Fields&Operators User`s Manual
Contents
1. 11 x 2 y 2 in layer 1 Change the name of the layer if you wish No other changes are necessary The gradient in layer 2 will automatically be recomputed for this function Fields amp Operators 21 Quick Start Menu 22 Fields Operators If you develop a way of looking at functions that you would like to use for many different functions consider placing your graph in the Quick Start menu The files listed in Quick Start are ordinary Fields amp Operators files but access to them is easy The idea is for these graphs to serve as templates for constructing graphs with a specific configuration Open the Quick Start sub menu from the File menu and select ADD CURRENT FILE TO QUICK START When Fields amp Operators was installed a separate directory was created for Quick Start files You may first save the file in this directory or not as you wish Story Board The Story Board gives you the control of a film director in preparing animations You can preview your animation modify the rotation and scaling of any frame delete frames and append individual frames or complete sequences In a later section we will discuss the mathematics behind the animation in which the horn is made to pass though the torus in the example THROUGH Horn Through Torus Here we will simply use that file to demonstrate the Story Board Open that file now and animate it to see the action before we manipulate it LT Y er A LAN WLLL2. If you have not made any changes you will be looking down along the z axis onto the graph of z sin x t cos y t From this point of view the graph is not very interesting Click one of the rotation buttons at the left of the window Hold the mouse down in a button for continuous rotation You can lock the rotation by holding the g key when you click on a rotation button The graph will continue to rotate in that direction even when you release the mouse Click to Stop but you may want to leave the rotation on R A you try time animation next ze gt ee Animation is one of the most powerful features of Fields amp Operators For now you should know that both real time and recorded animation are possible You will use recorded animation for graphs that take long to draw Since the definition of the graph we are working with already contains the variable you need only click on the top control bar m to start the animation 2 Andow For a quick reminder point to a control with the mouse without clicking A hint will appear at the bottom right of the main window After a few seconds a hint will also appear at the mouse position To get an idea of the kinds of graphs you will be able to create with Fields amp Operators let s take a look at some of the sample graphs that were copied to your hard disk when you installed the program Fields amp Operators 11 There are three ways of opening fil
3. These are a few miscellaneous examples They are chosen as suggestions for using Fields amp Operators Each example has an associated file on your disk Open the files and look at the Layers and Domain dialogs for details on how they are defined All of these examples were designed to be animated Light Ray LIGHTRAY A light ray is composed of perpendicular oscillating electric and magnetic fields The changing electric field induces a magnetic field and the changing magnetic field induces an electric field This is complex behavior even a graph such as that above can be confusing Animation can help in visualizing the action This graph is composed of 5 layers There is a curve for the outline of each field as well as the two fields The fifth layer is a single vector rotating in the y z plane Normalization is turned off for all vector fields so they are drawn to their correct lengths Conic Sections CONICS This example demonstrates the use of Polar coordinates You won t find an option for polar coordinates on the Coordinates dialog Polar coordinates can be though of as the two dimensional version of either cylindrical or spherical coordinates Cylindrical coordinates treat the elevation above the x y plane by giving the z coordinate while spherical coordinates give the angle of elev ation to the point In this example we use cylindrical coordinates In spherical coordinates the angle would be called 6 phi Fie
4. WOT I Our animation will first rotate the combined figure without the horn moving After 10 frames we will start the horn moving and change the direction of the rotation Rotation Per Before we even open the Story Board let s set the amount of rotation Frame per frame This effects the rotation buttons in the main window as well as the story board Open the ROTATION INCREMENT dialog from the Environment menu To have the graph rotate 180 Tr radians in a span of 10 frames enter an increment of 27 20 The horn is made to move by adding f to its position Since we want to hold the horn stationary for the first 10 frames we want to make sure t does not vary Open the Time dialog click E Set its values as shown Start Stop Intervals y We are finally ready to open the Story Board Click Story boara or select from the SHOW STORY BOARD from the Story Board menu Adding Frames When you open the Story Board window it is initially blank Click APPEND FROM ANIMATION SETTINGS The same option is also available on the Story Board menu The 10 frames all with the same value of t and the same viewing angle will appear Fields amp Operators 23 Rotation in the Story Board Let s rotate all frames starting at the first frame Click on the first frame to select it Now click one of the rotation or zoom buttons on the top of the window These work as they do in the main window All frames starting at the selected
