VC.04 2D Vector Fields and Their Trajectories
 LITERACY

\(Off[General::spell]; 
Off[General::spell1]; 

    \(Off[ParametricPlot3D::ppcom]; \)\)Off[ParametricPlot::ppcom]; Off[Plot::plnr]; \(SetOptions[Limit, Analytic -> True]; 
$Pre\  = \ Chop; \)

CMView = {2.7, 1.6, 1.2};

Vector Drawer

BeginPackage["Vector3D`", "Graphics`Colors`"];

\(Axes3D::"usage" = 
      "Axes3D[a, b] creates a Graphics3D object of cartesian axes with x, y, and z running from -a/3 to a, and with axes labels b units beyond the tips of the axes.  Axes3D[a] is Axes3D[a, a/8]."; 

    \(Perpend::"usage" = 
      "Perpend[a] returns a unit vector which is perpendicular to the vector a."; \)\)Arrow::"usage" = 
      "Arrow[a] produces a 2 or 3 dimensional vector from the origin to a. \
Arrow[a, Tail -> tail] gives a vector from tail to a."; Vector::"usage" = 
      "Vector[a, b] gives a 2D or 3D vector from a to b."; ArrowHead::"usage" = 
      "ArrowHead[a, vec] produces an arrowhead with its tip placed at the \
point a, pointing in the direction of the vector vec."; Tail::"usage" = 
      "Tail -> point puts the tail of the vector at point."; Norm::"usage" = 
      "Norm[vec] returns the Euclidean length of vec."; Normalize::"usage" = 
      "Normalize[vec] returns a unit vector in the direction of vec, \
provided vec is nonzero."; Aperture::"usage" = 
      "Aperture is the ratio of the radius of the base to the length of the \
head of a vector."; TipSize::"usage" = 
      "TipSize specifies an absolute size for the inner tip length of the \
head of a vector."; TipRatio::"usage" = 
      "TipRatio is the ratio of the inner length of the arrowhead to the \
length of the shaft of a vector."; HeadRatio::"usage" = 
      "HeadRatio is the ratio of the outer length of the arrowhead to the \
length of the shaft of a vector."; HeadSize::"usage" = 
      "HeadSize specifies an absolute size for the head of a vector."; 
    ScaleFactor::"usage" = 
      "ScaleFactor specifies the amount to scale a vector in length and may \
be either a number or a pure function."; ZeroVectorPointSize::"usage" = 
      "ZeroVectorPointSize is the size of the point used to represent a \
zero vector."; Options[Vector] = 
      Options[Arrow] = 
        Options[ArrowHead] = {HeadRatio -> 0.18, 
            TipRatio -> 0.14, Aperture -> 0.3, 
            ScaleFactor -> 1, HeadType -> Polygon, 
            ShaftQ -> True, EdgesQ -> True, 
            VectorColor -> RGBColor[0, 0, 1], ShaftWidth -> 0.005, 
            ZeroVectorPointSize -> 0.01}; Begin["`Private`"]; SetOptions[ParametricPlot, AspectRatio -> Automatic]; SetOptions[Plot, AspectRatio -> Automatic]; SetOptions[Graphics, AspectRatio -> Automatic]; Axes3D[u_, v_] := 
      Graphics3D[{{Blue, Line[{{-(u/3), 0, 0}, {u, 0, 0}}]}, 
          Text["x", {u + v, 0, 0}], {Blue, 
            Line[{{0, -(u/3), 0}, {0, u, 0}}]}, 
          Text["y", {0, u + v, 0}], {Blue, 
