CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.1VECTOR FIELDS 14.2LINE INTEGRALS 14.3INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 14.4GREEN’S THEOREM 14.5CURL AND DIVERGENCE 14.6SURFACE INTEGRALS 14.7THE DIVERGENCE THEOREM 14.8STOKES’ THEOREM

CHAPTER 14 Vector Calculus Slide 3 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.9APPLICATIONS OF VECTOR CALCULUS

DEFINITION 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.1 Slide 4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

THEOREM 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.1 Slide 5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that the vector field F(x, y) =  M(x, y), N(x, y)  is continuous on the open, connected region D ⊂. Then, the line integral is independent of path in D if and only if F is conservative on D. Recall that a vector field F is conservative whenever F = ∇ f, for some scalar function f (called a potential function for F).

THEOREM 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.2 Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that F(x, y) =  M(x, y), N(x, y)  is continuous in the open, connected region D ⊂ and that C is any piecewise-smooth curve lying in D, with initial point (x 1, y 1 ) and terminal point (x 2, y 2 ). Then, if F is conservative on D, with F(x, y) = ∇ f (x, y), we have

EXAMPLE 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.1A Line Integral That Is Independent of Path Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Show that for F(x, y) =  2xy − 3, x 2 + 4y , the line integral is independent of path. Then, evaluate the line integral for any curve C with initial point at (−1, 2) and terminal point at (2, 3).

EXAMPLE Solution 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.1A Line Integral That Is Independent of Path Slide 8 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS Closed Curves Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We consider a curve C to be closed if its two endpoints are the same. That is, for a plane curve C defined parametrically by C = {(x, y)|x = g(t), y = h(t), a ≤ t ≤ b}, C is closed if (g(a), h(a)) = (g(b), h(b)).

THEOREM 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.3 Slide 10 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that F(x, y) is continuous in the open, connected region D ⊂. Then F is conservative on D if and only if for every piecewise-smooth closed curve C lying in D.

14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS Simply-Connected Regions Slide 11 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A region D is simply-connected if every closed curve in D encloses only points in D.

THEOREM 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.4 Slide 12 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that M(x, y) and N(x, y) have continuous first partial derivatives on a simply-connected region D. Then, is independent of path in D if and only if M y (x, y) = N x (x, y) for all (x, y) in D.

EXAMPLE 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.2Testing a Line Integral for Independence of Path Slide 13 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determine whether or not the line integral is independent of path.

EXAMPLE Solution 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.2Testing a Line Integral for Independence of Path Slide 14 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The line integral is not independent of path.

14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS CONSERVATIVE VECTOR FIELDS Slide 15 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let F(x, y) =  M(x, y), N(x, y) , where we assume that M(x, y) and N(x, y) have continuous first partial derivatives on an open, simply-connected region D ⊂. The following five statements are equivalent, meaning that for a given vector field, either all five statements are true or all five statements are false. 1. F(x, y) is conservative on D.

14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS CONSERVATIVE VECTOR FIELDS Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2. F(x, y) is a gradient field in D (i.e., F(x, y) = ∇ f (x, y), for some potential function f, for all (x, y) ∈ D). 3.is independent of path in D. 4. for every piecewise-smooth closed curve C lying in D. 5. M y (x, y) = N x (x, y), for all (x, y) ∈ D.

14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS Conservative Force Fields in Space Slide 17 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For a three-dimensional vector field F(x, y, z), we say that F is conservative on a region D whenever there is a scalar function f (x, y, z) for which F(x, y, z) = ∇ f (x, y, z), for all (x, y, z) ∈ D. As in two dimensions, f is called a potential function for the vector field F.

EXAMPLE 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.3Showing That a Three-Dimensional Vector Field Is Conservative Slide 18 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Show that the vector field F(x, y, z) =  4xez, cos y, 2x 2 e z  is conservative on, by finding a potential function f.

EXAMPLE Solution 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.3Showing That a Three-Dimensional Vector Field Is Conservative Slide 19 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

EXAMPLE Solution 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.3Showing That a Three-Dimensional Vector Field Is Conservative Slide 20 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

THEOREM 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.5 Slide 21 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that the vector field F(x, y, z) is continuous on the open, connected region D ⊂. Then, the line integral is independent of path in D if and only if the vector field F is conservative on D; that is, F(x, y, z) = ∇ f (x, y, z), for all (x, y, z) in D, for some scalar function f (a potential function for F).

THEOREM 14.3 INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 3.5 Slide 22 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Further, for any piecewise-smooth curve C lying in D, with initial point (x 1, y 1, z 1 ) and terminal point (x 2, y 2, z 2 ), we have