MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 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 V VECTOR INTEGRAL CALCULUS 2 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 dir V w the direction u in 4 lVwl this greatest value I V w 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 where f x y c c constant then at every point xo yo the gradient vector V w at this point is perpendicular to the contour line passing through this point i e 5 the gradient field o f f is perpendicular to the contour curves o f f Example 1 Let w vw d 5 1 T m gj T T Using the definition 3 of gradient we find xi yj T The domain of V w is the xy plane with 0 O deleted and it is continuously differentiable in this region Since Ix i y jl T we see that IVwl 1 Thus all the vectors of the vector field V w 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 1 in that direction Finally the contour curves for w are circles centered at 0 O which are perpendicular to the vectors V w 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 distribution 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 x i y j 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 x y represents the velocity vector of the flow at that point that is its direction gives the direction of flow and its magnitude gives the speed of the flow Then there is the flow field defined by where 6 …
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