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HARVARD MATH 21A - Vector Fields

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4/7/2004, VECTOR FIELDS Math21a, O. KnillVECTOR FIELDS.Planar vector field.A vector field in the plane is a map, which assigns to each point (x, y)in the plane a vector F (x, y) = (M (x, y), N(x, y)).Vector field in space.A vector field in space is a map, which assigns to each point (x, y, z) inspace a vector F(x, y, z) = (M(x, y, z), N(x, y, z), P (x, y, z)).-2 -1 1 2-2-112PLANAR VECTOR FIELD EXAMPLES.1) F (x, y) = (y, −x) is a planar vector field which you see in a pictureon the right.2) F (x, y) = (x − 1, y)/((x − 1)2+ y2)3/2− (x + 1, y)/((x + 1)2+ y2)3/2is the electric field of positive and negative point charge. It is calleddipole field. It is shown in the picture above.-2 -1 1 2-2-112GRADIENT FIELD. 2D: If f (x, y) is a function of two variables, then F (x, y) = ∇f(x, y) is called a gradientfield. The same in 3D: gradient fields are of the form F (x, y, z) = ∇f(x, y, z).EXAMPLE. (2x, 2y, −2z) is the vector field which is orthogonal to hyperboloids x2+ y2− z2= const.EXAMPLE of a VECTOR FIELD. If H(x, y) is a function of two variables, then (Hy(x, y), −Hx(x, y)) is calleda Hamiltonian vector field. An example is the harmonic Oscillator H(x, y) = x2+ y2. Its vector field(Hy(x, y), −Hx(x, y)) = (y, −x) is the same as in example 1) above.WHEN IS A VECTOR FIELD A GRADIENT FIELD (2D)?F (x, y) = (M(x, y), N(x, y)) = ∇f(x, y) implies Nx(x, y) = My(x, y). If this does not hold at some point, F isno gradient field. We will see next week that the condition curl(F ) = Nx− My= 0 is also necessary for F tobe a gradient field.EXAMPLE. VECTOR FIELDS IN BIOLOGY.Let x(t) denote the population of a ”prey species” like tuna fish and y(t) is the population size of a ”predator”like sharks. We have x0(t) = ax(t) + bx(t)y(t) with positive a, b because both more predators and more preyspecies will lead to prey consumption. The rate of change of y(t) is −cy(t) + dxy, where c, d are positive.We have a negative sign in the first part because predators would die out without food. The second term isexplained because both more 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 + 0.2xyVolterra explained with such systems the oscillation of fish pop-ulations in the Mediterranean sea. At any specific point (x, y) =(x(t), y(t)), there is a curve r(t) = (x(t), y(t)) through thatpoint for which the tangent r0(t) = (x0(t), y0(t) is the vector(0.4x − 0.4xy, −0.1y + 0.2xy).VECTOR FIELDS IN PHYSICSNewton’s law m~r00= F relates the acceleration ~r00of a body withthe force F acting at the point. For example, if x(t) is the positionof a mass point in [−1, 1] attached at two springs and the massis m = 2, then the point experiences a force (−x + (−x)) = −2xso that mx00= 2x or x00(t) = −x(t). If we introduce y(t) = x0(t)of t, then x0(t) = y(t) and y0(t) = −x(t). Of course y is thevelocity of the mass point, so a pair (x, y), thought of as an initialcondition, describes the system so that nature knows what thefuture evolution of the system has to be given that data.-11We don’t yet know yet the curve t 7→ (x(t), y(t)), but we know thetangents (x0(t), y0(t)) = (y(t), −x(t)). In other words, we know adirection at each point. The equation (x0= y, y0= −x) is called asystem of ordinary differential equations (ODE). More generally,the problem when studying ODE’s is to find solutions x(t), y(t)of equations x0(t) = f(x(t), y(t)), y0(t) = g(x(t), y(t)). Here welook for curves x(t), y(t) so that at any given point (x, y), the tan-gent vector (x0(t), y0(t)) is (y, −x). You can check by differentia-tion that the circles (x(t), y(t)) = (r sin(t), r cos(t)) are solutions.They form a family of curves. Can you interpret these solutionsphysically?VECTOR FIELDS IN MECHANICSIf x(t) is the angle of a pendulum, then the gravity acting on it produces a forceF (x) = −gm sin(x), where m is the mass of the pendulum and where g is a constant.For example, if x = 0 (pendulum at bottom) or x = π (pendulum at the top), thenthe force is zero.The Newton equation ”mass times acceleration = Force” gives¨x(t) = −g sin(x(t)) .xThe equation of motion for the pendulum ¨x(t) = −g sin(x(t)) can be writtenwith y = ˙x also 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)). Writing down explicit formulas for (x(t), y(t)) is in this case notpossible with known functions like sin, cos, exp, log etc. However, one still canunderstand the curves:Curves on the top of the picture represent situations where the velocity y is large. They describe the pendulumspinning around fast in the clockwise direction. Curves starting near the point (0, 0), where the pendulum is ata stable rest, describe small oscillations of the pendulum.VECTOR FIELDS IN METEOROLOGY. On maps likehttp://www.hpc.ncep.noaa.gov/sfc/satsfc.gif one can see Isoterms,curves of constant temperature or pressure p(x, y) = c. These are levelcurves. The wind maps are vector fields. F (x, y) is the wind velocityat the point (x, y). The wind velocity F is not always normal to theisobares, the lines of equal pressure p. The scalar pressure field p andthe velocity field F depend on time. The equations which describe theweather dynamics are called the Navier Stokes equationsd/dtF + F · ∇F = ν∆F − ∇p + f, divF = 0(we will see what is ∆, div later.) It is a partial differential equationlike ux− uy= 0. Finding solutions is not trivial: 1 Million dollars aregiven to the person proving that the equations have smooth solutions


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HARVARD MATH 21A - Vector Fields

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