            Line[{{0, 0, -(u/3)}, {0, 0, u}}]}, 
          Text["z", {0, 0, u + v}]}]; Axes3D[u_] := Axes3D[u, (u/8)]; Norm[a_?VectorQ] := N[(Sqrt[a . a])]; Normalize[a_?VectorQ] := If[a . a == 0, a, (a/Norm)[a]]; Perpend[a_?VectorQ] := 
      Normalize[{a[[2]], 
            -a[[1]]}] /; 
        Length[a] == 2Perpend[a_?VectorQ] := 
      Normalize[
          If[a[[1]] == 0, {1, 0, 0}, {0, 
              a[[3]], 
              -a[[2]]}]] /; 
        Length[a] == 3Base3D = 
      Table[{Cos[((2 k π)/8.) ], Sin[((2 k π)/8.) ]}, {k, 0, 
          8}]; Vector[from_?VectorQ, to_?VectorQ, opts___] := 
      Module[{sq, sw, ht, hs, hr, tr, ap, sf, vc, zvps, diff = to - from, 
            tip}, {sq, sw, ht, hs, hr, ts, tr, ap, sf, vc, zvps} = 
            {ShaftQ, ShaftWidth, HeadType, HeadSize, HeadRatio, TipSize, 
                  TipRatio, Aperture, ScaleFactor, VectorColor, 
                  ZeroVectorPointSize} /. {opts} /. Options[Vector]; 
          If[from == to, 
            Return[Graphics[{PointSize[zvps], vc, Point[from]}]]]; 
          diff = If[NumberQ[N[sf]], sf diff, sf[diff]]; tip = from + diff; 
          If[NumberQ[hs], hr = (hs/Norm)[diff]; tr = 0.75 hr]; 
          If[NumberQ[ts], tr = (ts/Norm)[diff]]; 
          Graphics[{
              If[sq, {vc, Thickness[sw], Line[{from, tip - tr diff}]}, {}], 
              With[{trans = ap hr Norm[diff] Perpend[diff]}, {vc, 
                  Thickness[sw], 
                  ht[{tip, tip - hr diff + trans, tip - tr diff, 
                      tip - hr diff - trans, tip}]}]}]] /; 
        Length[from] == Length[to] == 2Vector[from_?VectorQ, to_?VectorQ, opts___] := 
      Module[{sq, eq, sw, hs, hr, tr, ap, sf, vc, zvps, diff = to - from, 
            tip}, {sq, eq, sw, hs, hr, ap, sf, vc, zvps} = 
            {ShaftQ, EdgesQ, ShaftWidth, HeadSize, HeadRatio, Aperture, 
                  ScaleFactor, VectorColor, ZeroVectorPointSize} /. {opts}
                   /. Options[Vector]; 
          If[from == to, 
            Return[Graphics3D[{PointSize[zvps], vc, Point[from]}]]]; 
          diff = If[NumberQ[N[sf]], sf diff, sf[diff]]; tip = from + diff; 
          If[NumberQ[hs], hr = (hs/Norm)[diff]]; tr = (hr/2); 
          Graphics3D[{
              If[sq, {vc, Thickness[sw], Line[{from, tip - tr diff}]}, {}], 
              If[eq, {}, EdgeForm[]], {SurfaceColor[vc], 
                With[{temp = Perpend[diff]}, 
                  (Polygon[Append[#1, tip]]&)/@
                    Partition[
                      (tip - hr diff + #1&)/@
                        ((ap hr Norm[diff] Base3D) . {temp, 
                              Normalize[temp\[Cross]diff]}), 2, 1]]}}]] /; 
        Length[from] == Length[to] == 3Arrow[a_, Tail -> b_, opts___] := Vector[b, a + b, opts]Arrow[a_, opts___] := Vector[Table[0, {Length[a]}], a, opts]\(ArrowHead[a_, b_, opts___] := 
      Vector[a - b, a, ShaftQ -> False, opts]; 

