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Notebook[{
Cell["Partial Derivatives, 3D Plots and Contour Plots", "Title"],

Cell["\<\
Tuesday, October 3
Created by Jason Sherman\
\>", "Subsubtitle"],

Cell["Recall that the command D[ ] is used for derivatives", "Text"],

Cell[BoxData[
    \(D[x^2, x]\)], "Input"],

Cell[BoxData[
    \(D[x^2, {x, 2}]\)], "Input"],

Cell["Partial derivatives are equally as easy:", "Text"],

Cell[BoxData[
    \(D[x^2 - y^2 + x*y, x]\)], "Input"],

Cell[BoxData[
    \(D[x^2 - y^2 + x*y, y]\)], "Input"],

Cell["Derivative wrt. x then wrt. y", "Text"],

Cell[BoxData[
    \(D[D[x^2 - y^2 + x*y, x], y]\)], "Input"],

Cell[BoxData[
    \(D[x^2 - y^2 + x*y, x, y]\)], "Input"],

Cell["\<\
What about a gradient?  \[Del]f = { \[PartialD]f / \[PartialD]x  ,  \
\[PartialD]f / \[PartialD]y  }.  We can define \[Del]f(x,y) as a vector \
function of x and y...  (we are a bit ahead of the class here)\
\>", "Text"],

Cell[BoxData[{
    \(f[x_, y_] = x^2 - y^2 - 2  x + 4  y + 6\), "\[IndentingNewLine]", 
    \(gradf[x_, y_] = {\ D[f[x, y], x\ ]\ , \ D[f[x, y], y]\ }\)}], "Input"],

Cell["Another way to get the gradient", "Text"],

Cell[BoxData[
    \(grdf[x_, y_]\  = \ D[f[x, y], {{x, y}}]\)], "Input"],

Cell["The Jacobian, a matrix of partial second derivatives", "Text"],

Cell[BoxData[
    \(D[x^2 - y^2 + x*y, \ {{x, y}, 2}]\  // MatrixForm\)], "Input"],

Cell["Plot3D", "Subsubsection"],

Cell["\<\
Plot3D works just as you would expect if you've used the 2-D version, Plot.\
\>", "Text"],

Cell[BoxData[
    \(Plot3D[x^2 - y^2, {x, \(-3\), 3}, {y, \(-3\), 3}]\)], "Input"],

Cell["Several helpful Options for Plot3D are shown below. ", "Text"],

Cell[BoxData[
    \(f[x_, y_] = Sqrt[x^2 + y^2]*Exp[\(-x^2\) - y^2]\)], "Input"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}]\)], "Input"],

Cell["Label Axes in the same way you would a 2D plot.", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      AxesLabel \[Rule] {"\<x-axis\>", "\<y-axis\>", "\<f(x,y)\>"}]\)], \
"Input"],

Cell["\<\
ViewPoint lets you change the angle from which you view the plot.  You should \
try using the ViewPoint Selector found in the menu-bar under Input.\
\>", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      ViewPoint \[Rule] {2, 0, 0.4}]\)], "Input"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2\), 2}, {y, \(-2\), 2}, 
      ViewPoint \[Rule] { .2,  .2, 2}]\)], "Input"],

Cell["\<\
We can plot more points using PlotPoints->N, and we can turn the Mesh off by \
Mesh->False\
\>", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      ViewPoint \[Rule] { .8,  .8, 2}, PlotPoints \[Rule] 50]\)], "Input"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      ViewPoint \[Rule] { .8,  .8, 2}, PlotPoints \[Rule] 50, 
      Mesh \[Rule] False]\)], "Input"],

Cell["FaceGrids can sometimes help", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      ViewPoint \[Rule] {1, 1, 1}, PlotPoints \[Rule] 100, 
      Mesh \[Rule] False, FaceGrids \[Rule] All]\)], "Input"],

Cell["You may want to turn the axes off", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      ViewPoint \[Rule] {1, 1, 1}, PlotPoints \[Rule] 100, 
      Mesh \[Rule] False, FaceGrids \[Rule] All, 
      Axes \[Rule] False]\)], "Input"],

Cell["\<\
Here are all the available options for Plot3D.  Look some of them up if you \
like.\
\>", "Text"],

