Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals

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1 Math 280 Calculus III Chapter 16 Sections: 16.1, Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1 Vector Fields Vector Fields Generally, a vector field is a function that assigns a vector to each point in its domain. A vector field on a three-dimensional domain in space might have a formula like F(x, y, z) = M(x, y, z)i + N(x, y, z)j + P(x, y, z)k. The field is continuous if the component functions M, N, and P are each continuous. The field is differentiable if each component function is differentiable. Examples of Vector Fields 1

2 The gradient field of a differentiable function f(x, y, z) is the field formed by all the gradient vectors of the function, and is represented as = + + At each point (x, y, z), the gradient field gives a vector pointing in the direction of greatest increase of f, with magnitude being the value of the directional derivative in that direction. 2

3 Examples of Gradient Fields If we attach the gradient vector f of a scalar function f(x, y, z) to each point of a level surface of the function, we obtain a three-dimensional field on the surface. If we attach the velocity vector to each point of a flowing fluid, we have a three-dimensional field defined on a region in space. The gradient vector of a differentiable scalar-valued function at a point gives the direction of greatest increase of the function. 3

4 Example 1 Suppose that the temperature T at each point (x, y, z) in a region of space is given by =100 and that F(x, y, z) is defined to be the gradient of T. Find the vector field F. Example 2 Find the gradient vector field of f (x, y,) = x 2 y y 3. Example 3 Match the vector field F to the correct plot below. F(x, y, z) = x i + y j + z k. 4

5 Section 16.2 Line Integrals. The idea is to decrease the size of each toward zero and so to increase the number of these arcs to infinity. This creates something similar to a Reimann sum. The line integral is the limit of this sum as n approaches infinity. Sometimes we need to calculate the total mass of a wire lying along a curve in space or to find the work done by a variable force acting along such a curve. In such cases, the notion of integration presented in Calculus I is insufficient to find the result. We need to integrate over a curve, C, rather than over an interval [a, b]. These more general integrals are called line integrals Let f(x, y, z) be a real-valued function that we want to integrate over the curve C within the domain of f. The key to integrating a line integral is to use the parameterization of f. If r(t) = g(t)i + h(t)j + k(t)k is the parameterization of f over the interval a t b, then the values of f along the curve are given by the composite function f(g(t), h(t), k(t)). 5

6 . In this parametrization, we often use the formula ( ) = +. 6

7 Example 2 The figure below shows a path from the origin to (1, 1, 1), the union of line segments C 1 and C 2. Integrate f(x, y, z) = x 3y 2 + z over C 1 C 2. We can generate a cylindrical surface by moving a straight line along C orthogonal to the plane (as shown in the figure). If z = f(x, y) is a nonnegative continuous function over a region in the plane containing the curve C, then the graph of f is a surface that lies above the plane. From the definition, where 0 as n. The line integral 7 is the area of the wall in the figure.

8 Example 3 Evaluate where F(x, y, z) = zi + xyj y 2 k along the curve C given by ( )= + +, 0 t 1. Example 4 A slender metal arch, denser at the bottom than top, lies along the semi-circle y 2 + z 2 = 1, z 0, in the yz-plane. Find the center of the arch s mass if the density at the point (x, y, z) on the arch is δ(x, y, z) = 2 z. 8

9 and we can write, where F= P i + Q j + R k. Example 5 Evaluate, where C consists of the line segment C 1 from (2, 0, 0) to (3, 4, 5), followed by the vertical line segment C 2 from (3, 4, 5) to (3, 4, 0). 9

10 Example 6 Evaluate the line integral 2, where C is the helix r(t) = (cos t)i + (sin t)j + t k, 0 t 2π. Example 7 Evaluate, where C is the circular helix given by the equations x = cos t, y = sin t, z = t, 0 t 2π. 10