End[]; 

    \(EndPackage[]; \)\)

Graphics`Colors`GosiaGreen = RGBColor[0,  0.392187,  0];

Spherical Setup

\(Clear[polar]; 

    polar[{x_, y_, z_}] := {\@\(x\^2 + y\^2 + z\^2\), ArcTan[y\/x], 
        ArcCos[z\/\@\(x\^2 + y\^2 + z\^2\)]}; 
Clear[X, length]; 

    length[X_] := \@\(X . X\); 
\(rr = 3; \)\)R = 5; scale = 0.8; red = RGBColor[1, 0, 0]; blue = RGBColor[0, 0, 1]; color1 = RGBColor[0.8, 0.2, 0.8]; color2 = RGBColor[0.2, 0.8, 0.2]; black = RGBColor[0, 0, 0]; \(Clear[axes]; 

    axes := Graphics3D[{Line[{X, {1.5\ rr, 0, 0}}], 
          Line[{X, {0, 1.5\ rr, 0}}], Line[{X, {0, 0, 2\ rr}}]}]; 

    Clear[xyplane]; 

    xyplane := 
      Graphics3D[
        Polygon[{{\(-2\)\ rr, \(-2\)\ rr, 0}, {\(-2\)\ rr, 2\ rr, 0}, {2\ rr, 
              2\ rr, 0}, {2\ rr, \(-2\)\ rr, 0}}]]; 
\(X = {0, 0, 0}; \)\)origin = Graphics3D[{black, Point[X]}]; Z = {2, 3, 5}; \(Clear[point]; 

    point := Graphics3D[{red, PointSize[0.03], Point[scale\ Z]}]; 

    \(Z0 = {0, 0, length[0.8\ Z]}; \)\)\(Clear[zpoint]; 
zpoint = Graphics3D[{black, Point[Z0]}]; 

    \(XY = scale\ Z\ {1, 1, 0}; \)\)\(Clear[xypoint]; 
xypoint := Graphics3D[{black, Point[XY]}]; 

    \(X0 = {length[scale\ Z\ {1, 1, 0}], 0, 0}; \)\)xpoint = Graphics3D[{black, Point[X0]}]; Y0 = {0, length[scale Z {1, 1, 0}], 0}; ypoint = Graphics3D[{black, Point[Y0]}]; \(Clear[ray]; 
ray := Graphics3D[{blue, Line[{X, scale\ Z}]}]; 

    Clear[ray1]; 
ray1 := Graphics3D[{color1, Line[{scale\ Z, Z}]}]; 

    Clear[xyray]; 
xyray := Graphics3D[{red, Line[{X, Z\ {1, 1, 0}}]}]; 

    \(rays := {ray, xyray}; \)\)xyaux = 
      (length[0.8 Z] (0.8 Z {1, 1, 0}))/
        length[0.8 Z {1, 1, 0}]; \(Clear[zarc]; 

    zarc = Graphics3D[{blue, 
          Line[Table[
              \(Cos[t]\ \) {0, 0, length[0.8\ Z]} + Sin[t]\ xyaux, {t, 0, 
                \(polar[0.8\ Z]\)[[3
                  ]],  .1}]]}]; 
Clear[zarc1]; 

    zarc1 = Graphics3D[{color1, 
          Line[Table[
              \(Cos[t]\ \) {0, 0, length[0.8\ Z]} + Sin[t]\ xyaux, {t, 
                \(polar[0.8\ Z]\)[[3]], 
                1,  .1}]]}]; 
Clear[zarc2]; 

    zarc2 = Graphics3D[{color1, 
          Line[Table[
              \(Cos[t]\ \) {0, 0, length[0.8\ Z]} + Sin[t]\ xyaux, {t, 
                \(-0.8\), 0,  .1}]]}]; 
Clear[xyarc]; 

    xyarc = Graphics3D[{blue, 
          Line[Table[
              Cos[t]\ X0 + Sin[t]\ Y0, {t, 0, 
                \(polar[XY]\)[[2
                  ]],  .1}]]}]; 
Clear[xyarc1]; 

    xyarc1 = Graphics3D[{color1, 
          Line[Table[Cos[t]\ X0 + Sin[t]\ Y0, {t, \(- .5\), 0,  .1}]]}]; 

    Clear[xyarc2]; 

    xyarc2 = Graphics3D[{color1, 
          Line[Table[
              Cos[t]\ X0 + Sin[t]\ Y0, {t, 
                \(polar[XY]\)[[2]], 
                \(polar[XY]\)[[2]] + 
                  0.3,  .1}]]}]; 
Clear[stick]; 

    stick = Graphics3D[{red, Line[{scale\ Z, XY}]}]; 

    \(xaxislabel = Text["x", {0.363, 0.26}]; \)\)yaxislabel = Text["y", {0.73, 0.3}]; zaxislabel = Text["z", { .5, 0.67}]; stickplot = 
      Show[xyplane, axes, ray, point, Boxed -> False, 
        ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; rho = Text["r", {0.53, 0.45}]; rlabel = 
      Show[xyplane, axes, ray, point, Boxed -> False, 
        ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {rho, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; phi = Text["s", {0.55, 0.57}]; slabel = 
      Show[xyplane, zpoint, axes, ray, point, zarc, zarc1, zarc2, 
        Boxed -> False, ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {phi, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; theta = Text["t", { .5, 0.3}]; tlabel = 
      Show[xyplane, axes, xypoint, xpoint, ray, stick, point, xyarc, xyarc1, 
        xyarc2, Boxed -> False, ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {theta, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; all = 
      Show[xyplane, axes, ray, stick, point, zpoint, xpoint, xypoint, xyarc, 
        xyarc1, xyarc2, zarc, zarc1, zarc2, Boxed -> False, 
        ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {theta, phi, rho, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity];