5. the domain drawing a graph can take several seconds This can seem painfully slow while you are rotating the graph or making other changes In that case try using FAST WIRE FRAME which limits the number of points in the graph overriding your setting in the domain dialogs You can set an independent Draw Quality for the button To set the quality for the Draw button click it with the Secondary right mouse button Typically you will set the best quality for Draw and use it on demand Now we ask you to spend some time examining the other example files on your own But first you should be aware of the general purpose of the main program dialogs Fields amp Operators 13 Coordinates Layers Domains Time and Animation Story Board Layer Summary Axes Allows you to select Rectangular Cylindrical or Spherical coordinates Each graph is composed of one or more layers The Layers dialog controls the mathematical definition and appearance of the layers There is much to explore here A Domain is an array of points distributed in one two or three dimensions Calculations for each layer can depend directly on values from the shared domain or from a private domain or from other layers which in turn depend on a domain Sets the range of values for during animation Open the dialog by clicking EA Lets you play the film director in controlling your animations Offers a summary of the ma
6. for functions of the form z F x y If we had selected Cylindrical or Spherical coordinates different variables would be used Using the DISPLAY AS drop down list you can select if your graph is to be drawn as a surface vector field or level curves There are restrictions on the use of level curves See Help Also level curves are usually slow to draw because of the extra computations required The choices available here depend on the Operator Later for example we will use the Gradient operator Since the gradient is a vector field no other choices would be available Apply To Appearance Page A New Layer Feel free to experiment with this setting You will have to close the Layers dialog to see the results Don t forget to rotate the graph as needed Return to the surface display to continue with this tour The APPLY TO drop down list allows you to compose functions by using the output from one layer as the input for another We currently have only one layer so the only choice for the source is the domain Click on the tab to turn to the Appearance page for layer 1 l Thick grid lines Shift forward Side 1 a eee See a Fill Fixed Color Color by value MEMO ERBE REN Side2 X Gide EHE MAP Fill Fixed Color Color by value BEBEBBE EEE MET A surface has 2 sides Mathematically this is not strictly be true Can you give an example Can yo
7. how your functions evolve All it takes is a single mouse click Part of understanding functions in two or three dimensions is generalizing the one dimensional derivative In multi variable calculus this leads to the differential operators Divergence Gradient Curl and Laplacian as well as vector derivatives and tangent planes Fields amp Operators computes these operators in closed form and displays the results as vector fields or surfaces whichever is appropriate New features in Fields amp Operators provide tools for exploring the differential geometry of curves the evolute and osculating circle The Touch Panel brings a new level of hands on control to mathematical graphics Try it You need not know anything about calculus to use and enjoy Fields amp Operators Even without the advanced features there will be enough to delight users with all levels of mathematical experience Our intention is to supply a tool for experimentation and most importantly for the simple enjoyment of mathematics Fields amp Operators 3 4 Fields amp Operators Installing Fields amp Operators Learning to Use Fields amp Operators Although Fields amp Operators is a powerful program with many features we hope that you find its command structure both convenient and natural After you spend some time with the Guided Tour in the next section you should be quite comfortable with Fields amp Operators Fields amp Operators inc
8. in each frame We reserve a discussion of the Tangent operator until later Operator Tangent to Tangent Plane 8 Name Display as Tangent plane E Point of tangency When you inspect the layer definition for the tangent operator pay attention to the TANGENT TO drop down list Also check the path for the tangent animation on the Tangent At dialog accessed through the button which appears on the Layer dialog when the operator is Tangent R2 sR1 SINCOS We have already seen an example of a function R2 gt R1 z F x y sin x cos y Note that this is a bona fide function Any vertical line intersects the graph at most once I e any point x y in the domain has only one result z associated with it Fields amp Operators 39 40 Fields Operators The notion of derivative should tell us how the output of a function at a given point changes when we change the input What can this mean if the input is two dimensional In this case we can ask about the directional derivatives How does the function change at a point if we change the input in a specific direction In a sense the Gradient field captures this information If u is a unit vector in some direction the derivative in that direction can be shown to be uegrad f The gradient at each point indicates the direction in the function changes fastest that is the direction in which the derivative is largest In the example printed above the vect
9. is illustrated graphically with the Tangent operator The next few paragraphs give some background into linear maps and derivatives Be assured that the graphs we will discuss are generated with only a few mouse clicks or key strokes The Tangent operator is not available with cylindrical or spherical coordinates Linear and Linear maps are the simplest functions to understand In one dimension Affine Maps alinear function is simply a multiplication In higher dimensions it is multiplication by a matrix For the sake of illustrating linear approximations via tangents we will use affine maps rather than linear maps An affine map combines a linear map stretch with an addition shift The one dimensional case is most familiar but it will actually be easier to begin with linear maps applied to a two dimensional domain The general form of a linear map from a two dimensional domain to three dimensions is f ax by fy cx dy fr extfy where a b c d e and f are constants Different constants give different maps If e and f are both 0 the result is a map from the plane to the plane A linear map is usually summarized by writing the coefficients in a matrix a a Q a 50 Fields amp Operators The J acobian Matrix a Obviously any linear transformation maps the origin to the origin Also lines are
