
Equation 9
If F = P i + Q j + R k is a vector field on and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, the divergence of F is the function of three variables defined by:

35. Observe that: Curl F is a vector field. Div F is a scalar field.
CURL F VS. DIV F
Observe that:
Curl F is a vector field.
Div F is a scalar field.

36. In terms of the gradient operator
DIVERGENCE
Equation 10
In terms of the gradient operator
the divergence of F can be written symbolically as the dot product of and F:

37. If F(x, y, z) = xz i + xyz j – y2 k find div F.
DIVERGENCE
Example 4
If F(x, y, z) = xz i + xyz j – y2 k find div F.
By the definition of divergence (Equation 9 or 10) we have:

38. If F is a vector field on , then curl F is also a vector field on .
DIVERGENCE
If F is a vector field on , then curl F is also a vector field on .
As such, we can compute its divergence.
The next theorem shows that the result is 0.

39. DIVERGENCE
Theorem 11
If F = P i + Q j + R k is a vector field on and P, Q, and R have continuous second-order partial derivatives, then div curl F = 0

40. By the definitions of divergence and curl,
Proof
By the definitions of divergence and curl,
The terms cancel in pairs by Clairaut’s Theorem.

41. Note the analogy with the scalar triple product: a . (a x b) = 0
DIVERGENCE
Note the analogy with the scalar triple
product: a . (a x b) = 0

42. Show that the vector field F(x, y, z) = xz i + xyz j – y2 k
DIVERGENCE
Example 5
Show that the vector field F(x, y, z) = xz i + xyz j – y2 k
can’t be written as the curl of another vector field, that is, F ≠ curl G
In Example 4, we showed that div F = z + xz and therefore div F ≠ 0.

43. This contradicts div F ≠ 0.
DIVERGENCE
Example 5
If it were true that F = curl G, then Theorem 11 would give: div F = div curl G = 0
This contradicts div F ≠ 0.
Thus, F is not the curl of another vector field.

44. DIVERGENCE
Again, the reason for the name divergence can be understood in the context of fluid flow.
If F(x, y, z) is the velocity of a fluid (or gas), div F(x, y, z) represents the net rate of change (with respect to time) of the mass of fluid (or gas) flowing from the point (x, y, z) per unit volume.

45. INCOMPRESSIBLE DIVERGENCE
In other words, div F(x, y, z) measures the tendency of the fluid to diverge from the point (x, y, z).
If div F = 0, F is said to be incompressible.

46. GRADIENT VECTOR FIELDS
Another differential operator occurs when we compute the divergence of a gradient vector field .
If f is a function of three variables, we have:

47. This expression occurs so often that we abbreviate it as .
LAPLACE OPERATOR
This expression occurs so often that we abbreviate it as
The operator is called the Laplace operator due to its relation to Laplace’s equation

48. LAPLACE OPERATOR
We can also apply the Laplace operator to a vector field F = P i + Q j + R k in terms of its components:

49. VECTOR FORMS OF GREEN’S THEOREM
The curl and divergence operators allow us to rewrite Green’s Theorem in versions that will be useful in our later work.

50. VECTOR FORMS OF GREEN’S THEOREM
We suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypotheses of Green’s Theorem.

51. VECTOR FORMS OF GREEN’S THEOREM
Then, we consider the vector field F = P i + Q j
Its line integral is:

52. VECTOR FORMS OF GREEN’S THEOREM
Regarding F as a vector field on with third component 0, we have:

53. VECTOR FORMS OF GREEN’S THEOREM
Therefore,

54. VECTOR FORMS OF GREEN’S TH.
Equation 12
Hence, we can now rewrite the equation in Green’s Theorem in the vector form

55. VECTOR FORMS OF GREEN’S TH.
Equation 12 expresses the line integral of the tangential component of F along C as the double integral of the vertical component of curl F over the region D enclosed by C.
We now derive a similar formula involving the normal component of F.

56. VECTOR FORMS OF GREEN’S TH.
If C is given by the vector equation
r(t) = x(t) i + y(t) j a ≤ t ≤ b
then the unit tangent vector (Section 13.2) is:

57. VECTOR FORMS OF GREEN’S TH.
You can verify that the outward unit normal vector to C is given by:

58. VECTOR FORMS OF GREEN’S TH.
Then, from Equation 3 in Section 12.2, by Green’s Theorem, we have:

59. VECTOR FORMS OF GREEN’S TH.

60. VECTOR FORMS OF GREEN’S TH.
However, the integrand in that double integral is just the divergence of F.
So, we have a second vector form of Green’s Theorem—as follows.

61. VECTOR FORMS OF GREEN’S TH.
Equation 13
This version says that the line integral of the normal component of F along C is equal to the double integral of the divergence of F over the region D enclosed by C.