Crisscross

\(Clear[x]; 

    crisscross = 
      Plot[{Cos[x], Sin[x]}, {x, π\/8, \(3\ π\)\/8}, 
        PlotStyle -> {{RGBColor[1, 0, 0], Thickness[0.02]}}, 
        Axes -> None, AspectRatio -> 1\/3, 
        DisplayFunction -> Identity]; \)

L.1)

Here's a rather detailed plot of a certain vector field:

\(Clear[Field, x, y]; 
Field[x_, y_] = {x, 2 - y}; 

    \(scalefactor = 0.2; \)\)Fieldplot = 
      Table[Arrow[scalefactor Field[x, y], Tail -> {x, y}], {x, -5, 
          6, (6/10)}, {y, -2, 6, (4/10)}]; Show[Fieldplot, Axes -> Automatic, 
      AxesLabel -> {"x", "y"}];

Visible in this plot are four families of trajectories in this vector field.
Pencil in one trajectory of each type.

L.2)

Here's a plot of a certain vector field shown with the plots of two curves:

\(Clear[Field, x, y]; 
Field[x_, y_] = 0.2\ {y, x}; 

    \(vectorfieldplot = 
      Table[Arrow[Field[x, y], Tail -> {x, y}], {x, \(-3\), 3, 0.5}, {y, 
          \(-3\), 3, 0.5}]; \)\){m[x_, y_], n[x_, y_]} = Field[x, y]; {a, b} = {2.5, -2}; starterpoint = {a, b}; Clear[Derivative, x, y, t];
equationx=x'[t]==m[x[t], y[t]];
equationy=y'[t]==n[x[t], y[t]];starterx = x[0] == a; startery = y[0] == b; endtime = 13; approxsolutions = 
      NDSolve[{equationx, equationy, starterx, startery}, {x[t], y[t]}, {t, 
          0, endtime}]; \(Clear[trajectory]; 

    trajectory[t_] = {
        x[t] /. approxsolutions[[1]], 
        y[t] /. approxsolutions[[1]]}; 

    \(trajectoryplot = 
      ParametricPlot[trajectory[t], {t, 0, endtime}, 
        PlotStyle -> {{Thickness[0.016]}}, 
        DisplayFunction -> Identity]; \)\)starterplot = Graphics[{PointSize[0.03], Point[starterpoint]}]; othercurve = 
      Plot[-0.32 (t^2), {t, -3.26, 2.5}, 
        PlotStyle -> Thickness[0.016], 
        DisplayFunction -> Identity]; Show[vectorfieldplot, starterplot, othercurve, trajectoryplot, 
      PlotRange -> All, DisplayFunction -> $DisplayFunction];

One of these curves is a genuine trajectory in the vector field. 
The other is bogus. 
Identify the trajectory and discuss the information that led to your choice.

L.3)

You are given that the point {2, 1} is on a certain trajectory in the vector field
       Field[x, y] = {x +  y, x -  y}.
Write down a vector that is tangent to this trajectory at the point {2, 1}.

L.4)

How do you know that when you run a trajectory in the vector field        
       Field[x, y] = {(y^2), (x^2)},
starting at any point you like, you are guaranteed that the trajectory is the graph of a function whose derivative is positive?

L.5)

Here's an ellipse C and the plot of a certain vector field Field[x, y] plotted with tail at {x, y} for a fairly generous selection of points {x, y} on the ellipse.

\(Clear[Field, m, n, x, y]; 
m[x_, y_] = x + y; 
\(n[x_, y_] = y - x; 
    \)\)Field[x_, y_] = {m[x, y], n[x, y]}; \(Clear[t]; 
x[t_] = 2 + 8\ Cos[t]; 
\(y[t_] = 1 + 4\ Sin[t]; \)\)curveplot = 
      ParametricPlot[{x[t], y[t]}, {t, 0, 2 π}, 
        PlotStyle -> {{Thickness[0.01]}}, 
        DisplayFunction -> Identity, 
        Epilog -> Text["C", {5, 4}]]; jump = (π/10); fieldvectors = 
      Table[Arrow[Field[x[t], y[t]], Tail -> {x[t], y[t]}], {t, 0, 
          2 π - jump, jump}]; Show[curveplot, fieldvectors, AspectRatio -> Automatic, 
      AxesLabel -> {"x", "y"}, 
      DisplayFunction -> $DisplayFunction];

Look at this plot, and then estimate whether the net flow of this vector field across C is from inside to outside, or is from outside to inside, and estimate whether the net flow of this vector field along this curve is clockwise or counterclockwise.