10. sphere in the illustration we have changed the maximum fin the domain to 5 rather than the default of 2 pi This allows us to look inside the spheres Operators When you use cylindrical or spherical coordinates the following operators are not available on the Operator pop up menu on the Layer dialogs Dot product Evolute Cross product Osculating Circle Tangent Plane Frenet Frame 34 Fields amp Operators Sa duex3 Fields amp Operators 35 Graphs and Dimension The first consideration in drawing a graph and thinking about functions and especially their derivatives is dimension Informally a function is a computation that takes one or more inputs and produces one or more outputs The set of possible inputs is called the Domain The number of inputs to the computation is the dimension of the domain The output of the function is the Range and the number of outputs is the dimension of the range Technically the graph of a function is the set of tuples of the form in in my Out Out gt OUt and so the dimension in which we draw the graph is the sum of the dimensions of the domain and range Fields amp Operators can help visualize cases with up to three dimensions in each of the domain and range Including animation can effectively add yet another dimension How can we visualize high dimensions in a three dimensional world or a two dimensional computer screen We will illustrate various combi
11. Bottle Microsoft Internet Explorer olx File Edit View Go Favorites Help Address KLEINBOT HTML 2 A new look at the Klein bottle This is a variation on a new way of representing a Klein bottole due to Tom Banchoff A figure 8 is twisted as it sweeps around a circle If we ignore the intersection this is a one sided surface This is a very cool animation Visit The Mathematics Deptartment Home Page If you are familiar with HTML HyperText Markup Language you can customize the page using any text editor 26 Fields amp Operators Sorry we cannot give guidance here on posting the page for public or private access Your University or local computer user s group can help If you would like to make the graphs you create widely available please contact us about posting them on our WWW site Fields amp Operators 27 Gradients with the Other Uses of the Touch Panel 28 Fields amp Operators Touch Panel In this section we present an alternative method for adding a gradient field to a surface Select RESET from the File menu or load any graph defining a surface where z F x y Rotate the graph to a convenient viewing angle Next click to open the Touch Panel The panel represents the domain As you move the mouse in the Touch Panel its coordinates in the domain are displayed in the corner of the Touch Panel window and the coordinates in 3 dimensions of the image of that point are sho
12. Operators In order for the curve to be smooth we use many subdivisions in the x dimension on the domain This would give too many vectors in the Frenet frame layer Instead we use a private domain with fewer subdivisions R2 gt R2 Jacobian This is a distortion of the flat plane The Jacobian at a point is the linear map that best approximates the underlying map at that point Again the Tangent Plane operator performs the computation Linear maps are discussed below Zz x Another way to visualize maps from is R to R as a two dimensional vector field We won t cover this here but see R3 gt R 3 below R2 gt R3 Sphere The simplest way to draw a sphere is to use the identity functions in Spherical coordinates We use Rectangular coordinates here to illustrate defining surfaces parametrically The Coordinate Functions We can think of the R2R3 case as taking a two dimensional sheet and distorting it to a surface in three dimensions This is more general than the surfaces obtainable as z F x y discussed above A sphere cannot be graphed as z F x y because it would not satisfy the vertical line test For simplicity we ll assume the radius of the sphere is 1 To see how to draw a sphere consider the circle shown a here All points on Fe the circle have the same z coordinate z sin 0 J gt The radius of the circle which is not the same as the radius of the sphere is r cos We get
13. PETER Painted horse 15 000 BC Lascaux France This manual and the accompanying software are protected by United States and international copyright laws Copyright Martin Lapidus All Rights Reserved Fields amp Operators 1 Contents Preliminary 1 Installing Fields amp Operators 5 Learning to Use Fields amp Operators 7 Guided Tour 9 A Surface and Its Gradient 15 Story Board 23 Publish Your Graphs on the World Wide Web 26 Gradients with the Touch Panel 28 Coloring Surfaces with Functions 29 Vector Fields and Integral Flows 31 Polar and Cylindrical Coordinates 33 Examples 35 Graphs and Dimension 37 R gt R XSIN 5X 38 R25R SINCOS 39 R gt 5R2 Circle 40 R R Helix 41 R2 5R2 Jacobian 42 R2 5R3 Sphere 42 R35R3 44 Other Examples 45 Light Ray LIGHTRAY 45 Conic Sections CONICS 45 Antenna 46 Passing Through a Torus Through 47 Derivatives and Tangent Planes 50 Fields amp Operators 3 1191d Kreul Fields amp Operators 1 Introduction Fields amp Operators is an interactive graphics program that allows you to experiment with curves surfaces and vector fields in two or three dimensions Creating graphs even complex ones with multiple interrelated parts is simple and intuitive No programming is ever required Because curves surfaces and vector fields often vary in time Fields amp Operators will let you generate animations to see
14. ame drawing on the Draw Quality menu Rotate the graph with each setting Don t forget that this surface and hence its gradient also are time dependent Earlier we saw how to use real time and recorded animation Watch how the field varies with the surface The vector field lies in the x y plane In this case the field intersects the surface A simple trick separates the two layers Change the sin x t cos y t 2 definition of the surface in layer 1 to sin x t cos y t 2 Adding 2 raises the surface but does not effect the gradient Later we will discuss controlling the number of vectors that are drawn For now you should take a look at the different controls that become available on the Appearance page of the Layers dialog when the layer is a vector field Be especially sure to look into vector scaling Click Help on the appearance page Before going on let s save the current graph in a disk file In fact it s a good idea to save work in progress regularly In the unlikely event of a computer crash you can at least resume with your last saved version Choose SAVE from the File menu The four files you have most recently accessed will appear at the bottom of the File menu for quick retrieval Fields amp Operators 19 Domains A Separate Domain 20 Fields amp Operators As you know the trigonometric functions are periodic Let s look at two full periods by changing the domain Select DOMAIN from the
