(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 223159, 6472]*) (*NotebookOutlinePosition[ 223850, 6496]*) (* CellTagsIndexPosition[ 223806, 6492]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Lab 2B - Parametrizing Surfaces", FontSize->24, FontWeight->"Bold", FontVariations->{"Underline"->True}], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: rogness@math.umn.edu" }], "Text", CellFrame->True, TextAlignment->Center, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontSize->16]], "Section"], Cell[TextData[{ "As in last week's lab, there is no calculus in this notebook. Last week \ you learned how to graph curves using ", StyleBox["ParametricPlot", FontWeight->"Bold"], " and ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ". You also had the chance to find the parametrizations for various \ curves. This week we'll do the same thing with surfaces.\n\nWe haven't \ rigorously defined surfaces yet, but you probably have an intuitive idea of \ what a surface is. It's like a piece of paper, or a sheet of rubber, but it \ doesn't have to be flat; it can be bent, curved, or even have holes in it. \ Sometimes a surface encloses a solid region in space. If so, we call it a ", StyleBox["closed", FontSlant->"Italic"], " surface. 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.33673 L .39994 .33928 L p F P 0 g s .242 .165 .566 r .78187 .61347 m .82263 .5588 L .83568 .61575 L p F P 0 g s .242 .165 .566 r .83568 .61575 m .79275 .67083 L .78187 .61347 L p F P 0 g s .929 .708 .583 r .2875 .69422 m .23134 .64467 L .24348 .58748 L p F P 0 g s .944 .653 .458 r .23134 .64467 m .2875 .69422 L .28895 .74852 L p F P 0 g s .92 .844 .753 r .26843 .53193 m .24499 .47661 L .28561 .43149 L p F P 0 g s .966 .843 .67 r .24499 .47661 m .26843 .53193 L .24348 .58748 L p F P 0 g s .537 .691 .938 r .58335 .32983 m .57009 .31792 L .61537 .30182 L p F P 0 g s .537 .691 .938 r .61537 .30182 m .64112 .32562 L .58335 .32983 L p F P 0 g s .431 .587 .906 r .65346 .35339 m .64112 .32562 L .69532 .33388 L p F P 0 g s .431 .587 .906 r .69532 .33388 m .71214 .37412 L .65346 .35339 L p F P 0 g s .264 .353 .767 r .75696 .45639 m .76342 .40564 L .80466 .44705 L p F P 0 g s .264 .353 .767 r .80466 .44705 m .7958 .50496 L .75696 .45639 L p F P 0 g s .648 .257 .289 r .45775 .84701 m .54706 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.45064 .29241 L p F P 0 g s .702 .844 .951 r .41912 .31308 m .45064 .29241 L .48759 .31326 L p F P 0 g s .783 .914 .928 r .41912 .31308 m .39994 .33928 L .34257 .35373 L p F P 0 g s .56 .77 .977 r .55164 .30959 m .53007 .30583 L .53563 .27735 L p F P 0 g s .56 .77 .977 r .53563 .27735 m .57881 .28499 L .55164 .30959 L p F P 0 g s .379 .016 .217 r .6346 .83965 m .7133 .8145 L .69244 .85242 L p F P 0 g s .379 .016 .217 r .69244 .85242 m .6226 .87425 L .6346 .83965 L p F P 0 g s .601 .809 .98 r .53007 .30583 m .50787 .3071 L .49108 .27994 L p F P 0 g s .653 .836 .971 r .45064 .29241 m .49108 .27994 L .50787 .3071 L p F P 0 g