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No doubt about it. The net flow of this vector field across this curve is from inside to \ outside.\ \>", "SmallText"] }, Closed]], Cell["B.2.b) ", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Describe the net flow across the ellipse parameterized by\n ", Cell[BoxData[ \(P[t] = {6 Cos[t], 4 Sin[t]}\)]], " \nwith ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] 2 \[Pi]\)]], " of a fluid whose velocity is given by the vector field\n ", Cell[BoxData[ \(Field[x, y] = {y - \ 1, y + \ x}\)]], "." }], "Text"], Cell["Answer:", "Subsubsection"], Cell["Enter the vector field:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y];\)\), "\n", \(\(m[x_, y_] = y - 1;\)\), "\n", \(\(n[x_, y_] = y + x;\)\), "\n", \(Field[x_, y_] = {m[x, y], n[x, y]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-1\) + y, x + y}\)], "Output"] }, Open ]], Cell["Here's a look at the curve and the vector field:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(vectorfieldplot = 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The net flow of this vector field across this curve is from inside to \ outside.\ \>", "SmallText"] }, Closed]], Cell["B.2.c) ", "Subsubsection"], Cell[TextData[{ "Describe the net flow across and along the ellipse parameterized by\n \ ", Cell[BoxData[ \(P[t] = {5 Cos[t], 3 Sin[t]}\)]], " \nwith ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] 2 \[Pi]\)]], " of a fluid whose velocity is given by the vector field\n ", Cell[BoxData[ RowBox[{\(Field[x, y]\), "=", RowBox[{"{", RowBox[{\(x - \ y\), ",", StyleBox[\(y\/2\), FontSize->16]}], "}"}]}]]], "." }], "Text", Evaluatable->False], Cell["Answer:", "Subsubsection"], Cell["Enter the vector field:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y];\)\), "\n", \(\(m[x_, y_] = x - y;\)\), "\n", \(\(n[x_, y_] = y\/2;\)\), "\n", \(Field[x_, y_] = {m[x, y], n[x, y]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x - y, y\/2}\)], "Output"] }, Open ]], Cell["Here's a look the curve and the vector field:", 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Looking at the plot, you can see: that: \[Rule] The net flow of this vector field across this curve is from inside to \ outside. \[Rule] The net flow of this vector field along this curve is \ counterclockwise.\ \>", "SmallText"] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["T.4) The 2D electric field", "Subsection", CellTags->"VC.04.T4"], Cell[TextData[{ "In the electric interpretation, a vector field\n ", Cell[BoxData[ \(ElectricField[x, y] = {m[x, y], n[x, y]}\)]], "\nis a vector such that\n ", Cell[BoxData[ \(ElectricField[x, y] . U\)]], "\nmeasures the instantaneous voltage drop when you leave ", Cell[BoxData[ \({x, y}\)]], " in the direction of a unit vector ", Cell[BoxData[ \(U\)]], ". \nThis is the same as saying ", Cell[BoxData[ \(ElectricField[x, y]\)]], " represents the force the field places on a unit charge at the point ", Cell[BoxData[ \({x, y}\)]], "." }], "Text"], Cell["T.4.a)", "Subsubsection"], Cell[TextData[{ "Here's a plot of the electric field resulting from a point charge of size \ ", Cell[BoxData[ \(2\)]], " at ", Cell[BoxData[ \({0.50, 1.50}\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ElectricField, x, y];\)\), "\n", \(\(q = 2;\)\), "\n", \(\({a, b} = {0.50, 1.50};\)\), "\n", \(ElectricField[x_, y_] = \(q\ {x - a, y - b}\)\/\(\((x - a)\)\^2 + \((y - b)\)\^2\)\)}], \ "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(2\ \((\(-0.5`\) + x)\)\)\/\(\((\(-0.5`\) + x)\)\^2 + \((\(-1.5`\) + \ y)\)\^2\), \(2\ \((\(-1.5`\) + y)\)\)\/\(\((\(-0.5`\) + x)\)\^2 + \ \((\(-1.5`\) + 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How do you interpret this plot?\ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The electric field vector, ", Cell[BoxData[ \(ElectricField[x, y]\)]], ", plotted as above, with tail at ", Cell[BoxData[ \({x, y}\)]], " points in the direction of biggest voltage drop, and the length of the \ vector represents the size of the voltage drop in that direction.\nReason: If \ you leave ", Cell[BoxData[ \({x, y}\)]], " in the direction of a unit vector ", Cell[BoxData[ \(U\)]], ", then the instantaneous voltage drop is\n ", Cell[BoxData[ \(ElectricField[x, y] . U\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar] \[LeftDoubleBracketingBar]U\ \[RightDoubleBracketingBar] Cos[angle\ between]\)\)\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar] Cos[angle\ between]\)\)\)]], "\nbecause ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]U\[RightDoubleBracketingBar] = 1\)]], ".\nThis tells you that the instantaneous voltage drop is as big as \ possible when\n ", Cell[BoxData[ \(ElectricField[x, y] . U\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar] Cos[angle\ between]\)\)\)]], "\nis as big as possible. This happens when \n ", Cell[BoxData[ \(Cos[angle\ between] = 1\)]], ", \nwhich is the same as saying that ", Cell[BoxData[ \(angle\ between = 0\)]], ".\nThe upshot:\nTo get the biggest instantaneous voltage drop, you take\n \ ", Cell[BoxData[ RowBox[{"U", "=", StyleBox[\(ElectricField[x, y]\/\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar]\), FontSize->16]}]]], ".\nThe direction of the biggest instantaneous voltage drop is the \ direction of \n ", Cell[BoxData[ \(ElectricField[x, y]\)]], ".\nThe size of the biggest instantaneous voltage drop is\n ", Cell[BoxData[ RowBox[{\(ElectricField[x, y]\), ".", StyleBox[\(ElectricField[x, y]\/\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar]\), FontSize->16]}]]], "\n ", Cell[BoxData[ RowBox[{"=", StyleBox[\(\[LeftDoubleBracketingBar]ElectricField[x, y]\ \[RightDoubleBracketingBar]\^2\/\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar]\), FontSize->16]}]]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar]\)\)\)]], ".\n \nThis means that ", Cell[BoxData[ \(ElectricField[x, y]\)]], " points in the direction of the biggest instantaneous voltage drop, and \ that the instantaneous voltage drop in this direction is ", Cell[BoxData[ \(\[LeftDoubleBracketingBar]ElectricField[x, y]\[RightDoubleBracketingBar]\)]], "." }], "SmallText"] }, Closed]], Cell["T.4.b)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "What do you get when you put a charge of size \n ", Cell[BoxData[ \(1.5\)]], " at ", Cell[BoxData[ \({\(-0.875\), 0.375}\)]], ", \nand a charge of size \n ", Cell[BoxData[ \(\(-1.5\)\)]], " at ", Cell[BoxData[ \({0.875, 0.375}\)]], "? " }], "Text"], Cell["Answer:", "Subsubsection"], Cell["\<\ You get something pretty interesting. Start by adding the electric fields resulting