15. anel Demo This file is also available as DIFFERENTIAL GEOMETRY under Quick Start on the File menu 12 Fields amp Operators Drawing Quality Shaded Solid and Wire Frame Exploring on Your Own The graph shows a sine curve its evolute and the osculating circle at a point If you are not familiar with the language of differential geometry keep going anyway There is a brief description in Help The position at which the osculating circle rests could have been managed with animation but instead it has been defined to work with the Touch panel The file OSCULATI Curve with Osculating Circle 1 contains the animated version Click Touch The simplest way to learn about the purpose of the Touch Panel is to try it Click on the slider and drag it If we were working with a two dimensional surface rather than a one dimensional curve the Touch Panel would have given two dimensional dragging The Draw Quality menu allows you to select Shaded Solid or Wire Frame transparent graphs On some systems there may be a difference in speed among these three styles Shaded is slowest Wire Frame fastest Of course these settings only have an effect with surfaces If your graph has only curves or vectors shaded or solid surfaces are meaningless With shading you can specify up to 3 light sources via the Light Sources dialog For details see Help If you work with very complicated graphs with many subdivisions in
16. e oo re er Ree eevee t H Es e fe Re te ee eee HS Polar and Cylindrical identity Functions Coordinates Fields amp Operators can work with three types of coordinate systems rectangular cylindrical and spherical You select the coordinate system from the Coordinates dialog Coordinates CO Rectangular Cylindrical Ol Spherical r p p Z Z rand 9 are used differently in cylindrical and spherical coordinates When you change coordinate systems Fields amp Operators will reset all domain layer and other definitions to their default values For this reason you will be asked to confirm your choice when you change coordinates Your choice of coordinate system determines the allowable variables in function definitions on the Layer dialogs For example if you are working with spherical coordinates you cannot use the variables x y or z This would lead to an error message When entering functions you must type out theta or phi You cannot enter Greek letters A useful way of thinking about domains in cylindrical or spherical coordinates is to consider the identity functions These are the defaults in the layer definitions Fields amp Operators 33 W Mf pes TTL ZZ The identity functions with cylindricaland sphericalcoordinates In each of these figures we have defined two r subdivisions The default is one r subdivision giving one cylinder or one sphere For the double
17. ee the total of 30 frames In the second half the horn should appear to move Click in frame 11 to select it Hold g while you click one time Now we will introduce an abrupt change of direction at frame 20 Select frame 20 and click and hold E without q Play or record the animation from this story board Other Options There are a few Story Board options we haven t looked at here The Story Board menu has an option for adding a single frame from the main window to the Story Board If you click a frame with the secondary right mouse button a menu pops up with additional options See Help for details Changing The Story Board manages the rotation scale and time settings for each Definitions frame It does not store the underlying definitions as set on the Layers dialog If you were to change the domain and layer definitions without resetting the Story Board the sequence of actions would be applied to the new set of functions Fields amp Operators 25 Publish Your Graphs on the World Wide Web Publishing your graphs on the World Wide Web couldn t be easier Select SAVE AS HTML from the EXPORT sub menu on the File menu This will build and save a Web page consisting of your graph along with the text from the Notes window The graph will be saved in GIF format in a separate file and a reference to that GIF file will be included in the Web page You will of course need a Web browser to see the page E Klein
18. es Fields amp Operators remembers the latest 4 files you have opened These will be listed at the bottom of the File menu There is also a collection of files that will be useful starting points when you make you own graphs These are available on the QUICK START sub menu of the File menu For now use File Open and select the file DRUM from the Examples folder When you open this file you will be notified that notes have been attached To open the Notes window select it from the View menu The notes window is a simple text editor for you to leave notes that are saved as part of the file Recorded The note for this file informs you that this graph is defined for Animation animation For this tour we have purposefully chosen a graph that is slow to draw and will give best results with recorded animation Select RECORD ANIMATION from the Animation menu You will be asked to enter a file name After that Fields amp Operators will draw and store a number of frames for the animation When it is finished Select PLAY ANIMATION from the Animation menu If you haven t tried the various playback controls EIN IO her o Remember that each control has a hint associated with it When you are done with this recorded animation click Close Playback Later we will look at the Story Board This will give you much more control over you animations Touch Panel The next sample graph will introduce the Touch Panel Open the file TUCHDEMO Touch P
