Lecture 19 Vector fields Dan Nichols nichols@math.umass.edu MATH 233, Spring 218 University of Massachusetts April 1, 218 (2) Chapter 16 Chapter 12: Vectors and 3D geometry Chapter 13: Curves and vector functions r(t) = x(t), y(t) Chapter 14-15: Surfaces and functions of multiple variables z = f(x, y) Chapter 16: Vector fields, or vector functions of two variables F(x, y) = P (x, y), Q(x, y)
(3) Vector calculus We write a vector function of two variables like this: F(x, y) = P (x, y), Q(x, y) Input is two variables x, y Output is a 2D vector F(x, y) Really just two scalar functions of x, y bundled together as components of a vector F can also be viewed as a vector field: Definition Let D be a region in R 2. A vector field in R 2 is a function that assigns to each point (x, y) in D a two-dimensional vector F(x, y). Input is a point (x, y) in a region of R 2 Output is a 2D vector F(x, y) Simply: for each (x, y) in the domain, draw the arrow F(x, y) at that point. F(x, y) = 1, 1 (constant vector field) 2 (4) Vector fields 1 1 2 2 1 1 2
(5) Vector fields F(x, y) = x 4 y, log(1 + x) 2 1 1 2 1 2 3 4 Example 1: Draw the vector field F(x, y) = y, x. (6) Vector fields: example 2 1 1 2 2 1 1 2
(7) Vector fields: example Example 2: Draw the vector field F(x, y) = x + 1, xy. 2 1 1 2 3 2 1 1 2 (8) Vector fields We ll mostly work with 2D vector fields because they are easier to draw and visualize. But you can also have vector fields in more dimensions. A 3D vector field is like a 3D arrow at each point in R 3 : F(x, y, z) = P (x, y, z), Q(x, y, z), R(x, y, z)
(9) Vector fields Vector fields are extremely important in many areas of science and engineering. You can use them to describe the velocity of a fluid that is flowing Or the force (e.g. gravity) at each point in space. (1) Velocity fields Suppose some object is moving with position function r(t) and we know that at every point (x, y) along its path, its velocity (or tangent vector) is F(x, y). Or, for every value of t, r (t) = F(x(t), y(t)). Or, if we write F(x, y) = P (x, y), Q(x, y), then dx dy = P (x(t), y(t)) dt Example: pinecone floating in a river. Its velocity is always determined by the flow of the water at its current location. You can find such a path by drawing a curve that follows the arrows of a vector field diagram. dt = Q(x(t), y(t)). -1.5-1 -.5.5 1 1.5.5 -.5-1 -1.5
(11) Velocity fields: example Example 3: Suppose the velocity vector of the wind at point (x, y) in a 2D space is y, cos x. A leaf is being blown by the wind so that the leaf s velocity is always equal to the wind velocity at its current position. Describe the leaf s path of motion if it starts from the point ( π 2, 1). Do the same if it instead starts at the point (2π, ). π 2π 3π 2 1-1 -2 (12) Velocity fields Ocean currents near Florida (aoml.noaa.gov)
(13) Force fields Sometimes there is some force (e.g. gravity or magnetism) that affects everything in some region of 2D or 3D space. Force is a vector, with magnitude and direction The force F on an object depends on the object s position (x, y) (or (x, y, z)). So this force can be described as a vector field F(x, y) (or F(x, y, z)). (14) Force fields Suppose there s a big object of mass M centered at the origin, and a small object of mass m located at the point (x, y, z). Let r = x, y, z be the vector from the big object to the small one. Let r = r = x 2 + y 2 + z 2 denote the distance between the objects Newton s Law of Gravitation: the magnitude of the gravitational force exerted on the objects is F = GMm r 2 (G 6.67 1 11 Nm 2 /kg 2 is the gravitational constant) The force pulls the small object directly towards the big one, so the gravitational force vector F on the small object points in the direction of r. M r m
(15) Force fields We know that F has magnitude F = GMm r 2 We know that it points in the direction of r = x, y, z So the force vector field is F(x, y, z) = GMm r 2 = GMm r 2 ( ˆr) ( 1 r = GMm r r 3 ) r GMm = (x 2 + y 2 + z 2 x, y, z. ) 3/2 (16) Force fields Gravitational force field in R 2 of a large mass located at the origin: -1 1 1.5 -.5-1 Suppose an object is at rest somewhere in space and the only force acting on it is the gravitational pull from this large mass. Where will it go?
(17) Gravity Use Newton s second law: F = ma = m d2 r dt GMm (x 2 + y 2 ) F(x, y) = m d2 x, y dt2 d2 x, y = m x, y 3/2 dt2 This gives us a set of ordinary differential equations: d 2 x dt 2 = GMx d 2 y (x 2 + y 2 ) 3/2 dt 2 = GMy (x 2 + y 2 ) 3/2 Solve these equations (taking into account initial position and velocity of the object) to determine the path of motion r(t) = x(t), y(t). This is (usually) very hard! (18) Gravity: example Example 4: Let C be the circle in the xy-plane centered at the origin with radius R. For any α >, the vector function r(t) = R cos( αt), R sin( αt) traces C counterclockwise at constant speed. Show that for some α >, this is a solution to the gravitational force ODEs: d 2 x dt 2 = GMx d 2 y (x 2 + y 2 ) 3/2 dt 2 = GMy (x 2 + y 2 ) 3/2 What is the period (time it takes to complete one orbit) in terms of G, M, and R?