s .601 .809 .98 r .49108 .27994 m .53563 .27735 L .53007 .30583 L p F P 0 g s .966 .843 .67 r .24348 .58748 m .2168 .52855 L .24499 .47661 L p F P 0 g s .997 .811 .532 r .2168 .52855 m .24348 .58748 L .23134 .64467 L p F P 0 g s .087 0 .4 r .79275 .67083 m .83568 .61575 L .8336 .67327 L p F P 0 g s .087 0 .4 r .8336 .67327 m .79101 .72608 L .79275 .67083 L p F P 0 g s .254 .465 .872 r .71214 .37412 m .69532 .33388 L .74328 .35495 L p F P 0 g s .254 .465 .872 r .74328 .35495 m .76342 .40564 L .71214 .37412 L p F P 0 g s .081 .169 .656 r .7958 .50496 m .80466 .44705 L .83338 .49691 L p F P 0 g s .081 .169 .656 r .83338 .49691 m .82263 .5588 L .7958 .50496 L p F P 0 g s .877 .917 .851 r .28561 .43149 m .29211 .38004 L .34257 .35373 L p F P 0 g s .931 .954 .775 r .29211 .38004 m .28561 .43149 L .24499 .47661 L p F P 0 g s .37 .621 .949 r .64112 .32562 m .61537 .30182 L .65836 .29896 L p F P 0 g s .37 .621 .949 r .65836 .29896 m .69532 .33388 L .64112 .32562 L p F P 0 g s .854 .498 .378 r .37368 .82884 m .30196 .79711 L .28895 .74852 L p F P 0 g s .805 .35 .156 r .30196 .79711 m .37368 .82884 L .3901 .86485 L p F P 0 g s .736 .318 .257 r .46491 .88065 m .3901 .86485 L .37368 .82884 L p F P 0 g s .944 .653 .458 r .28895 .74852 m .23319 .70099 L .23134 .64467 L p F P 0 g s .924 .542 .255 r .23319 .70099 m .28895 .74852 L .30196 .79711 L p F P 0 g s .699 .932 .965 r .45064 .29241 m .41912 .31308 L .3696 .31553 L p F P 0 g s .783 .914 .928 r .34257 .35373 m .3696 .31553 L .41912 .31308 L p F P 0 g s .813 .986 .885 r .3696 .31553 m .34257 .35373 L .29211 .38004 L p F P 0 g s .091 0 .057 r .7133 .8145 m .776 .77661 L .74758 .81968 L p F P 0 g s .091 0 .057 r .74758 .81968 m .69244 .85242 L .7133 .8145 L p F P 0 g s .331 .672 .976 r .61537 .30182 m .57881 .28499 L .60481 .27402 L p F P 0 g s .331 .672 .976 r .60481 .27402 m .65836 .29896 L .61537 .30182 L p F P 0 g s 0 .253 .762 r .76342 .40564 m .74328 .35495 L .78225 .38866 L p F P 0 g s 0 .253 .762 r .78225 .38866 m .80466 .44705 L .76342 .40564 L p F P 0 g s .997 .811 .532 r .23134 .64467 m .20301 .58521 L .2168 .52855 L p F P 0 g s .974 .707 .297 r .20301 .58521 m .23134 .64467 L .23319 .70099 L p F P 0 g s 0 0 .452 r .82263 .5588 m .83338 .49691 L .84742 .55321 L p F P 0 g s 0 0 .452 r .84742 .55321 m .83568 .61575 L .82263 .5588 L p F P 0 g s .533 .031 .009 r .46491 .88065 m .54458 .88384 L .54095 .9041 L p F P 0 g s .399 0 0 r .54458 .88384 m .6226 .87425 L .60499 .89635 L p F P 0 g s .399 0 0 r .60499 .89635 m .54095 .9041 L .54458 .88384 L p F P 0 g s .931 .954 .775 r .24499 .47661 m .25129 .41762 L .29211 .38004 L p F P 0 g s .947 .941 .612 r .25129 .41762 m .24499 .47661 L .2168 .52855 L p F P 0 g s .613 .08 0 r .3901 .86485 m .46491 .88065 L .47547 .90152 L p F P 0 g s .533 .031 .009 r .54095 .9041 m .47547 .90152 L .46491 .88065 L p F P 0 g s 0 0 .122 r .79101 .72608 m .8336 .67327 L .81563 .7285 L p F P 0 g s 0 0 .122 r .81563 .7285 m .776 .77661 L .79101 .72608 L p F P 0 g s .567 .899 .982 