L.6)

Here's a circle and the plot of a certain vector field Field[x, y] plotted with tails at {x, y} for a generous selection of points {x, y} on the circle.

\(Clear[Field, m, n, x, y]; 
m[x_, y_] = \(-x\) - y; 

    \(n[x_, y_] = \(-y\) + x; \)\)Field[x_, y_] = {m[x, y], n[x, y]}; \(Clear[t]; 
x[t_] = 1 + 2\ Cos[t]; 
\(y[t_] = 2 + 2\ Sin[t]; \)\)curveplot = 
      ParametricPlot[{x[t], y[t]}, {t, 0, 2 π}, 
        PlotStyle -> {{Thickness[0.01]}}, 
        DisplayFunction -> Identity, 
        Epilog -> Text["C", {-0.7, 1.7}]]; jump = (π/12); scalefactor = 0.6; fieldvectors = 
      Table[Arrow[scalefactor Field[x[t], y[t]], 
          Tail -> {x[t], y[t]}], {t, 0.1, 2 π + 0.1 - jump, jump}]; 
    Show[curveplot, fieldvectors, AspectRatio -> Automatic, 
      AxesLabel -> {"x", "y"}, 
      DisplayFunction -> $DisplayFunction];

Here are tangential and normal components of what you see above:

Clear[tangent, tancomponent];
tangent[t_]={x'[t],y'[t]};
tancomponent[t_] = 
        ((Field[x[t], y[t]] . tangent[t] tangent[t])/tangent)[t] . 
            tangent[t]; tancomps = 
      Table[Arrow[scalefactor tancomponent[t], Tail -> {x[t], y[t]}], {
          t, 0.1, 2 π + 0.1 - jump, jump}]; tans = Show[curveplot, tancomps, AspectRatio -> Automatic, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; normcomps = 
      Table[Arrow[scalefactor (Field[x[t], y[t]] - tancomponent[t]), 
          Tail -> {x[t], y[t]}], {t, 0.1, 2 π + 0.1 - jump, jump}]; 
    norms = 
      Show[curveplot, normcomps, AspectRatio -> Automatic, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; Show[GraphicsArray[{tans, norms}], 
      DisplayFunction -> $DisplayFunction];

Look at these plots, and then estimate whether the net flow of this vector field across the circle is from inside to outside, or is from outside to inside, and estimate whether the net flow of this vector field along this circle is clockwise or counterclockwise.

L.7)

Here's a circle and the plot of a different vector field Field[x, y] plotted with tails at {x, y} for a generous selection of points {x, y} on the circle.

\(Clear[Field, m, n, x, y]; 
m[x_, y_] = y; 
\(n[x_, y_] = x; \)\)Field[x_, y_] = {m[x, y], n[x, y]}; \(Clear[t]; 
x[t_] = Cos[t]; 
\(y[t_] = Sin[t]; \)\)curveplot = 
      ParametricPlot[{x[t], y[t]}, {t, 0, 2 π}, 
        PlotStyle -> {{Thickness[0.01]}}, 
        DisplayFunction -> Identity, 
        Epilog -> Text["C", {0.7, 0.5}]]; jump = (π/12); scalefactor = 1; fieldvectors = 
      Table[Arrow[scalefactor Field[x[t], y[t]], 
          Tail -> {x[t], y[t]}], {t, 0.1, 2 π + 0.1 - jump, jump}]; 
    Show[curveplot, fieldvectors, AspectRatio -> Automatic, 
      AxesLabel -> {"x", "y"}, 
      DisplayFunction -> $DisplayFunction];

Here are tangential and normal components of what you see above:

