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Chapter 16 – Vector Calculus. 16.5 Curl and Divergence. Objectives: Understand the operations of curl and divergence Use curl and divergence to obtain vector forms of Green’s Theorem. Vector Calculus. Here, we define two operations that: Can be performed on vector fields.
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Chapter 16 – Vector Calculus 16.5 Curl and Divergence • Objectives: • Understand the operations of curl and divergence • Use curl and divergence to obtain vector forms of Green’s Theorem 16.5 Curl and Divergence
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. • However, one produces a vector field whereas the other produces a scalar field. 16.5 Curl and Divergence
Definition - Curl • Suppose F = Pi + Qj + Rk is a vector field on 3 and the partial derivatives of P, Q, and R all exist, then the curl of F is the vector field defined by: 16.5 Curl and Divergence
Curl • As a memory aid, let’s rewrite Equation 1 using operator notation. • We introduce the vector differential operator (“del”) as: 16.5 Curl and Divergence
Curl 16.5 Curl and Divergence
Curl • Thus, the easiest way to remember Definition 1 is by means of the symbolic expression 16.5 Curl and Divergence
Theorem 3 • If f is a function of three variables that has continuous second-order partial derivatives, then 16.5 Curl and Divergence
Conservative Vector Field • 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. 16.5 Curl and Divergence
Theorem 4 • If F is a vector field defined on all of 3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. • NOTE: Theorem 4 is the 3-D version of Theorem 6 in Section 16.3 • NOTE: This theorem says that it is true if the domain is simply-connected—that is, “has no hole.” 16.5 Curl and Divergence
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 16.1). 16.5 Curl and Divergence
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. 16.5 Curl and Divergence
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. 16.5 Curl and Divergence
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 16.8 as a consequence of Stokes’ Theorem. 16.5 Curl and Divergence
Example 1 – pg. 1068 # 21 • Show that any vector field of the form where f, g, and hare differentiable functions, is irrotational. 16.5 Curl and Divergence
Definition - Divergence • If F = Pi + Qj + Rk is a vector field on 3and ∂P/∂x, ∂Q/∂y, and ∂R/∂zexist, the divergence of F is the function of three variables defined by: 16.5 Curl and Divergence
Divergence • In terms of the gradient operator the divergence of F can be written symbolically as the dot product of del and F: 16.5 Curl and Divergence
Curl versus Divergence • Observe that: • Curl F is a vector field. • Div F is a scalar field. 16.5 Curl and Divergence
Theorem 11 • If F = Pi + Qj + Rk is a vector field on 3 and P, Q, and R have continuous second-order partial derivatives, then div curl F = 0 16.5 Curl and Divergence
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. 16.5 Curl and Divergence
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. 16.5 Curl and Divergence
Example 2 – pg. 1068 # 22 • Show that any vector of the form is incompressible. 16.5 Curl and Divergence
Example 3 – pg. 1068 • Find the following for the given vector field: • The curl • The divergence 16.5 Curl and Divergence
Gradient Vector Fields • Another differential operator occurs when we compute the divergence of a gradient vector field f. • If f is a function of three variables, we have: 16.5 Curl and Divergence
Laplace Operator • This expression occurs so often that we abbreviate it as 2f. • The operator is called the Laplace operator due to its relation to Laplace’s equation 16.5 Curl and Divergence
Green’s Theorem – Vector Form • Hence, we can now rewrite the equation in Green’s Theorem in the vector form using curl as equation 12: 16.5 Curl and Divergence
Green’s Theorem – Vector Form • 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 write a similar formula involving the normal component of F and the divergence. (see book for proof) 16.5 Curl and Divergence
Green’s Theorem – Vector Form • 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. 16.5 Curl and Divergence
Example 4 – pg. 1068 • Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 16.5 Curl and Divergence