(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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We use the notation Du f (a, b). In lecture last time we showed that, if u = u1i + u2j, then ;[s] 47:0,0;42,1;44,0;45,1;46,0;48,1;49,0;85,2;86,0;99,1;100,0;102,1;103,0;143,1;145,0;146,1;147,0;149,1;150,0;156,1;157,0;159,1;160,0;179,1;180,0;182,1;183,0;205,2;206,0;230,1;231,3;233,1;234,0;236,1;237,0;239,1;240,0;284,2;285,0;288,1;289,4;290,2;291,0;294,1;295,4;296,2;297,0;304,-1; 5:21,14,10,AGaramond,0,12,0,0,0;18,14,10,AGaramond,2,12,0,0,0;5,14,10,AGaramond,1,12,0,0,0;1,22,14,AGaramond,65,12,0,0,0;2,18,12,AGaramond,64,10,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 104; pictureWidth = 175; pictureHeight = 33] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.188571 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.188571 scale 1 string 175 33 1 [175 0 0 33 0 0] { currentfile 1 index readhexstring pop } false 3 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FFFFFFFFFFC7FF8FFFFFFFFF03FFFFE3FE3FFFF1FF1FFFFFFFFFFFFF8E3F FFFE3FE3FFFF1FF1FFFC7FC7FFFE3FFFFFFFE38FFFFF8FF8FFFFC7FC7FFF 1FF1FFFFFFF8 FFFFFFFFFFC7FF8FFFFFFFF803FFFFFC01FFFFF0001FFFFFFFFFFFFC003F FFFFC01FFFFF0001FFFC7FC7FFFE3FFFFFFF000FFFFFF007FFFFC0007FFF 1FF1FFFFFFF8 FFFFFFFFFFF8FF8FFFFFFFFFE07FFFFFFFFFFFFE38FFFFFFFFFFFFFFF1C7 FFFFFFFFFFFFE38FFFFFFFFFFFFFFFFFFFFFFC71FFFFFFFFFFFFF8E3FFFF FFFFFFFFFFF8 FFFFFFFFFFF8007FFFFFFFFFFC0FFFFFFFFFFFFE38FFFFFFFFFFFFFFFE07 FFFFFFFFFFFFE38FFFFFFFFFFFFFFFFFFFFFFF81FFFFFFFFFFFFF8E3FFFF FFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFFFE3FFFFFFFFFFFFF1FFFFF FFFFFFFFFFF8 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] The vector fx(a, b)i + fy(a, b)j is called the gradient of f, denoted by Ñf (a, b). ;[s] 30:0,0;11,1;12,2;13,0;14,1;15,0;17,1;18,0;19,3;20,0;23,1;24,2;25,0;26,1;27,0;29,1;30,0;31,3;32,0;47,1;55,0;59,1;60,0;73,4;74,1;76,0;77,1;78,0;80,1;81,0;84,-1; 5:14,16,12,Chicago,0,12,0,0,0;11,16,12,Chicago,2,12,0,0,0;2,19,13,Chicago,66,10,0,0,0;2,16,12,Chicago,1,12,0,0,0;1,18,13,Symbol,0,12,0,0,0; :[font = text; inactive; preserveAspect] Recall that the dot product of two vectors is defined by v×w = ||v|| ||w|| cosq. Use this definition to answer the following questions. Which unit vector u gives the largest possible Du f (a, b)? ;[s] 21:0,0;57,1;58,2;59,1;60,0;65,1;66,0;71,1;72,0;78,2;79,0;155,1;156,0;184,3;185,4;187,3;188,0;190,3;191,0;193,3;194,0;197,-1; 5:9,16,12,Chicago,0,12,0,0,0;5,16,12,Chicago,1,12,0,0,0;2,18,13,Symbol,0,12,0,0,0;4,16,12,Chicago,2,12,0,0,0;1,24,16,Chicago,65,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Which unit vector u gives the most negative ("smallest") Du f (a, b)? ;[s] 11:0,0;18,1;19,0;57,2;58,3;60,2;61,0;63,2;64,0;66,2;67,0;70,-1; 4:5,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,1,12,0,0,0;4,16,12,Chicago,2,12,0,0,0;1,24,16,Chicago,65,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] A unit vector u that has Du f (a, b) = 0 must be related in what way to Ñf (a, b)? ;[s] 18:0,0;14,1;15,0;25,2;26,3;28,2;29,0;31,2;32,0;34,2;35,0;72,4;73,2;75,0;76,2;77,0;79,2;80,0;83,-1; 