Clear[tangent, tancomponent];
tangent[t_]={x'[t],y'[t]};
tancomponent[t_] = 
        ((Field[x[t], y[t]] . tangent[t] tangent[t])/tangent)[t] . 
            tangent[t]; tancomps = 
      Table[Arrow[scalefactor tancomponent[t], Tail -> {x[t], y[t]}], {
          t, 0.1, 2 π + 0.1 - jump, jump}]; tans = Show[curveplot, tancomps, AspectRatio -> Automatic, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; normcomps = 
      Table[Arrow[scalefactor (Field[x[t], y[t]] - tancomponent[t]), 
          Tail -> {x[t], y[t]}], {t, 0.1, 2 π + 0.1 - jump, jump}]; 
    norms = 
      Show[curveplot, normcomps, AspectRatio -> Automatic, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; Show[GraphicsArray[{tans, norms}], 
      DisplayFunction -> $DisplayFunction];

Look at these plots and then estimate whether the net flow of this vector field across the circle is from inside to outside, from outside to inside, or zero, and estimate whether the net flow of this vector field along this circle is clockwise or counterclockwise or zero.

L.8)

True or false:
When you plot a given vector field at points {x, y} on a given curve, then all the field vectors you plot turn out to be perpendicular to the curve.

L.9)

Here is a rectangle C with eight labeled points:

{a, b} = {-2, -1}; {h, k} = {6, 4}; sizer = 0.03; Show[
      Graphics[{Thickness[0.01], 
          Line[{{a, b}, {a + h, b}, {a + h, b + k}, {a, b + k}, {a, b}}]}], 
      Graphics[{PointSize[sizer], Point[{a, b}]}], 
      Graphics[{PointSize[sizer], Point[{a + (h/2), b}]}], 
      Graphics[{PointSize[sizer], Point[{a + h, b}]}], 
      Graphics[{PointSize[sizer], Point[{a + h, b + (k/2)}]}], 
      Graphics[{PointSize[sizer], Point[{a + h, b + k}]}], 
      Graphics[{PointSize[sizer], Point[{a + (h/2), b + k}]}], 
      Graphics[{PointSize[sizer], Point[{a, b + k}]}], 
      Graphics[{PointSize[sizer], Point[{a, b + (k/2)}]}], spacer = 3; 
      Graphics[Text[{a, b}, {a, b}, {0, spacer}]], 
      Graphics[Text[{a + (h/2), b}, {a + (h/2), b}, {0, spacer}]], 
      Graphics[Text[{a + h, b}, {a + h, b}, {0, spacer}]], 
      Graphics[Text[{a + h, b + (k/2)}, {a + h, b + (k/2)}, {
            -(spacer/2), -1}]], 
      Graphics[Text[{a + h, b + k}, {a + h, b + k}, {0, -spacer}]], 
      Graphics[Text[{a + (h/2), b + k}, {a + (h/2), b + k}, {0, -spacer}]], 
      Graphics[Text[{a, b + k}, {a, b + k}, {0, -spacer}]], 
      Graphics[Text[{a, b + (k/2)}, {a, b + (k/2)}, {(spacer/2), -1}]], 
      Axes -> True, AxesLabel -> {"x", "y"}, 
      AspectRatio -> Automatic, 
      PlotRange -> {{-6, 10}, {-6, 6}}];

Go with the vector field 
       Field[x, y] = {y +  x, -x}. 
For each labeled point {x, y}, pencil in the field vector Field[x, y] with tail at {x, y}. 
On the basis of your plot, estimate whether the net flow of this vector field across the edge of this rectangle is from inside to outside, or is from outside to inside, and estimate whether the net flow of this vector field along this curve is clockwise or counterclockwise.

L.10)

Your boss gives you a differential equation 
       y'[x]=f[x, y[x]] with y[0] = 2,
and asks you to give an approximate plot of its solution.  
As a literate calculus person, you know that when you start the trajectory of a certain vector field at {0, 2}, then the plot of this trajectory will be the same as the plot of the solution you want.
The question here is:
What vector field do you go with?

L.11)

Can two trajectories of a vector field ever cross over each other like this:

\(Clear[x]; 

    Plot[{Cos[x], Sin[x]}, {x, π\/8, \(3\ π\)\/8}, 
      PlotStyle -> {{Thickness[0.01]}}, Axes -> None, 
      AspectRatio -> 1\/3]; \)

Why or why not?

