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UIUC MATH 241 - 16_05

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1 1 Copyright © Cengage Learning. All rights reserved. 16 Vector CalculusCopyright © Cengage Learning. All rights reserved. 16.5 Curl and Divergence3 3 Curl and Divergence Learning objectives:  compute the curl of a vector field  show that a given three-dimensional vector field is not conservative by computing its curl  show that a given three-dimensional vector field is conservative (i.e. the gradient of a function of three variables) by finding its domain and computing its curl  find a potential function of a three-dimensional conservative vector field4 4 Curl and Divergence Learning objectives (continued):  compute the divergence of a vector field  show that a given vector field is not the curl of another vector field by computing its divergence  compute the Laplace operator of a function and the Laplace operator of a vector field  compute the line integral of the tangential component of a vector field using the vector form of Green’s Theorem  compute the line integral of the normal component of a vector field using the vector form of Green’s Theorem5 5 Curl6 6 Curl If F = P i + Q j + R k is a vector field on and the partial derivatives of P, Q, and R all exist, then the curl of F is the vector field on defined by Let’s rewrite Equation 1 using operator notation. We introduce the vector differential operator ∇ (“del”) as7 7 Curl It has meaning when it operates on a scalar function to produce the gradient of f : 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:8 8 Curl So the easiest way to remember Definition 1 is by means of the symbolic expression9 9 Curl Recall that the gradient of a function f of three variables is a vector field on and so we can compute its curl. The following theorem says that the curl of a gradient vector field is 0.10 10 Curl Since a conservative vector field is one for which F = ∇f, Theorem 3 can be rephrased as follows: If F is conservative, then curl F = 0. This gives us a way of verifying that a vector field is not conservative. (In other words, if curl F is not zero, then F cannot be conservative.)11 11 Curl The converse of Theorem 3 is not true in general, but the following theorem says the converse is true if F is defined everywhere. (More generally it is true if the domain is simply-connected, that is, “has no hole.”)12 12 Curl The reason for the name curl is that the curl vector is associated with rotations. See your Stewart text for details. Another reason occurs when F represents the velocity field in fluid flow. 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), and the length of this curl vector is a measure of how quickly the particles move around the axis (see Figure 1). Figure 113 13 Curl If curl F = 0 at a point P, then the fluid is free from rotations at P and F is called irrotational at P. In other words, there is no whirlpool or eddy at P. If curl F = 0, then 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. (More details when we discuss Stokes’ Theorem.)14 14 Divergence15 15 Divergence If F = P i + Q j + R k is a vector field on and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, then the divergence of F is the function of three variables defined by Observe that curl F is a vector field but div F is a scalar field.16 16 Divergence In terms of the gradient operator ∇ = (∂/∂x) i + (∂/∂y) j + (∂/∂z) k, the divergence of F can be written symbolically as the dot product of ∇ and F:17 17 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. Again, the reason for the name divergence can be understood in the context of fluid flow. (More details will follow in Section 16.9 on the Divergence Theorem.)18 18 Divergence If F(x, y, z) is the velocity of a fluid (or gas), then 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. 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, then F is said to be incompressible. Another differential operator occurs when we compute the divergence of a gradient vector field ∇f.19 19 Divergence If f is a function of three variables, we have and this expression occurs so often that we abbreviate it as ∇2f. The operator ∇2 = ∇  ∇ is called the Laplace operator because of its relation to Laplace’s equation20 20 Divergence We can also apply the Laplace operator ∇2 to a vector field F = P i + Q j + R k in terms of its components: ∇2F = ∇2P i + ∇2Q j + ∇2R k21 21 Vector Forms of Green’s Theorem22 22 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. We suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypotheses of Green’s Theorem. Then we consider the vector field F = P i + Q j.23 23 Vector Forms of Green’s Theorem Its line integral is and, regarding F as a vector field on with third component 0, we have24 24 Vector Forms of Green’s Theorem Therefore and we can now rewrite the equation in Green’s Theorem in the vector form25 25 Vector Forms of Green’s Theorem 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. If C is given by the vector equation r(t) = x(t) i + y(t) j a ≤ t ≤ b then the unit tangent vector is26 26 Vector Forms of Green’s Theorem The outward unit normal vector to C is given by (See Figure 2.) Figure 227 27 Vector Forms of Green’s Theorem Then from the equation we have28 28 Vector Forms of Green’s Theorem by Green’s Theorem. But the integrand in this double integral is just the divergence of F.29 29 Vector Forms of Green’s Theorem So we have a second vector form of


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UIUC MATH 241 - 16_05

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