Math S21a: Multivariable calculus Oliver Knill, Summer 2011Lecture 19: VectorfieldsA vector field in the plane is a map, which assigns to each point (x, y) in the planea vector~F (x, y) = hP (x, y), Q(x, y)i. A vector field in space is a map, which assignsto each point (x, y, z) in space a vector~F (x, y, z) = hP (x, y, z), Q(x, y, z), R(x, y, z)i.For example~F (x, y) = hx −1, yi/((x −1)2+ y2)3/2−hx + 1, yi/((x + 1)2+ y2)3/2is the electric fieldof positive and negative point charge. It is called dipole field. It is shown in the picture belowIf f(x, y) is a function of two var iables, then~F (x, y) = ∇f(x, y) is called a gradientfield. Gradient fields in space are of the form~F (x, y, z) = ∇f (x, y, z).When is a vector field a gradient field?~F (x, y) = hP (x, y), Q(x, y)i = ∇f(x, y) implies Qx(x, y) =Py(x, y). If this does not hold at some point, F is no gradient field.Clairot test: If Qx(x, y) − Py(x, y) is not zero at some point, then~F (x, y) =hP (x, y), Q(x, y)i is not a gradient field.We will see next week that the condition curl(F ) = Qx− Py= 0 is also necessary for F to bea g radient field. In class, we see more examples on how to construct the potential f fr om thegradient field F .1 Is the vector field~F (x, y) = hP, Qi = h3x2y + y + 2, x3+ x − 1i a gradient field? Solution:the Clairot test shows Qx− Py= 0. We integrate the equation fx= P = 3x2y + y + 2 andget f(x, y) = 2x + xy + x3y + c(y). Now take the derivative of this with respect to y to getx + x2+ c′(y) and compare with x3+ x − 1. We see c′(y) = −1 and so c(y) = −y + c. Wesee the solutionx3y + xy − y + 2x .2 Is the vector field~F (x, y) = hxy, 2xy2i a gradient field? Solution: No: Qx− Py= 2y2− xis not zero.Vector fields are important in differential equations. We look at some examples in populationdynamics and mechanics. You can skip this part. This is more motivational.3 Let x(t) denote the population of a ”prey species” like tuna fish and y(t) is the popula t ionsize of a ”predator” like sharks. We have x′(t) = ax(t) + bx(t)y(t) with positive a, b becauseboth more predators and more prey species will lead to prey consumption. The rate of changeof y(t) is −cy(t) + dxy, where c, d are positive. We have a negative sign in the first partbecause predators would die out without food. The second term is explained because bothmore predators as well as more prey leads to a growth of predators through reproduction.A concrete example is the Volterra-Lodka system˙x = 0.4x − 0.4xy˙y = −0.1y + 0.2xy .Volterra explained with such systems the oscillation of fish populations in the Mediterraneansea. At any specific point ~r (x, y) = hx(t), y(t)i, there is a curve = ~r(t) = hx(t), y(t)i throughthat point for which the tangent ~r′(t) = (x′(t), y′(t) is the vector h0.4x − 0.4xy, −0.1y +0.2xyi.4 A class vector fields important in mecha nics are Hamiltonian fields: If H(x, y) is a functionof two variables, then hHy(x, y), −Hx(x, y)i is called a Hamiltonian vector field. Anexample is the harmonic oscillator H(x, y) = x2+y2. Its vector field (Hy(x, y), −Hx(x, y)) =(y, −x). The flow lines of a Hamiltonian vector fields are located on the level curves of H(as you have shown in th homewo r k with the chain rule).5 Newton’s law m~r′′= F relates the acceleration ~r′′of a body with t he force F acting atthe point. For example, if x(t) is the position of a mass point in [−1, 1] attached at twosprings and the mass is m = 2, t hen t he point experiences a force (−x + (−x)) = −2x sothat mx′′= 2x or x′′(t) = −x(t). If we introduce y(t) = x′(t) of t, then x′(t) = y(t) andy′(t) = −x(t). Of course y is the velocity of the mass point, so a pair (x, y), thought of asan initial condition, describes the system so that nature knows what the future evolution ofthe system has to be given that data.6 We don’t yet know yet the curve t 7→ ~r(t) = hx(t), y(t)i, but we know the tangents ~r′(t) =hx′(t), y′(t)i = hy(t), −x(t)i. In other words, we know a direction at each point. Theequation (x′= y, y′= −x) is called a system o f ordinary differential equations (ODE’s)More generally, the problem when studying ODE’s is to find solutions x(t), y(t) of equationsx′(t) = f (x(t), y(t)), y′(t) = g(x(t), y(t)). Here we look for curves x(t), y(t) so that at anygiven point (x, y), the tangent vector (x′(t), y′(t)) is (y, −x). You can check by differentiationthat t he circles (x(t), y(t)) = (r sin(t), r cos(t)) are solutions. They form a family of curves.-117 If x(t) is the angle of a pendulum, then the gravity acting on it produces a force G(x) =−gm sin(x), where m is the mass of the p endulum and where g is a constant. For example,if x = 0 (pendulum at bottom) or x = π (pendulum at the top), then the force is zero.The Newton equation ”mass times acceleration = force” gives¨x(t) = −g sin ( x(t)) .8 The equation of motion for the pendulum ¨x(t) = −g sin(x(t)) can be written with y = ˙xalso asddt(x(t), y(t)) = (y(t), −g sin(x(t)) .Each possible motion of the pendulum x(t) is described by a curve ~r(t) = (x(t), y(t)). Writ -ing down explicit formulas for (x(t), y(t)) is in this case not possible with known functionslike sin, cos, exp, log etc. However, one still can understand the curves.xCurves on the top of the picture represent situatio ns where the velocity y is large. They describethe pendulum spinning around fast in the clockwise direction. Curves starting near the point(0, 0), where the pendulum is at a stable rest, describe small oscillations of the pendulum.Vector fields in weather forecast On weather maps, one can see isoterms, curves of constanttemperature or isobars, curves p(x, y) = c of constant pressure. These are level curves. The windvelocity~F (x, y) is close but not always exactly perpendicular to the isobars, t he lines of equalpressure p. In reality, the scalar pressure field p and the velocity field~F also depend on time. Theequations which describe the weather dynamics are called the Navier Stokes equationsddt~F +~F ·∇~F = ν∆~F −∇p + f, div~F = 0(where ∆ and div are defined later. This is an other example of a partial differential equation.It is one of the millenium problems to prove that these equations have smoot h solutions in space.Homework1 The vector field~F (x, y) = hx/r3, y/r3i appears in electrostatics, where r =√x2+ y2is thedistance to the charge. Find a function f(x, y) such
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