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 V8 Vector Fields in Space Just as in Section V1 we considered vector fields in the plane so now we consider vector fields in three space These are fields given by a vector function of the type Such a function assigns the vector F xo yo zo to a point so yo zo where M N and P are all defined We place the vector so its tail is at so yo zo and in this way get the vector field Such a field in space looks a little like the interior of a haystack As before we say F is continuous in some domain D of 3 space we will usually use LLdomain rather than region when referring to a portion of 3 space if M N and P are continuous in that domain We say F is continuously differentiable in the domain D if all nine first partial derivatives Mz My Mz Nx Ny Nz Px PY pz exist and are continuous in D Again as before we give two physical interpretations for such a vector field The three dimensional force fields of different sorts gravitational electrostatic electromagnetic all give rise to such a vector field at the point so yo zo we place the vector having the direction and magnitude of the force which the field would exert on a unit test particle placed at the point The three dimensional flow fields and velocity fields arising from the motion of a fluid in space are the other standard example We assume the motion is steady state i e the direction and magnitude of the flow at any point does not change over time We will call this a three dimensional flow As before we allow sources and sinks places where fluid is being added to or removed from the flow Obviously we can no longer appeal to people standing overhead pouring fluid in at various points they would have to be aliens in four space but we could think of thin pipes inserted into the domain at various points adding or removing fluid The velocity field of such a flow is defined just as it was previously v x y z gives the direction and magnitude speed of the flow at x y z The flow field F dv where d x y z is the density may be similarly interpreted dir F the direction of flow 2 IF1 mass transport rate per unit area at x y z in the flow direction that is IF1 is the rate per unit area at which mass is transported across a small piece of plane perpendicular to the flow at the point x y z The derivation of this interpretation is exactly as in Sections V1 and V3 replacing the small line segment A1 by a small plane area AA perpendicular to the flow Example 1 Find the three dimensional electrostatic force field F arising from a unit positive charge placed at the origin given that in suitable units F is directed radially outward from the origin and has magnitude l l p 2 where p is the distance from the origin 2 V VECTOR INTEGRAL CALCLUS Solution The vector x i y j and has magnitude p Therefore zk with tail at x y z is directed radially outward E x a m p l e 2 a Find the velocity field of a fluid rotating with constant angular velocity w around the z axis in the direction given by the right hand rule right hand fingers curl in direction of flow when thumb points in the k direction b Find the analogous field if the flow is rotating about the y axis Solution a The flow doesn t depend on z it is really just a two dimensional problem whose solution is the same as before section V1 Example 4 F x y z w yi x j b If the axis of flow is the y axis the flow will have no j component and will not depend on y However by the right hand rule the flow in the xz plane is clockwise when the positive x and z axes are drawn so as to give a right handed system Thus E x a m p l e 3 Find the three dimensional flow field of a gas streaming radially outward with constant velocity from a source at the origin of constant strength Solution This is like the corresponding two dimensional problem section V1 Example 3 except that the area of a sphere increases like the square of its radius Therefore to maintain constant velocity the density of flow must decrease like llp2as you go out from the origin letting 6 be the density and ci be constants we get Notice that in the three dimensional case this field is the same as the one in Example 1 above with the magnitude falling off like l p2 For the two dimensional case the analogue of a point fluid source at the origin is not a point charge at the origin but a uniform charge along a vertical wire both give the field whose magnitude falls off like l l r Exercises Section 6A
View Full Document
Unlocking...