L.12)

You have a differential equation 
       y'[x]=f[x, y[x]].
If you have a formula for f[x, y] and you start with y[0] = 1, then you can use NDSolve to plot out a function y[x] with y[0] = 1 and
       y'[x]=f[x, y[x]].
On the other hand if you start with y[0] = 2, then you can use NDSolve to plot out a function y[x] with y[0] = 2 and
       y'[x]=f[x, y[x]].
The question here is: 
Can the plots of these two solutions ever cross over each other like this:

Clear[x]Plot[{Cos[x], Sin[x]}, {x, (π/8), ((3 π)/8)}, 
      PlotStyle -> {{Thickness[0.01]}}, Axes -> None, 
      AspectRatio -> (1/3)];

Why or why not?

L.13)

Here is a plot of the gradient field of a certain function near its maximizer, which is plotted as a heavy dot:

\(Clear[f, x, y, r, t, gradf]; 

    f[x_, y_] = 
      E\^\(-\((x\ y - 1)\)\^2\)\/\(1 + 0.8\ \((y - 1)\)\^2 + 
          \((1 - x)\)\^2\); 

    \(gradf[x_, y_] = {∂\_x f[x, y], ∂\_y f[x, y]}; 
    \)\)maximizer = {1, 1}; max = Graphics[{PointSize[0.05], Point[maximizer]}]; scalefactor = 0.07; h = 0.1; gradfieldplot = 
      Table[Arrow[scalefactor gradf[x, y], Tail -> {x, y}], {x, 
          0.95 - h, 0.95 + h, (h/6)}, {y, 1 - h, 1 + h, (h/6)}]; Show[max, gradfieldplot, Axes -> Automatic, 
      AxesLabel -> {"x", "y"}];

Pencil in a couple of trajectories.
Where are the trajectories in this gradient field trying to go?
Why are they trying to go there?

L.14)

Here is a plot of the negative gradient field of a certain function near its minimizer, which is plotted as a heavy dot:

\(Clear[f, gradf, x, y]; 

    f[x_, y_] = 1.2\ E\^\(x\^2\) + 2.9\ E\^\(y\^2\) - 8.2\ x\ y; 

    \(gradf[x_, y_] = {∂\_x f[x, y], ∂\_y f[x, y]}; 
    \)\)minimizer = {0.989761, 0.771567}; min = Graphics[{PointSize[0.05], Point[minimizer]}]; \(Clear[neggradf]; 
neggradf[x_, y_] = \(-gradf[x, y]\); 

    \(scalefactor = 0.013; \)\)h = 0.2; neggradfieldplot = 
      Table[Arrow[scalefactor neggradf[x, y], Tail -> {x, y}], {x, 
          0.9 - h, 0.9 + h, (h/6)}, {y, 0.7 - h, 0.7 + h, (h/6)}]; neggrads = 
      Show[min, neggradfieldplot, Axes -> Automatic, 
        AxesLabel -> {"x", "y"}];

Pencil in a couple of trajectories.
Where are the trajectories in this negative gradient field trying to go?
Why are they trying to go there?

L.15)

Suppose f[(Subscript[x, 0]), (Subscript[y, 0])] > f[x, y] for all other {x, y} near {(Subscript[x, 0]), (Subscript[y, 0])}.  If you center a small circle C at {(Subscript[x, 0]), (Subscript[y, 0])}, why do you expect that the net flow of gradf[x, y] across C is from outside to inside?
Suppose f[(Subscript[x, 0]), (Subscript[y, 0])] < f[x, y] for all other {x, y} near {(Subscript[x, 0]), (Subscript[y, 0])}.  If you center a small circle C at {(Subscript[x, 0]), (Subscript[y, 0])}, why do you expect that the net flow of gradf[x, y] across C is from inside to outside?

L.16)

True or False?
Trajectories through the negative gradient field of 
       f[x, y] = (x^2) +  3  (y^2) 
all hustle their buns to the minimizing point {0, 0} where they stop dead.

L.17)

What do you think is the relation between between trajectories of the negative gradient field and Mathematica's FindMinimum instruction?

L.18)

True or false?
Trajectories through the gradient field of f[x, y] try to seek out maximizers of f[x, y], but they sometimes get sidetracked by a pretender.
Trajectories through the negative gradient field of f[x, y] try to find minimizers of f[x, y] but they sometimes get sidetracked by a pretender.