11 Line Integrals of Vector Fields 11

12 . Example 8. 12

13 Example 9 Find the work done by the force field F(x, y, z) = xi + yj + zk in moving an object along the curve C parameterized by r(t) = (cos πt)i + (t 2 )j + (sin πt) k, 0 t 1. We sometimes want to write a line integral of a scalar function with respect to one of the coordinates, such as. This integral is not the same as the arc length integral defined above. Let F = M(x, y, z)i be a defined vector field over the curve C parameterized by r(t) = g(t)i + h(t)j + k(t)k, a t b. With this notation, we have x = g(t) and then dx = g (t) dt. And = (,, ) ( ) = (,, ). We now DEFINE the line integral of M over C with respect to the coordinate x as (,, ), h,, ). Similar processes can be applied to the j and k components giving the following rules. 13

14 Section 16.3 The Fundamental Theorem for Line Integrals Proof of Theorem 1 A gravitational field G is a vector field that represents the effect of gravity at a point in space due to the presence of a massive object. The gravitational force on a body of mass m placed in the field is given by F=mG. Similarly, an electric field E is a vector field in space that represents the effect of electric forces on a charged particle placed within it. The force on a body of charge q placed in the field is given by F=qE. In gravitational and electric fields, the amount of work it takes to move a mass or charge from one point to another depends on the initial and final positions of the object not on which path is taken between these positions. In this section we study vector fields with this property and the calculation of work integrals associated with them. Example 1 Find the work done by the gravitational field ( )= In moving a particle with mass m from the point (3, 4, 12) to the point (2, 2, 0) along a Piecewise-smooth curve C. 14

15 Example 1 Suppose the force field F= f is the gradient of the function 1 (,, )= Find the work done by F in moving an object along a smooth curve C joining (1, 0, 0) to (0, 0, 2) that does not pass through the origin. 15

16 Figure Four connected regions. In (a) and (b), the regions are simply connected. In (c) and (d), the regions are not simply connected because C1 and C2 cannot be contracted to a point inside the regions containing them. 16

17 Proof of Theorem 3 Part 1 17

18 Proof of Theorem 3 Part 2 gh Example 3 Show that F = (2x 3)i zj + (cos z)k is not conservative. Example 4 Show that F = (e x cos y + yz)i + (xz e x sin y)j + (xy + z)k is conservative over its natural domain and find a potential function for it. Continued on next page 18

19 19

20 Example 5 Show that the vector field = satisfies the equations in the Component Test, but is not conservative over its natural domain. Explain why this is possible. 20

21 Example 6 Show that y dx + x dy + 4 dz is exact and evaluate the integral (,, ) + +4 (,, ) Over any path from (1, 1, 1) to (2, 3, 1). 21

22 Section 16.4 Green s Theorem in the Plane Green s Theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. (See Figure 1. We assume that consists of all points inside as well as all points on C.) In stating Green s Theorem we use the convention that the positive orientation of a simple closed curve C refers to a single counterclockwise traversal of C. Thus if C is given by the vector function r(t), a t b, then the region is always on the left as the point r(t) traverses C. (See Figure 2.) Figure 1 Positive Orientation Figure 2 Negative Orientation Example 3 Verify Green s Theorem for the vector field 22 See figure below.

23 23

24 24

25 Other regions to which Green s Theorem applies. 25

26 Example 6 Evaluate +3, where C is the boundary of the semiannular region D in the upper half-plane between the circles x 2 + y 2 = 1 and x 2 + y 2 = 4. 26

27 Section 16.5 Curl and Divergence If F = Mi + Nj + Pk is a vector field and the partial derivatives of M, N, and P all exist, then the curl of F is the vector field defined by = + +. If we define the symbolic operator = + +, we get the formula = =. Example 1 If = + find curl F. Theorem If f is a function of three variables that has continuous second-order partial derivatives, then curl ( f) = 0. 27

28 Theorem If F is a vector field defined on R 3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. Example 2 Show that the vector field = + is not conservative. Example 3 (a) Show that = is a conservative field. (b) Find a function f such that F = f. 28

29 The divergence of a vector field = (,, ) + (,, ) + (,, ) is the scalar function = = + +. Proof Example 4 Show that the vector field = + cannot be written as athe curl of another vector field, that is, F curl G. Green s Theorem can be written in the vector forms, and = ( ) = (, ) So 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. 29

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