Vector Analysis 15 Copyright © Cengage Learning. All rights reserved.

Vector Fields Copyright © Cengage Learning. All rights reserved. 15.1

3 Understand the concept of a vector field. Determine whether a vector field is conservative. Find the curl of a vector field. Find the divergence of a vector field. Objectives

4 Vector Fields

5 In Ch 12, studied vector valued functions – assign a vector to a real number. -- represent curves and motion along a curve (in general in 3D). In this chapter, we study functions that assign a vector to a point in the plane or a point in space are called vector fields, and they are useful in representing various types of force fields and velocity fields.

6 The gradient is one example of a vector field. 2D example: f(x, y, z) = x 2 + y 2 =>  f(x, y, z) = f x (x, y, z)i + f y (x, y, z)j = 2xi + 2yj = Remember from Ch 13 that the graphical interpretation of this field is a family of vectors in xy plane each of which points in the direction of maximum increase in function z=f(x,y) 3D example: f(x, y, z) = x 2 + y 2 + z 2 then gradient of f  f(x, y, z) = f x (x, y, z)i + f y (x, y, z)j + f z (x, y, z)k = 2xi + 2yj + 2zk = Vector field in space is a vector field in 3D space. Note that the component functions for this particular vector field are 2x, 2y, and 2z. Vector Fields

7 A vector field F(x, y, z) = M(x, y, z)i + N(x, y, z)j + P(x, y, z)k is continuous at a point if and only if each of its component functions M, N, and P is continuous at that point. Some common physical examples of vector fields are velocity fields, gravitational fields, and electric force fields. Vector Fields

8 1.Velocity fields describe the motions of systems of particles in the plane or in space. For instance, Figure 15.1 shows the vector field determined by a wheel rotating on an axle. Notice that the velocity vectors are determined by the locations of their initial points—the farther a point is from the axle, the greater its velocity. Velocity fields are also determined by the flow of liquids through a container or by the flow of air currents around a moving object, as shown in Figure Figure 15.1 Vector Fields

9 2. Gravitational fields are defined by Newton’s Law of Gravitation, which states that the force of attraction exerted on a particle of mass m 1 located at (x, y, z) by a particle of mass m 2 located at (0, 0, 0) is given by where G is the gravitational constant and u is the unit vector in the direction from the origin to (x, y, z). Vector Fields In Figure 15.3, you can see that the gravitational field F has the properties that F(x, y, z) always points toward the origin, and that the magnitude of F(x, y, z) is the same at all points equidistant from the origin (i.e. on a sphere centered at origin). A vector field with these two properties is called a central force field.

10 Using position vector r = xi + yj + zk for the point (x, y, z), you can write the gravitational field F as Vector Fields

11 3. Electric force fields are defined by Coulomb’s Law, which states that the force exerted on a particle with electric charge q 1 located at (x, y, z) by a particle with electric charge q 2 located at (0, 0, 0) is given by where r = xi + yj + zk, u = r/||r||, and c is a constant that depends on the choice of units for ||r||, q 1, and q 2. Vector Fields

12 Note that an electric force field has the same form as a gravitational field. That is, Such a force field is called an inverse square field. Vector Fields

13 Example 1 – Sketching a Vector Field Sketch some vectors in the vector field given by F(x, y) = –yi + xj. Solution: You could plot vectors at several random points in the plane. However, it is more enlightening to plot vectors of equal magnitude. This corresponds to finding level curves in scalar fields. In this case, vectors of equal magnitude lie on circles.

14 Example 1 – Solution To begin making the sketch, choose a value for c and plot several vectors on the resulting circle. For instance, the following vectors occur on the unit circle of radius 1. These and several other vectors in the vector field are shown in Figure Figure 15.4 cont’d F(x, y) = –yi + xj

15 Matlab/Octave xm = -3; xM = 3; dx = xM/8; [X,Y] = meshgrid(xm:dx:xM); FX = -Y; FY = X ; contour(X,Y,sqrt(FX.^2+FY.^2)); hold on quiver(X,Y,FX,FY) colormap hsv hold off

16 My example Here level curves of the norm scalar field are ellipses. FX = X; FY = 3*Y ;

17 Book Ex 3, p 1061 Note that at the edge of the tube v = 0

18 dx = 0.2; [X,Y] = meshgrid(-3:dx:3,-3:dx:3); FX = 0; FY = 0; FZ = 16-X.^2-Y.^2; Z = FZ; quiver3(X,Y,Z,FX,FY,FZ,dx);

19 Conservative Vector Fields

20 Conservative Vector Fields Note in my example above (see right) that all vectors are orthogonal to level curve from which they emanate. Because of the properties of gradient, it is natural to ask if the corresponding vector field F = is the gradient of some differential scalar function f. Some vector fields can be represented as the gradients of differentiable functions and some cannot—those that can are called conservative vector fields. The vector field F = is conservative. To see this let f(x,y) = x^2/2 + (3/2)y^2 => grad(f) =

21 Every inverse square field is conservative. To see this, let and where u = r/||r||. Because cont’d Example 4(b) – Conservative Vector Fields it follows that F is conservative.

22 Conservative Vector Fields The following important theorem gives a necessary and sufficient condition for a vector field in the plane to be conservative. Note The sufficiency of this condition is proved later in 15.4 Also note that this theorem is only valid for simply connected domains -- no holes (more in 15.4).

23 My example: Note that solution for f is comparable to an indefinite integral. It represents a family of potential functions, any two of which differ by only a constant. To find the constant and thus the unique solution, we need to specify initial condition. FX=3+2*X.*Y; FY = X.^2-3*Y.^2 ;

24

25 My example: Not conservative

26 Curl of a Vector Field Theorem 15.1 has a counterpart in 3D. 1 st need definition of the curl of a vector field in space:

27 The cross product notation used for curl comes from viewing the gradient  f as the result of the differential operator  acting on the function f. In this context, you can use the following determinant form as an aid in remembering the formula for curl.

28 Proof: Same as in 2D and we still have to prove sufficiency in 15.4 Note that conservative fields are also called irrotational

29 Example 7 – Finding the Curl of a Vector Field Find curl F of the vector field given by F(x, y, z) = 2xyi + (x 2 + z 2 )j + 2yzk. Is F irrotational? Solution: The curl of F is given by curl F(x, y, z) =   F(x, y, z) Because curl F = 0, F is irrotational.

30 continue:

31 My example

32 My example: Not a conservative field.

33 Divergence of a Vector Field

34 Divergence of a Vector Field You have seen that the curl of a vector field F is itself a vector field. Another important function defined on a vector field is divergence, which is a scalar function.

35 The dot product notation used for divergence comes from considering  as a differential operator, as follows. Divergence of a Vector Field

36 Example 9 – Finding the Divergence of a Vector Field Find the divergence at (2, 1, –1) for the vector field Solution: The divergence of F is At the point (2, 1, –1), the divergence is

37 My example

38 Divergence of a Vector Field 86, 88, 89 are variation of product rule. Component by component proofs are long. Easier just to apply product rule and create proper vector or scalar delta operators.