5:8,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,1,12,0,0,0;7,16,12,Chicago,2,12,0,0,0;1,24,16,Chicago,65,12,0,0,0;1,18,13,Symbol,0,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] In the remainder of this lab you will use Mathematica to investigate the gradient graphically. ;[s] 3:0,0;42,1;53,0;95,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = section; inactive; preserveAspect] Part 2: A Simple Example :[font = text; inactive; preserveAspect] Consider the function that measures the distance from a point (x, y) in the plane to the origin: ;[s] 5:0,0;63,1;64,0;66,1;67,0;97,-1; 2:3,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 158; pictureWidth = 85; pictureHeight = 14] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.164706 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.164706 scale 1 string 85 14 1 [85 0 0 14 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFC01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF007 FFFE FFFFFFFFFFFFFF8FFFE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8 FFFE FFFFFFFFF1FFFFF1FFFFFFC7FFFFFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFF 1FFE FFFFC7FF8FF8FF81FFFF8000FFFFFFFFFFFF1FFFC7FC7FFFFFFFFFFFFE00 1FFE FFFFF8FF8FFF1FF1FFFF8FF8FFFE0003FFFF1FFFF8FF8FFFFFFFFFFFFE3F E3FE FFFFF8FC7FFFE38FFFFF8FF81FFFFFFFFFFF03FFFF1C7FFFFFF1FFFFFE3F E3FE FFFFFF1C7FFFFF8FFFFFF1FF1FFFFFFFFFF8E3FFFFFC0007FFF1FFFFFFC7 0000 FFFFFF1C7FFFFC71FFFFF1FF1FFE0003FFF8E3FFFFE38FFFFC0007FFFFC7 E07E FFFFFFE07FFFFC7FC7FFFE3F03FFFFFFFFC0E3FFFFE3F03FFFF1FFFFFFF8 FC0E FFFFFF007FFFFC7FC7FFFE3F03FFFFFFFFF8E3FFFFE3FE3FFFF1FFFFFFF8 FF8E FFFFFFFC0FFFFFFFFFFFFFF8FFFFFFFFFFFFFC7FFFFFFFC7FFFFFFFFFFFF FFF0 FFFFFFFF81FFFFFFFFFFFFF8FFFFFFFFFFFFFC7FFFFC7FC7FFFFFFFFFFFF 1FF0 FFFFFFFFF1FFFFFFFFFFFFC7FFFFFFFFFFFFFC0000000000000000000000 0000 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFE pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] Compute the gradient of this function, and write your answer below. (You may perform the calculations in Mathematica or by hand, whichever is more convenient.) ;[s] 3:0,0;106,1;117,0;161,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] Ñf (x, y) = ;[s] 7:0,1;1,2;3,0;4,2;5,0;7,2;8,0;13,-1; 3:3,16,12,Chicago,0,12,0,0,0;1,18,13,Symbol,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] Suppose we are at a point (x, y) and we wish to increase our distance from the origin as quickly as possible. Which way does common sense tell us we should walk? ;[s] 5:0,0;27,1;28,0;30,1;31,0;163,-1; 2:3,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Based on your answer above, the gradient Ñf (a, b) should be parallel to what vector? ;[s] 8:0,0;41,1;42,2;44,0;45,2;46,0;48,2;49,0;86,-1; 3:4,16,12,Chicago,0,12,0,0,0;1,18,13,Symbol,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] We can use Mathematica to illustrate this idea. First load the package PlotField by entering the command ;[s] 5:0,1;11,2;22,1;72,0;81,1;106,-1; 3:1,13,10,Courier,1,12,0,0,0;3,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] <