from the two point \ charges:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ElectricField1, ElectricFieldF2, CombinedElectricField, x, y];\)\), "\n", \(\(q1 = 1.5;\)\), "\n", \(\({a1, b1} = {\(-0.875\), 0.375};\)\), "\n", \(\(ElectricField1[x_, y_] = \(q1\ {x - a1, y - b1}\)\/\(\((x - a1)\)\^2 + \((y - b1)\)\^2\ \);\)\), "\n", \(\(q2 = \(-1.5\);\)\), "\n", \(\({a2, b2} = {0.875, 0.375};\)\), "\n", \(\(ElectricField2[x_, y_] = \(q2\ {x - a2, y - b2}\)\/\(\((x - a2)\)\^2 + \((y - b2)\)\^2\ \);\)\), "\n", \(\(CombinedElectricField[x_, y_] = ElectricField1[x, y] + ElectricField2[x, y];\)\n\), "\n", \(\(Show[Graphics[{Red, PointSize[0.08], Point[{a1, b1}]}], Graphics[{Blue, PointSize[0.08], Point[{a2, b2}]}], Table[Arrow[CombinedElectricField[x, y], Tail \[Rule] {x, y}], {x, \(-3\), 3}, {y, \(-2\), 4}], Axes \[Rule] Automatic];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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its gradient field?" }], "Text"], Cell["G.4.a.ii)", "Subsubsection"], Cell[TextData[{ "This time go with the function\n ", Cell[BoxData[ RowBox[{\(f[x, y]\), "=", StyleBox[\(2\/\(1\ + \ \((y\ - \ 2)\)\^2\ + \ 1.5\ \((x\ - \ 1)\)\^2\)\), FontSize->16]}]]], ".\nSay why you are sure that ", Cell[BoxData[ \({x, y} = {1, 2}\)]], " maximizes ", Cell[BoxData[ \(f[x, y]\)]], ". Then plot the (scaled) gradient field of ", Cell[BoxData[ \(f[x, y]\)]], " near ", Cell[BoxData[ \({1, 2}\)]], ".\nDescribe what you see and explain why you see it." }], "Text"], Cell["G.4.a.iii)", "Subsubsection"], Cell[TextData[{ "You are given a certain function ", Cell[BoxData[ \(f[x, y]\)]], " and the information that \n ", Cell[BoxData[ \({a, b, f[a, b]}\)]], " \nsits on the top of a crest of the surface given by \n ", Cell[BoxData[ \(z = f[x, y]\)]], ". \nYou center a small circle at ", Cell[BoxData[ \({a, b}\)]], " and plot the gradient field of ", Cell[BoxData[ \(f[x, y]\)]], " on this circle. \nIs the net flow of the gradient field of ", Cell[BoxData[ \(f[x, y]\)]], " across this circle from inside to outside or from outside to inside?" }], "Text"], Cell["G.4.b.i) Minimizers", "Subsubsection"], Cell[TextData[{ "This time go with \n ", Cell[BoxData[ \(f[x, y] = e\^\(\(1\/9\) \((x\^2 + \ 2 y\^2)\)\)\)]], "\[CenterDot] ." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ 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What are the trajectories in this gradient field trying to get away \ from?\ \>", "Text"], Cell["G.4.b.ii)", "Subsubsection"], Cell[TextData[{ "You are given a certain function ", Cell[BoxData[ \(f[x, y]\)]], " and the information that \n ", Cell[BoxData[ \({a, b, f[a, b]}\)]], " \nsits on the bottom of a dip of the surface given by \n ", Cell[BoxData[ \(z = f[x, y]\)]], ". \nYou center a small circle at ", Cell[BoxData[ \({a, b}\)]], " and plot the gradient field of ", Cell[BoxData[ \(f[x, y]\)]], " on this circle. Is the net flow of the gradient field of ", Cell[BoxData[ \(f[x, y]\)]], " across this circle from inside to outside or from outside to inside?" }], "Text"], Cell["G.4.c) Crests and dips", "Subsubsection"], Cell[TextData[{ "Here's a plot of the surface\n ", Cell[BoxData[ RowBox[{"z", "=", RowBox[{\(f[x, y]\), "=", StyleBox[\(\(Sin[2\ x]\ + \ Sin[3\ y]\)\/\(0.5\ + \ x\^2\ + \ y\^2\)\), FontSize->16]}]}]]], "\nfor ", Cell[BoxData[ \(\(-1\) \[LessEqual] x \[LessEqual] 1\)]], " and ", Cell[BoxData[ \(\(-1\) \[LessEqual] y \[LessEqual] 1\)]], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x, y];\)\), "\n", \(\(f[x_, y_] = \(Sin[2\ x] + Sin[3\ y]\)\/\(0.5 + x\^2 + y\^2\);\)\), "\n", \(\(Plot3D[f[x, y], {x, \(-1\), 1}, {y, \(-1\), 1}, AxesLabel \[Rule] {"\", "\", "\"}, ViewPoint \[Rule] CMView];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .63304 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix 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.29187 .32082 .24849 .31844 Metetra .297 .694 .976 r .24849 .31844 .29187 .32082 .25929 .30028 .21544 .30035 Metetra .388 .716 .988 r .21544 .30035 .25929 .30028 .22565 .2759 .18134 .27833 Metetra .517 .643 .912 r .60664 .42266 .64596 .42529 .62444 .39794 .58483 .39415 Metetra .5 .635 .914 r .58483 .39415 .62444 .39794 .60236 .37011 .56247 .36493 Metetra .465 .634 .928 r .56247 .36493 .60236 .37011 .57971 .34404 .53953 .33712 Metetra .383 .639 .956 r .53953 .33712 .57971 .34404 .55647 .32327 .51597 .31403 Metetra .133 .59 .924 r .51597 .31403 .55647 .32327 .53253 .31229 .49168 .29979 Metetra .367 0 0 r .49168 .29979 .53253 .31229 .50774 .31456 .46647 .29773 Metetra .587 .078 .009 r .46647 .29773 .50774 .31456 .48189 .3289 .44015 .30739 Metetra .641 .182 .147 r .44015 .30739 .48189 .3289 .45484 .34775 .41262 .32301 Metetra .659 .19 .128 r .41262 .32301 .45484 .34775 .42663 .36116 .38392 .33617 Metetra .638 .113 0 r .38392 .33617 .42663 .36116 .39738 .36308 .35417 .34078 Metetra .512 0 0 r .35417 .34078 .39738 .36308 .36718 .35309 .32348 .33517 Metetra 0 .337 .689 r .32348 .33517 .36718 .35309 .33606 .33393 .29187 .32082 Metetra .039 .517 .884 r .29187 .32082 .33606 .33393 .30397 .30888 .25929 .30028 Metetra .214 .606 .95 r .25929 .30028 .30397 .30888 .2708 .28059 .22565 .2759 Metetra .446 .619 .924 r .64596 .42529 .68623 .43279 .66506 .40829 .62444 .39794 Metetra .396 .595 .925 r .62444 .39794 .66506 .40829 .64333 .38411 .60236 .37011 Metetra .295 .561 .928 r .60236 .37011 .64333 .38411 .62103 .36265 .57971 .34404 Metetra .05 .465 .883 r .57971 .34404 .62103 .36265 .59815 .34752 .55647 .32327 Metetra .44 0 0 r .55647 .32327 .59815 .34752 .57461 .34293 .53253 .31229 Metetra .701 .192 .022 r .53253 .31229 .57461 .34293 .55026 .35141 .50774 .31456 Metetra .737 .31 .236 r .50774 .31456 .55026 .35141 .52489 .37007 .48189 .3289 Metetra .744 .339 .283 r .48189 .3289 .52489 .37007 .49834 .38958 .45484 .34775 Metetra .751 .322 .231 r .45484 .34775 .49834 .38958 .47062 .39968 .42663 .36116 Metetra .735 .24 .061 r .42663 .36116 .47062 .39968 .44185 .39562 .39738 .36308 Metetra .612 .044 0 r .39738 .36308 .44185 .39562 .41212 .37876 .36718 .35309 Metetra 0 .235 .646 r .36718 .35309 .41212 .37876 .38145 .35306 .33606 .33393 Metetra 0 .437 .851 r .33606 .33393 .38145 .35306 .34982 .32233 .30397 .30888 Metetra .157 .544 .926 r .30397 .30888 .34982 .32233 .31711 .28929 .2708 .28059 Metetra .403 .622 .941 r .68623 .43279 .72747 .44141 .70672 .42042 .66506 .40829 Metetra .322 .588 .94 r .66506 .40829 .70672 .42042 .68542 .40078 .64333 .38411 Metetra .151 .519 .916 r .64333 .38411 .68542 .40078 .66357 .38499 .62103 .36265 Metetra 0 .311 .726 r .62103 .36265 .66357 .38499 .64116 .37643 .59815 .34752 Metetra .628 .065 0 r .59815 .34752 .64116 .37643 .61813 .37852 .57461 .34293 Metetra .734 .271 .153 r .57461 .34293 .61813 .37852 .59432 .39219 .55026 .35141 Metetra .743 .335 .276 r .55026 .35141 .59432 .39219 .56949 .41276 .52489 .37007 Metetra .745 .338 .279 r .52489 .37007 .56949 .41276 .54348 .43025 .49834 .38958 Metetra .744 .289 .172 r .49834 .38958 .54348 .43025 .51626 .43529 .47062 .39968 Metetra .682 .14 0 r .47062 .39968 .51626 .43529 .48793 .42483 .44185 .39562 Metetra .42 0 0 r .44185 .39562 .48793 .42483 .45862 .40152 .41212 .37876 Metetra 0 .392 .825 r .41212 .37876 .45862 .40152 .42836 .37001 .38145 .35306 Metetra .139 .509 .911 r .38145 .35306 .42836 .37001 .39713 .33429 .34982 .32233 Metetra .254 .571 .939 r .34982 .32233 .39713 .33429 .36485 .29706 .31711 .28929 Metetra .415 .663 .962 r .72747 .44141 .76956 .44706 .74929 .42902 .70672 .42042 Metetra .327 .636 .963 r .70672 .42042 .74929 .42902 .72849 .41309 .68542 .40078 Metetra .133 .566 .924 r .68542 .40078 .72849 .41309 .70718 .40165 .66357 .38499 Metetra .253 0 0 r .66357 .38499 .70718 .40165 .68534 .39759 .64116 .37643 Metetra .575 .014 0 r .64116 .37643 .68534 .39759 .6629 .40325 .61813 .37852 Metetra .657 .188 .126 r .61813 .37852 .6629 .40325 .6397 .41819 .59432 .39219 Metetra .663 .229 .207 r .59432 .39219 .6397 .41819 .6155 .43709 .56949 .41276 Metetra .645 .185 .145 r .56949 .41276 .6155 .43709 .5901 .45082 .54348 .43025 Metetra .559 .005 0 r .54348 .43025 .5901 .45082 .56344 .45164 .51626 .43529 Metetra 0 .368 .686 r .51626 .43529 .56344 .45164 .5356 .43742 .48793 .42483 Metetra .165 .569 .933 r .48793 .42483 .5356 .43742 .50672 .41093 .45862 .40152 Metetra .321 .607 .95 r .45862 .40152 .50672 .41093 .47686 .37663 .42836 .37001 Metetra .385 .62 .945 r .42836 .37001 .47686 .37663 .44602 .33838 .39713 .33429 Metetra .416 .635 .945 r .39713 .33429 .44602 .33838 .41415 .29882 .36485 .29706 Metetra .483 .727 .978 r .76956 .44706 .81234 .44682 .79258 .43037 .74929 .42902 Metetra .419 .725 .988 r .74929 .42902 .79258 .43037 .77231 .41628 .72849 .41309 Metetra .277 .701 .969 r .72849 .41309 .77231 .41628 .75156 .40661 .70718 .40165 Metetra 0 .552 .767 r .70718 .40165 .75156 .40661 .7303 .40361 .68534 .39759 Metetra .332 0 0 r .68534 .39759 .7303 .40361 .70848 .40869 .6629 .40325 Metetra .441 0 0 r .6629 .40325 .70848 .40869 .68593 .42078 .6397 .41819 Metetra .422 0 0 r .6397 .41819 .68593 .42078 .6624 .43514 .6155 .43709 Metetra .267 0 0 r .6155 .43709 .6624 .43514 .63769 .44452 .5901 .45082 Metetra .168 .706 .789 r .5901 .45082 .63769 .44452 .6117 .44281 .56344 .45164 Metetra .501 .82 1 r .56344 .45164 .6117 .44281 .58448 .42803 .5356 .43742 Metetra .566 .756 .967 r .5356 .43742 .58448 .42803 .55613 .40209 .50672 .41093 Metetra .57 .713 .939 r .50672 .41093 .55613 .40209 .52676 .36853 .47686 .37663 Metetra .562 .694 .929 r .47686 .37663 .52676 .36853 .49637 .33075 .44602 .33838 Metetra .552 .69 .931 r .44602 .33838 .49637 .33075 .46494 .29123 .41415 .29882 Metetra .568 .782 .979 r .81234 .44682 .85567 .4398 .83642 .42351 .79258 .43037 Metetra .541 .801 .993 r .79258 .43037 .83642 .42351 .81668 .40941 .77231 .41628 Metetra .477 .824 .998 r .77231 .41628 .81668 .40941 .79647 .3992 .75156 .40661 Metetra .331 .803 .917 r .75156 .40661 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