19. for the layer are available for other layers but the layer is not drawn as part of the graph The torus is completed in the second layer by making use of a second wrapping operation The computation is similar but this time it is applied to the cylinder of the first layer and is perpendicular to the first wrap The axis for the cylinder is thought of as the angle parameter for the torus gins of radius x 1 Cylinder shifted The Horn Operator Apply to Show Layer Dei E ee Name Torus Display as x 1 x cos z z 1 x sin z The effect ofthe moving horn is created by taken successive slices from the surface below It is similar to the cylinder we defined in layer 1 except that its radius changes along its axis In this case it is centered on the y axis and the radius is the hyperbolic cosine of y cosh y is close to 7 when y is near 0 and increases exponentially for large y We divided by 3 to make it somewhat thinner The surface is a catenoid In the full catenoid above y is between 2 and 2 For the animation we take pieces of length 2 and move the pieces by adding t Look at the Domain and Animation dialogs Fields amp Operators 49 Derivatives and Tangent Planes The definition of derivative that is easiest to generalize to high dimensions is that the derivative at a point is the best linear approximation to the given function at that point This
20. h Dept of Engineering at a workshop of Computer Applications in Electromagnetics Education It represents the antenna pattern for a five element half wavelength spaced uniformly excited linear array Drawing a Torus It is defined in spherical coordinates as f abs sin t pi 2 cos theta t sin pi 2 cos theta where t 5 Notice that if t this long expression reduces to 1 giving a sphere of radius in the graph Animation reveals how the lobes evolve as t changes from to 5 Passing Through a Torus Through 4 N ARAN ANN This examples shows a horn passing through a torus We present it because it involves two separate tricks for generating such animations The first is in using two layers to draw the torus The second is how the horn is made to move through the torus Writing down a formula that maps a rectangular domain onto a torus can be an interesting though tedious exercise Using two layers makes it simple Fields amp Operators 47 48 Fields Operators oy z 5 cos x 5 sin x y The first step is to wrap the rectangle into a cylinder of radius 1 2 The diagram shows the map from two dimensions to three dimensions Operator Apply to O Show Laye Display as Suae lej From Layer 1 Cylinder Since this is only an intermediate step the layer is hidden Note the radio button at the top of the dialogo This means the computations
21. have already seen an example of a one dimensional tangent in the XSIN 5X example There the function was of the form y F x Exactly the same techniques are used even if the curve defined parametrically with a two dimensional range e g a circle or a three dimensional range e g a helix When drawing tangent lines be sure that the Tangent layer uses a one dimensional domain This is automatic if it shares the same domain as the curve but be careful if you define a private domain for it
22. imensional domain You may want to open the Domain dialog to see how it is defined Although this field appears as several slices 1t is defined in a single layer Next add a new layer and choose the Flow operator The integration is performed numerically essentially as a Riemann sum The two parameters that control the summation are entered on the Flow Parameters dialog The dialog is accessed through the OPTIONS button which appears on the Layer dialog whenever the Flow operator is selected Open that dialog and enter the values show below Smaller values of the increment give more accurate results but will slow computation Fields amp Operators 31 A Related Example 32 Fields amp Operators Operator Integral flow of Show Layer Flow Integrate 2 1 Layer One O Hide Layer Name Display as Flow ines Te Based at Increment Iterations Make sure that the BASED AT control is set to the domain This determines where each flow line begins eS An example representing a two dimensional time dependent vector field with its flow is given in the files OSCILAT1 Oscillator It is a os ee a ee classic example ee from the theory A or of differential aa 377 7 Dany 30777 AU AS o 7 7 equations AAA AN A ANA showing the My a Ad NAAA ig Ne OO phase space ofa R R WENN HN simple pendulum wien N Tee ye SANS A oscillator eve eK Be ee zer kr TTD e
23. lds amp Operators 45 Conics 46 Fields Operators To specify a point in polar coordinates we imagine a radius r drawn from the origin to the point The point is identified by giving the length of r and the angle 0 theta between the x axis and r Notice that any point can be identified in infinitely many ways because one or more full circles n 27 can be added to any Polar Coordinates angle measurement The conic sections ellipses parabolas and hyperbolas can all be described by a simple class of formulas in polar coordinates The general form of a conic section is r d e 1 e cos 0 0 lt e lt l the result is an ellipse When e 1 the result isa parabola and when 1 lt e the result is an hyperbola What is the geometric interpretation of d Here we use d 1 y where e is the O eccentricity When QA x Any time a family of functions depends on some parameter it is likely that animation can add insight into the functions Also consider using animation whenever a constant appears in a function definition We can watch a family of conics evolve by generating an animation in which we use r for the eccentricity fr 1 t cos theta In the animation t varies from 0 to 2 The domain has one z subdivision and one r subdivision Can you generalize this animation to surfaces Antenna This example is based on graphs presented by Warren Stutzman and John McKeeman of Virginia Tec
