(*^
::[	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.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

	FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";

	MacintoshStandardFontEncoding; 
	
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; 
	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  18, "Times"; 
	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6,  14, "Times"; 
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  18, "Times"; 
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  14, "Times"; 
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12,  12, "Times"; 
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5,  12, "Courier"; 
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; 
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; 
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  12, "Courier"; 
	fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic,  10, "Geneva"; 
	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = leftheader, inactive, L2,  12, "Times"; 
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; 
	fontset = leftfooter, inactive, L2,  12, "Times"; 
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = special1; inactive; preserveAspect]
                               The Gradient and the Directional Derivative

The cell below shows how the Mathematica  procedure Plot3D can be used to study a surface.  Evaluate it now.

;[s]
8:0,1;74,0;105,2;116,0;128,1;134,0;168,3;183,0;186,-1;
4:4,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
f[x_, y_] := x * y

Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1},
       PlotRange -> {-1, 1}]
:[font = special1; inactive; preserveAspect]
The next  cell reproduces the graph in the browser window showing what happens if you walk one unit in various different directions starting from the same point.  Evaluate it now.
;[s]
3:0,0;163,1;178,0;180,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
Clear[Surface, Curve, Circ]

x0 := 0
y0 := 0

Surface = Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1},
                 PlotRange -> {-1, 1},
                 DisplayFunction -> Identity]
Circ = ParametricPlot3D[
             {x0 + Cos[t], y0 + Sin[t],
              f[x0 + Cos[t], y0 + Sin[t]]},
              {t, 0, 2 Pi},
              DisplayFunction -> Identity]
                 
Curve[u_] := ParametricPlot3D[
              {x0 + t u[[1]],
               y0 + t u[[2]],
               f[x0 + t u[[1]], y0 + t u[[2]]]},
              {t, 0, 1},
              DisplayFunction -> Identity]
              
Show[Surface, Circ, 
     Curve[{1/Sqrt[2], 1/Sqrt[2]}], 
     Curve[{-1/Sqrt[2], 1/Sqrt[2]}], 
     Curve[{1/Sqrt[2], -1/Sqrt[2]}], 
     Curve[{-1/Sqrt[2], -1/Sqrt[2]}], 
             DisplayFunction -> $DisplayFunction]
   

:[font = special1; inactive; preserveAspect]
The next cell looks at two different two-dimensional curves obtained by walking along a surface starting at the same point but walking in two different directions.  Evaluate it now. 
;[s]
3:0,0;165,1;180,0;183,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
OneCurve[u_] :=

Plot[f[x0 + t u[[1]], y0 + t u[[2]]], {t, 0, 1},
     PlotRange -> {-1/2, 1/2},
     AspectRatio -> Automatic]
     
OneCurve[{1/Sqrt[2], 1/Sqrt[2]}]
     
OneCurve[{-1/Sqrt[2], 1/Sqrt[2]}]     
:[font = special1; inactive; preserveAspect]
The last cell draws a vector field showing the gradient vector at each point.  Evaluate it now.
;[s]
3:0,0;79,1;94,0;96,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
<<Graphics`PlotField`

PlotVectorField[{2x, -2y},
                {x, -1, 1}, {y, -1, 1}]
^*)