MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.V. VECTOR INTEGRAL CALCULUS V1. Plane Vector Fields 1. Vector fields in the plane; gradient fields. We consider a function of the type where M and N are both functions of two variables. To each pair of values (xo, yo) for which both M and N are defined, such a function assigns a vector F(xo, yo) in the plane. F is therefore called a vector function of two variables. The set of points (x, y) for which F is defined is called the domain of F. To visualize the function F(x, y), at each point (xo, yo) in the domain we place the corresponding vector F(xo, yo) so that its tail is at (xo, yo). Thus each point of the domain is the tail end of a vector, and what we get is called a vector field. This vector field gives a picture of the vector function F(x, Y). Conversely, given a vector field in a region of the xy-plane, it determines a vector function of the type (I), by expressing each vector of the field in terms of its i and j components. Thus there is no real distinction between "vector function" and "vector field". Mindful of the applications to physics, in these notes we will mostly use "vector field". We will use the same symbol F to denote both the field and the function, saying "the vector field F" , rather than "the vector field corresponding to the vector function F". We say the vector field F is continuous in some region of the plane if both M(x, y) and N(x, y) are continuous functions in that region. The intuitive picture of a continuous vector field is that the vectors associated to points sufficiently near (xo, yo) should have direction and magnitude very close to that of F(xo, yo) -in other words, as you move around the field, the vectors should change direction and magnitude smoothly, without sudden jumps in size or direction. In the same way, we say F is differentiable in some region if M and N are differentiable, that is, if all the partial derivatives exist in the region. We say F is continuously differentiable in the region if all these partial derivatives are themselves continuous there. In general, all the commonly used vector fields are continuously differentiable, except perhaps at isolated points, or along certain curves. But as you will see, these points or curves affect the properties of the field in very important ways.2 V. VECTOR INTEGRAL CALCULUS Where do vector fields arise in science and engineering? One important way is as gradient vector fields. If is a differentiable function of two variables, then its gradient is a vector field, since both partial derivatives are functions of x and y. We recall the geometric interpretation of the gradient: (4) dir Vw = the direction u in lVwl = this greatest value where I= Vw . u is the directional derivative of w in the direction u. ds u Another important fact about the gradient is that if one draws the contour curves of f (x, y), which by definition are the curves f(x, y) = c, c constant, then at every point (xo, yo), the gradient vector Vw at this point is per- pendicular to the contour line passing through this point, i.e., (5) the gradient field off is perpendicular to the contour curves off . Example 1. Let w = d m = T. Using the definition (3) of gradient, we find xi vw = 5.-1 +gj +yj = T T T The domain of Vw is the xy-plane with (0,O) deleted, and it is con- tinuously differentiable in this region. Since Ix i + y jl = T, we see that IVwl = 1 . Thus all the vectors of the vector field Vw are unit vectors, and they point radially outward from the origin. This makes sense by (4), since the definition of w shows that dwlds should be greatest in the radially outward direction, and have the value 1in that direction. Finally, the contour curves for w are circles centered at (0, O), which are perpendicular to the vectors Vw everywhere, as (5) predicts. 2. Force and velocity fields. Continuing our search for ways in which vector fields arise, here are two physical situations which are described mathematically by vector fields. We shall refer to them often in the sequel, using our physical intuition to suggest the sort of mathematical properties that vector fields ought to have.V1. PLANE VECTOR FIELDS 3 Force fields. From physics, we have the two-dimensional electrostatic force fields arising from a distri- bution of static (i.e., not moving) charges in the plane. At each point (xo, yo) of the plane, we put a vector representing the force which would act on a unit positive charge placed at that point. In the same way, we get vector fields arising from a distribution of masses in the xy-plane, representing the gravitational force acting at each point on a unit mass. There are also the electromagnetic fields arising from moving electric charges and/or a distribution of magnets, representing the magnetic force at each point. Any of these we shall simply refer to as a force field. Example 2. Find the two-dimensional electrostatic force field F arising from a unit positive charge placed at the origin, given that F is directed radially away from the origin and that it has magnitude c/r2. Solution. Since the vector xi + yj with tail at (x, y) is directed radially outward and has magnitude r, it has the right direction, and we need only change its magnitude to c/r2. We do this by multiplying it by c/r3, which gives Flow fields and velocity fields A second way vector fields arise is as the steady-state flow fields and velocity fields. Imagine a fluid flowing in a horizontal shallow tank of uniform depth, and assume that the flow pattern at any point is purely horizontal and not changing with time. We will call this a two-dimensional steady-state flow or for short, simply a flow. The fluid can either be compressible (like a gas), or incompressible (like water). We also allow for the possibility that at various points, fluid is beiong added to or subtracted from the flow; for instance, someone could be standing over the tank pouring in water at a certain point, or over a certain area. We also allow the density to vary from point to point, as it would for an unevenly heated gas. With such a flow we can associate two vector fields. There is the velocity field v(x, y) where the vector v(x, y) at the point
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