24. ludes an extensive Help facility There you can easily find answers to most of your questions about Fields amp Operators There are many places in this manual that we refer you to Help There are many sample graphs on your disk After you take the Guided Technical Tour browse through the sample files Many of these files include brief Support notes in the User Notes window Much of the time you will find that the graphs you want to create are variations on some of the included samples If you have any questions please feel free to call or e mail Don t wait until you have a problem to call we are always eager to hear your comments and suggestions 520 749 0200 support LascauxSoftware com Lascaux Graphics 5354 N Buckhorn Dr Tucson AZ 85750 USA http www LascauxSoftware com Fields amp Operators 7 2 Guided Tour 10 Fields amp Operators Starting Fields amp Operators Animation This guided tour is intended to introduce you to the major features of Fields amp Operators Follow along at the computer and feel free to experiment The tour will start with a quick look at some features and examples to give you an overview After that we will look at several specific examples with step by step instructions S You can start Fields amp Operators from the Start Menu The graph that appears when you start Fields amp Operators depends on how you have configured the startup default More on this later
25. main On the domain dialog click on the tab for layer 2 We see that currently this layer is using the shared domain To obtain a private domain for this layer click PRIVATE Exercise Axes Changing and Updating Functions Shared Private Copy Shared Domain To make sure that the minimum and maximum size of this layer match the underlying vector field begin by making a copy of the shared domain Click COPY SHARED DOMAIN Finally change the number of x and y subdivisions to 11 Try viewing the surface as Level curves Better yet add a new layer with the same definitions as the surface and display it as level curves keeping the original surface and vector field in the graph as well This graph is stored in the file SURFGRAD Surface With Gradient The coordinate axes help in visualizing the orientation of a graph in three dimensions and in _ judging its size Fields amp Operators lets you easily control many aspects of the axes appearance To open the Axes dialog select AXES from the View menu or the pop up menu Experiment with the settings Now that you have the domain and viewing angle and other parameters adjusted to give an interesting view of a function and its gradient you may want to investigate a different function and its gradient retaining the same design Save the current graph before making any changes We will change the function in layer 1 Enter the function F
26. mapped to lines and parallel lines are mapped to parallel lines A typical linear transformation is illustrated For a map not necessarily linear in rectangular coordinates from R2 to R3 the Jacobian matrix is defined as Of Of dx ody dof If dx dy Of 47 dx ody All partial derivativesare evaluated at P For each point P in the domain this is just a matrix of numbers so it corresponds to a linear map The linear map it defines is the best linear approximation at P As mentioned linear maps always pass through the origin To draw the tangent plane we must use an affine version shifting the plane to the point of tangency by adding the coordinates of the image of P in the surface The component functions for the plane are af Os Ox sd Ox y Q Fields amp Operators 51 Transformations and Equations of a Plane One Dimensional Maps 52 Fields amp Operators d f d fy Fy x Jy y Q d f d fy 0z a 0z y 0 where Q is the image of P When you compare these expressions to the equation for the tangent plane found in text books keep in mind that Fields amp Operators works with transformations applied to the domain 1t is a computation applied to x and y coordinates producing other coordinates The equation of a plane as it appears in most texts is an equation that is valid for points that lie in the plane We leave it to you to reconcile these two points of view We
27. mply the input to the computation Derivative and Tangent Vectors Frenet Frame In the domain x varies from 0 to27 The circle we have drawn is not the graph of a function Recalling the technical definition of a graph we expect the graph in this case to be three dimensional Points in the graph are of the form x cos x sin x Can you describe the graph How can you modify the layer to illustrate it Incidentally drawing a circle is even easier using Cylindrical coordinates R1 5R3 Helix This is a simple extension of the Circle example above Here we take a one dimensional line and lay it down as a path in three dimensions Dividing by 5 in the f component compresses the graph In general a curve in three dimensions is described parametrically by FAX A x fax How can we present the derivative of this function from R1 to R3 As always the derivative should describe how the output points on the curve change as the input the x parameter changes This is exactly what the tangent vectors at points y along the curve tell us The Frenet frame is a set of local coordinate axes at each point of a curve It is composed of the tangent the normal and the bi normal See Help for details When you select the Frenet frame Operator you can choose with vectors will appear In this illustration we included the tangent and normal but not the bi normal Fields amp Operators 41 42 Fields
28. nations of dimension through some examples These examples are stored on your disk Besides illustrating this discussion they are useful as starting points for your own graphs Fields amp Operators 37 R1_sR1 XSIN 5X x sin 5x The most familiar functions have one input and one output Our example graphs f x x sin 5 x From the Layers Dialog The function is defined in layer 1 Notice that f is a copy of the domain x The f component is not used since we are working only in the x y plane Minimum Maximum Subdivisions xe fa fc ye le Jf 7 a Jeo Jh____ SharedA1 x sin 5x A2 Envelope 1 43 Envelop 2 From the Domain Dialog 38 Fields amp Operators The Domain dialog shows that the entire domain lies on the x axis Notice that y and z from the domain do not appear as input in any of the function definitions There is one subdivision in each of those dimensions If we had defined multiple subdivisions in either the y or z dimensions the resulting image would not be changed However the same graph would be drawn multiple times The x axis would be scanned for each y value in the domain This is a needless repetition it would only slow the drawing In layers two and three we have added the lines y x and y x to illustrate the envelope that results from the multiplication by x The fourth layer draws a tangent line to the curve When you animate the graph the tangent line will move along the graph
