16 Vector Calculus cid 18 f1 x y cid 19 f2 x y 16 1 Vector Fields De nition A vector eld in the plane is a function F x y from a domain D R2 to the set of vectors in the plane We write F x y f1 x y i f2 x y j cid 104 f1 x y f2 x y cid 105 f1 f2 are the scalar components of F is at the point x y and the nose at cid 0 x f1 x y y f2 x y cid 1 A vector eld F x y consists of a set of arrows one for every pair x y The tail of each arrow When plotting choose enough points x y to make the overall picture clear Notation and terminology Vector elds like vectors are typed in bold They can be hand written either underlined F preferred or with an arrow cid 126 F The magnitude of a vector eld is the scalar function F x y returning the length of the vector F x y at each point x y The radial vector eld x its magnitude r is the scalar quantity r cid 112 x2 y2 Some authors use x for the same eld y has the common short hand r This is congruent with the fact that Examples For the following note that the computer typically shrinks the lengths of vectors in order to draw more for clarity cid 18 x 4 cid 19 1 1 F x y cid 19 cid 18 x y 1 cid 112 x2 y2 1 r r 2 F x y 1 22y 4 224x 22y 4 224x cid 18 y cid 19 x 3 F x y 1 cid 112 x2 y2 cid 18 y cid 19 x 4 F x y Note that examples 2 and 4 have domain D R2 0 0 Vector elds in real life Vector elds typically represent one of two physical ideas Velocity Fields A particle at location r has velocity F r Force Fields A particle at location r experiences a force F r A map of wind patterns could be interpreted as either each arrow represents 1 The wind velocity at a point how fast the air is moving 2 The force of the wind on a stationary observer Examples Here are three examples of vector elds and associated data being used to usefully sum marize data Ocean Currents Wind Patterns Animated Wind Patterns 2 22y 4 224x 22y 4 224x Vector Fields in 3D De nition A vector eld in three dimensions is a function F x y z from a domain D R3 to the set of vectors in 3 space We write f1 x y z f2 x y z f3 x y z f1 x y z i f2 x y z j f3 x y z k F x y z cid 104 f1 x y z f2 x y z f3 x y z cid 105 where f1 f2 f3 are the scalar components of F Plotting vector elds in three dimensions is not advisable by hand use a computer package When you have to visualize a 3D vector eld see if you can build up your understanding by covering up say the vertical component if you know what the eld is doing horizontally then vertically you can put these together in your mind Just as in 2D we have the short hands r Notation what we called when computing spherical polar intergals In vector calculus r usually means r regardless of the dimension Examples y z cid 16 x cid 17 and r r cid 112 x2 y2 z2 this last is 1 cid 112 x2 y2 1 y x 1 1 F x y z Unit vector eld Horizontal i j parts rotating clockwise compare with examples 3 4 above Vertical k part pointing upwards If a feather was dropped into this vector eld it would follow an ascending helix 1 cid 112 x2 y2 z2 x y z 1 r r 2 F x y z Also a unit vector eld All vectors point away from origin length 1 Domain excludes origin 3 De nition The gradient eld of a scalar function f x y is the vector eld Gradient Fields cid 18 fx x y cid 19 fy x y F x y f x y f is a potential function for F If F f for some f we say that F is conservative The de nition is similar in 3D f x y z cid 32 fx x y z cid 33 fy x y z fz x y z Notation is often used in Physics for potential functions Physicists also typically choose a potential function to be the negative of ours F f a good reason will be given later Examples 1 xy y x 2 x2 cos y sin x 3 Since cid 16 2x cos y cos x cid 17 x2 sin y 2 x2 y2 1 2 etc we have x x2 y2 1 2 2x 1 x2 y2 1 2 x2 y2 1 2 x y 4 In 3 dimensions This can be written r 1 r r in polar co ordinates x2 y2 z2 1 2 x2 y2 z2 3 2 cid 16 x y z cid 17 which can be written r 1 1 r3 r or 1 1 3 r if you prefer Level Curves Surfaces Recall that the gradient vector f points in the direction of greatest increase of f Otherwise said f points orthogonally to the level curves of f level surfaces in 3D Example The gradient eld 3x2 y2 is orthogonal to the ellipses with equations 3x2 y2 C C 0 constant cid 16 6x cid 17 2y 4 22y 4 224x Are all vector elds conservative No in fact most are not We will see that potential functions play a similar role to that of anti derivatives with respect to the Fundamental Theorem in single variable calculus While any continuous function has an anti derivative most continuous vector elds do not have a potential function Conservative vector elds are in a probabilistic sense very rare They are however very common in important theories from Physics cid 19 cid 18 x x Example F is not conservative If it were conservative there would exist a function f for which fx x fy We can attempt to partially integrate both of these equations When integrating with respect to x the constant of integration g y is an arbitrary function the most general expression whose x derivative is zero x2 g y fx x f x y fy x f x y xy h x 1 2 for some arbitrary functions g h If f is to exist and F is to be conservative we must be able to choose funcrions g y h x so that the above expressions for f are both true This is clearly impossible whence F is non conservative Later we will see a much easier way to show that a vector eld is non conservative 5