Cell[BoxData[
    \(Options[Plot3D]\)], "Input"],

Cell["Contour Plots", "Subsubsection"],

Cell["\<\
Again, contour plots work pretty much as you would expect.  Here is an \
example.\
\>", "Text"],

Cell[BoxData[
    \(ContourPlot[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}]\)], "Input"],

Cell["We can increase the number of contours...", "Text"],

Cell[BoxData[
    \(ContourPlot[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      Contours \[Rule] 20]\)], "Input"],

Cell["...and increase the resolution of the plot.", "Text"],

Cell[BoxData[
    \(ContourPlot[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
      Contours \[Rule] 20, PlotPoints \[Rule] 100]\)], "Input"],

Cell["\<\
We can also turn off the contour lines, making a neat looking picture.  This \
is sometimes called a \"Density Plot\"\
\>", "Text"],

Cell[BoxData[
    \(p1 = 
      ContourPlot[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
        Contours \[Rule] 40, ContourLines \[Rule] False, 
        PlotPoints \[Rule] 200]\)], "Input"],

Cell[TextData[{
  "Contour Plots don't have ",
  StyleBox["axes",
    FontSlant->"Italic"],
  ".  They have a ",
  StyleBox["frame.  ",
    FontSlant->"Italic"],
  "You can turn the axes and/or the frame on and off"
}], "Text"],

Cell[BoxData[
    \(Show[p1, Frame \[Rule] False]\)], "Input"],

Cell["\<\
This one has no frame, but it does have axes.  See the difference?\
\>", "Text"],

Cell[BoxData[
    \(Show[p1, Frame \[Rule] False, Axes \[Rule] True]\)], "Input"],

Cell["\<\
Just like you could label axes, you can label a frame.  The option \
FrameLabel->{} lets you label the top, bottom, left, and right side of your \
graph.\
\>", "Text"],

Cell[BoxData[
    \(Show[p1, 
      FrameLabel \[Rule] {"\<Bottom\>", "\<Left\>", "\<Top\>", \
"\<Right\>"}]\)], "Input"],

Cell[BoxData[
    \(Show[p1, 
      FrameLabel \[Rule] {"\<Bottom\>", "\<Left\>", "\<Top\>", "\<Right\>"}, 
      RotateLabel \[Rule] False]\)], "Input"],

Cell["\<\
Note that 2D plots can have a frame instead of a label as well, it is just \
that by default, they have axes.\
\>", "Text"],

Cell[BoxData[
    \(Plot[x^2, {x, \(-2\), 2}, Axes \[Rule] False, Frame \[Rule] True, 
      FrameLabel \[Rule] {"\<x\>", "\<y\>", "\<y = x^2\>", "\<Frames are \
fun\>"}]\)], "Input"],

Cell["Fun Stuff!", "Subsubsection"],

Cell["Back to a 3D Plot.", "Text"],

Cell[BoxData[
    \(p1 = 
      Plot3D[f[x, y], {x, \(-2.5\), 2.5}, {y, \(-2.5\), 2.5}, 
        PlotPoints \[Rule] 100, Mesh \[Rule] False]\)], "Input"],

Cell["\<\
Wouldn't it be nice to be able to rotate this without using ViewPoint?  We \
can!
We need to run a command first.  It will effect all 3D plots in your \
notebook.\
\>", "Text"],

Cell[BoxData[
    \(<< RealTime3D`\)], "Input"],

Cell[BoxData[
    \(Show[p1]\)], "Input"],

Cell[BoxData[
    \(Plot3D[\(-x\)/Sqrt[x^2 + y^2], {x, \(-3\), 3}, {y, \(-3\), 3}, 
      PlotPoints \[Rule] 50]\)], "Input"],

Cell[BoxData[
    \(Plot3D[\(-x\)/Sqrt[x^2 + y^2], {x, \(-3\), 3}, {y, \(-3\), 3}, 
      PlotPoints \[Rule] 150, Mesh \[Rule] False]\)], "Input"],

Cell["\<\
Notice how this is neat!, but it gets rid of axes labels... To get things \
back to normal, do:\
\>", "Text"],

Cell[BoxData[
    \(<< Default3D`\)], "Input"]
},
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WindowSize->{1842, 1052},
WindowMargins->{{25, Automatic}, {Automatic, 0}},
ShowSelection->True,
StyleDefinitions -> "DemoText.nb"
]

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