29. one will rotate in the direction you selected If drawing all these frames is too slow on your processor you can select FAST WIRE FRAME from the Draw Quality drop down list For even greater speed you can check AXES ONLY This will show only the orientation and size of your graph and so will be very fast Recall that axes are defined on the Axes dialog Rotate the image in frame so that it more or less matches the orientation in the illustration above Notice that all frames rotated by the same amount This is not the effect we want for animation For animated rotation we want the second frame to rotate more than the first and the third more than the second etc Make sure frame 1 is selected and then hold g while you click a one time Each frame should rotate the way we want Story Board k e qUe Ne N Ry 4 a N en WESEN sane wur KR Lie Riess as Baa HH LOL wi y NC 24 Fields Operators We are now ready to preview our progress so far Click PLAY FROM STORY BOARD on the Story Board menu If the drawing in the main window is too slow you can create a recorded animation instead by selecting RECORD FROM STORY BOARD Now would be a good time to save the graph Select SAVE AS and give it a new file name For the next 20 frames we do want f to vary Open the Time dialog and let vary from to 3 in 20 intervals Click APPEND FROM ANIMATION SETTINGS and open the Story Board again to s
30. ors of the gradient field are drawn normalized all scaled to same length This makes for neater drawings but can be misleading unless understood The Appearance page of the Layers dialog gives options for controlling vector scaling The tangent plane also gives information about the derivative The tangent at a point is not a line but a two dimensional plane Examples of tangents to a surface are given in Derivatives and Tangents below R1 R2 Circle Before looking at this example let s see how not to draw a circle Theoretically we can graph the top half of a circle of radius by letting x vary from to 1 and letting y V 1 x On the dialog this would be entered as 1 x2 5 This is unsatisfactory for two reasons First the spacing between points on the circle is not uniform Why Try it Second it only gives the top half of the circle We would have to add a second layer for bottom half defining y V J x A circle is not the graph of any function y x because most x values in the domain have two corresponding y values Instead we draw a circle parametrically The angle from 0 to 27 is the one dimensional parameter x and y in the range are both computed from the angle C C Notice that we use x as the angle parameter We cannot use theta when working in rectangular coordinates The trick in defining functions parametrically is to not let the terminology determine how you think about functions x is si
31. ou select RESET See Help It can become tedious to always start off looking down on the x y plane Buttons on the rotation palette allow you to specify a preferred viewing angle Rotate the graph in the usual way to a convenient angle then click da This will save the angle to your disk Now anytime you want to return to this preferred angle just click EN All graphs consist of one or more layers defined through the Layers dialog There are several ways of opening this dialog Click Layers or click in the main program window with the secondary right mouse button This is the Pop Up menu Select LAYERS from this menu Fields amp Operators 15 Coordinate Functions Display As 16 Fields amp Operators Rectangular coordinates Operator Apply to Show Layer z sin x t cos y t Layer One The dialog has two pages Definitions and Appearance The tabs at the top flip the pages As you might expect this is where you enter the functions that define the layer Notice that the OPERATOR is set to Function Layer 1 is special in that entering functions is the only operator available We will soon see the power that becomes available when you can select an operator The dialog shows that the three coordinate functions are currently defined as Fo x Fy y F sin x t cos y t Notice that in this case x and y from the domain are not changed This is the standard format
32. the complete circle if we take points r cos 8 r sin 8 z where 0 lt O lt 2T s of sphere 1 Ay Substituting for r this is cos b cos 8 cos sin sin b To generate the full sphere we let 6 vary from 7 2 to 7 2 Finally we recall that in Rectangular coordinates the variables are called x and y rather than 6 and x cos x cos y y cos x sin y From the Layers Dia log Fields amp Operators 43 44 Fields Operators Minimum Maximum Subdivisions lo pr m 7 yo e e 2 e E From the Domain Dialog If you find it difficult to visualize the angle parameters experiment by reducing the size on the domain to yield only a partial sphere For other examples of defining surfaces parametrically see Passing Through a Torus below Another method of visualizing functions from R2 to R3 is analogous to the next example In fact one slice from that example serves The vectors in one slice are based on a two dimensional domain and point in three dimensional space R3 gt R3 The vector field in the file 3DFIELD 3D Field is an example of a function from R3 to R3 We used that file when we looked at integral flows above At each point in a three dimensional domain we place a three dimensional vector that represents the value of the function at that point Without use of animation this is the highest dimensional situation that Fields amp Operators can handle Other Examples
33. thematical definitions in each layer of your graph For defining the appearance of coordinate axes Open each of the files in the Examples directory Most have short notes attached Feel free to experiment and make changes Save any changes you make with a different file name so as not to overwrite the examples As you explore each graph open the Layer Summary window The tour will continue with a step by step walk through in building a number of graphs 14 Fields amp Operators Preferred Viewing Angle Layer Dialog A Surface and Its Gradient In this part we will begin our detailed look at Fields amp Operators by drawing a number of interesting graphs In passing we will touch all of the major program features and controls We will focus on the graph of a surface and its gradient Even if you are not yet familiar with vector calculus keep reading We will work with surfaces where z is a function of x and y z F x y Other surfaces where x y and z are all functions of other variables are equally easy to draw Many examples appear in other sections For functions of the form z F x y the gradient is a vector field in the x y plane At each point it indicates the direction in which the function changes fastest The computation is described in Help Select RESET from the File menu This will return to the default graph of sin x t cos y t that we started with You can control the graph that appears when y