(19) Gravity This model for gravity ignores something important: the big object is also pulled slightly by the little object. So as the big object moves, the force field changes with respect to time Two-body problem. Much more complicated, harder to solve This is why the shape of planetary orbits around the sun are not circles; they are actually ellipses Kepler s laws: two objects revolve around their common barycenter (center of mass) in elliptical orbits (2) Gravity Suppose an object is subject to gravitational force from two large objects, e.g. binary stars. -2-1 1 2 2 Three-body problem: each object pulls the other two. Sundman (1912): you can write solutions as power series, but they converge too slowly to be useful. 1-1 -2
(21) Velocity fields vs. force fields When a vector field represents velocity (e.g. ocean currents, wind), objects in the field move by following the arrows, because the vector field controls the first derivative of r(t). Initial position of an object completely determines its path of motion. When a vector field represents a force (e.g. gravity, magnetism), finding the path of motion is more complicated because the vector field controls the second derivative of r(t). Object s path of motion depends on both its initial position and its initial velocity. (22) The gradient We ve already seen vector fields in a different context. For a (scalar) function f(x, y), the gradient is a vector field: f(x, y) = f x (x, y), f y (x, y). Now suppose F is some vector field. We say F is conservative if F is the gradient of some scalar function, i.e. if F = f for some scalar function f. In this case, we call f a potential function for F (or just a potential) The term conservative comes from conservation of energy. For example: F(x, y) = y 2, 2xy + 1 is conservative, and f(x, y) = xy 2 + y is a potential function for this field But G(x, y) = x, x is not conservative, there is no potential function. (We ll see why soon... )
(23) The gradient z = xe x2 y 2 1 2 2 1 1 2 2 (24) Conservative vector fields How can we tell if a given vector field F is conservative? There s no good test that works for every field, but in R 2 there s an easy way to see that some fields are not conservative If F is conservative, how do we find a potential f? These questions will become very important later when we learn how to compute the line integral of a vector field along a curve.
(25) Conservative vector fields Let F(x, y) = P (x, y), Q(x, y) be a vector field in R 2. Here s a quick way to tell if F is NOT conservative. Suppose there s some function f(x, y) such that f(x, y) = f x (x, y), f y (x, y) = F(x, y) Then f x (x, y) = P (x, y) and f y (x, y) = Q(x, y). Clairaut s theorem: if f,f x,f y are continuous, then y f x(x, y) = x f y(x, y) and therefore P y = Q x So if P y Q x, then F(x, y) = P (x, y), Q(x, y) is not conservative. (26) Indefinite partial integrals The indefinite partial integral of f(x, y) with respect to x is the function* F (x, y) such that xf (x, y) = f(x, y) (AKA partial antiderivative w.r.t x) As with a single-variable indefinite integral, we add an integration constant C to show that there are infinitely many antiderivatives. For example: ˆ e xy dx = 1 y exy + C(y). Note that since we re only integrating with respect to x, the integration constant can depend on y. So we write C(y) instead of C. Indefinite partial integral of that same function with respect to y: ˆ e xy dy = 1 x exy + C(x).
(27) Conservative vector fields We want to determine if F = P (x, y), Q(x, y) is conservative. That is, does there exist some f(x, y) such that f(x, y) = f x (x, y), f y (x, y) = P (x, y), Q(x, y)? If P (x, y) = xf(x, y), then ˆ f(x, y) = P (x, y) dx = F (x, y) + C(y) If Q(x, y) = y f(x, y), then ˆ f(x, y) = Q(x, y) dy = G(x, y) + D(x) Idea: compute these indefinite partial integrals, see if they are compatible. That is, see if there s a way to choose C(y) and D(x) such that the expressions above are equal. (28) Conservative vector fields: example Example 5: Show that F is conservative and find a potential function f. F(x, y) = sin(x + y) sin y + 2x, sin(x + y) x cos y 1 ˆ ˆ P (x, y) dx = cos(x + y) x sin y + x 2 + C(y) Q(x, y) dy = cos(x + y) x sin y y + D(x) We can make these two indefinite partial integrals agree if we choose C(y) = y and D(x) = x 2. So f(x, y) = cos(x + y) x sin y + x 2 y is a potential for F.
(29) Conservative vector fields: example Example 6: Prove that F(x, y) = 2xy 1, 2xy + 2y is not conservative by computing partial integrals. ˆ P (x, y) dx = x 2 y x + C(y) ˆ Q(x, y) dy = xy 2 + y 2 + D(x) Notice that these parts don t match up. There s no way to choose C(y) and D(x) to make the two partial integrals equal. (3) Conservative vector fields: example Example 7: Determine whether each vector field is conservative. Find a potential for each conservative field. F(x, y) = y, x G(x, y) = y, x H(x, y) = x, x 2 U(x, y) = 1 y, x y 2 V(x, y) = y 2 e xy, e xy + xye xy not conservative conservative: g(x, y) = xy not conservative conservative; u(x, y) = x y. not conservative
(31) Homework Paper homework #19 due Thursday WebAssign homework 15.4 (actually 15.3) due tomorrow night, 11:59 PM Midterms will be handed back on Thursday