13 VECTOR CALCULUS

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.

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

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.

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

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:

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:

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

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.

CURL

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

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.

CURL Example 1

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.

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.

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

GRADIENT VECTOR FIELDS Proof By Clairaut’s Theorem,

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.

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.

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.

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.”

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

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.

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 .

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.

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

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)

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)

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

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).

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.

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.

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 12.8 as a consequence of Stokes’ Theorem.

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:

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.

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:

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:

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.

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

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

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

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.

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.

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.

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.

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:

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

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:

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.

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.

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

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

VECTOR FORMS OF GREEN’S THEOREM Therefore,

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

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.

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:

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

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

VECTOR FORMS OF GREEN’S TH.

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.

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.