34. ts where the maximum function value is found You can edit the color distribution by clicking EDIT COLORS O Fixed Color Color by value To illustrate this feature we will color each side of the surface differently For Side 1 enter the function sin x t cos y t which is the function that defines the surface For Side 2 enter x 2 y 2 to draw a circular pattern Close the dialog and click to start animation If the action is too slow try a recorded animation Fields amp Operators 29 30 Fields amp Operators The technique we used for the circular pattern shows that the coloring needed not be related to the defining function of the surface Also the functions may be time dependent Note your color depth setting typically 8 bit 256 colors 16 bit or 32 bit influences the appearance of continuous colors Because this is Windows or video driver configuration we will not advise you here Vector Fields and Integral Flows An integral flow represents a numerical solution to the differential equation given by a vector field It describes how a particle would move under the influence of the vector field Each point in the domain is an initial point for a solution Solutions are generated by following the direction of the field at each point To generate an integral flow we will begin with the sample graph called 3DFIELD 3D Field y x Z This is the first time we have used a three d
35. u draw one See the KLEIN Klein bottle example You can set the fill and grid colors for each side here There are several other options most will be discussed below For full details turn to Help Return to the Definitions page Now we will add the gradient to the graph Click New Layer Initially the new layer is a duplicate of the previous layer except for its name We will change the functions of the new layer to reflect the gradient of layer 1 Select Gradient from the OPERATOR drop down list Fields amp Operators 17 18 Fields Operators Gradient Je 1 Lover one E Gradiem Vectorfiet E Fields amp Operators will automatically compute new functions The function definitions cannot be modified If you do want to edit the computed functions change the Operator to FUNCTION Notice that the selection in the DISPLAY As drop down list was automatically changed to VECTOR FIELD The gradient is applied to a surface scalar field and produces a vector field Because the display is a vector field an additional drop down list BASED AT appears for specifying the placement of the arrows of the field In this case we want the field to be arrayed throughout the domain so the default choice is correct Save Your Work Close the dialog to draw the newly defined graph The vector at each point indicates the direction in the domain in which the function values change fastest Try both solid and wire fr
36. view menu or click Minimum Maximum Subdivisions xka fa ft ye Jl tg i Shared 1 4 Surface A2 Gradient JA Surface 4 2 Gradient The tabs at the bottom of the dialog indicate that we are now defining the shared domain This is the domain accessed by all layers unless a layer specifically requests a private domain We will look at private domains below Notice that the x and y values of the domain vary between 4 and 4 Change the minimum x value to 2 pi and the maximum x value to 2 pi Do the same for the minimum and maximum y coordinates This example illustrates an important feature The expressions for the minimum and maximum values can include functions even time dependent ones The variables x y and z in rectangular coordinates may not be used here For example a domain can be made to shrink and expand by entering Minimum 2 sin t Maximum 2 sin t Since the z coordinate does not currently appear in any of the function definitions on the Layer dialog the minimum and maximum z values are ignored However be sure that the domain has exactly one z subdivision Also experiment with increasing the number of subdivisions in the x and y dimensions Higher values give smoother graphs There are so many vectors in the field that the graph appears cluttered We could go to the shared domain and reduce the number of subdivisions but then the surface would not be as smooth The solution is to introduce a private do
37. wn in the main graph window Touch Panel 1 A surf IX Gradient 1 Asurtace S Gradient ane y 4 00 4 00 x 4 00 y 4 00 Check GRADIENT or TANGENT PLANE or both A new layer will be added for the operator you selected Now when you move within the touch panel the field or plane will follow your touch on the surface For more on the Touch Panel see Help Having Fields amp Operators add layers automatically is just one way to use the Touch Panel You may include the expressions mx or my in any expression in the domain dialog or layer function definitions These will be interpreted as the x and y coordinates of the point in the domain pointed to by the mouse For example if you have a fast processor try F mx x 42 my y 2 Coloring Surfaces with Functions We will next look at a powerful variation of the simple shading we ve already seen Make sure the Draw Quality is set to SHADED before continuing Select RESET from the File menu so you will be working with the surface z sin x t cos y t Open the Appearance page of the Layer dialog For Side 1 click COLOR BY VALUE A continuous color bar appears as well as an edit box for you to enter a function The range of colors on the color bar will be mapped to the values of the function you enter For example the leftmost color on the bar will be used for points where the minimum function value is found the rightmost color is used for poin
Download Pdf Manuals
Related Search