r .49108 .27994 m .45064 .29241 L .41538 .28504 L p F P 0 g s .699 .932 .965 r .3696 .31553 m .41538 .28504 L .45064 .29241 L p F P 0 g s .068 .45 .881 r .69532 .33388 m .65836 .29896 L .69682 .31049 L p F P 0 g s .068 .45 .881 r .69682 .31049 m .74328 .35495 L .69532 .33388 L p F P 0 g s .345 .742 .984 r .57881 .28499 m .53563 .27735 L .54096 .26263 L p F P 0 g s .345 .742 .984 r .54096 .26263 m .60481 .27402 L .57881 .28499 L p F P 0 g s .176 0 0 r .6226 .87425 m .69244 .85242 L .66194 .87875 L p F P 0 g s .176 0 0 r .66194 .87875 m .60499 .89635 L .6226 .87425 L p F P 0 g s .43 .825 .984 r .53563 .27735 m .49108 .27994 L .47495 .2665 L p F P 0 g s .567 .899 .982 r .41538 .28504 m .47495 .2665 L .49108 .27994 L p F P 0 g s .924 .542 .255 r .30196 .79711 m .24977 .75374 L .23319 .70099 L p F P 0 g s .796 .31 0 r .24977 .75374 m .30196 .79711 L .3267 .83737 L p F P 0 g s .805 .35 .156 r .3901 .86485 m .3267 .83737 L .30196 .79711 L p F P 0 g s .43 .825 .984 r .47495 .2665 m .54096 .26263 L .53563 .27735 L p F P 0 g s .813 .986 .885 r .29211 .38004 m .32552 .33165 L .3696 .31553 L p F P 0 g s .761 .975 .707 r .32552 .33165 m .29211 .38004 L .25129 .41762 L p F P 0 g s .62 .054 0 r .3267 .83737 m .3901 .86485 L .41422 .88876 L p F P 0 g s .613 .08 0 r .47547 .90152 m .41422 .88876 L .3901 .86485 L p F P 0 g s .599 .951 .854 r .41538 .28504 m .3696 .31553 L .32552 .33165 L p F P 0 g s 0 0 .493 r .80466 .44705 m .78225 .38866 L .80961 .43409 L p F P 0 g s 0 0 .493 r .80961 .43409 m .83338 .49691 L .80466 .44705 L p F P 0 g s 0 0 .105 r .83568 .61575 m .84742 .55321 L .84517 .61333 L p F P 0 g s 0 0 .105 r .84517 .61333 m .8336 .67327 L .83568 .61575 L p F P 0 g s .947 .941 .612 r .2168 .52855 m .22272 .4653 L .25129 .41762 L p F P 0 g s .843 .793 .296 r .22272 .4653 m .2168 .52855 L .20301 .58521 L p F P 0 g s .974 .707 .297 r .23319 .70099 m .20513 .64399 L .20301 .58521 L p F P 0 g s .81 .461 0 r .20513 .64399 m .23319 .70099 L .24977 .75374 L p F P 0 g s 0 0 0 r .69244 .85242 m .74758 .81968 L .70632 .85256 L p F P 0 g s 0 0 0 r .70632 .85256 m .66194 .87875 L .69244 .85242 L p F P 0 g s 0 .435 .801 r .65836 .29896 m .60481 .27402 L .62825 .27845 L p F P 0 g s 0 .435 .801 r .62825 .27845 m .69682 .31049 L .65836 .29896 L p F P 0 g s 0 0 0 r .776 .77661 m .81563 .7285 L .7818 .77839 L p F P 0 g s 0 0 0 r .7818 .77839 m .74758 .81968 L .776 .77661 L p F P 0 g s 0 .091 .586 r .74328 .35495 m .69682 .31049 L .72836 .33699 L p F P 0 g s 0 .091 .586 r .72836 .33699 m .78225 .38866 L .74328 .35495 L p F P 0 g s .27 .771 .741 r .47495 .2665 m .41538 .28504 L .38369 .29264 L p F P 0 g s .599 .951 .854 r .32552 .33165 m .38369 .29264 L .41538 .28504 L p F P 0 g s .2 0 0 r .47547 .90152 m .54095 .9041 L .53623 .90925 L p F P 0 g s .052 0 0 r .54095 .9041 m .60499 .89635 L .58214 .9037 L p F P 0 g s .052 0 0 r .58214 .9037 m .53623 .90925 L .54095 .9041 L p F P 0 g s .243 0 0 r .41422 .88876 m .47547 .90152 L .48921 .9074 L p F P 0 g s .2 0 0 r .53623 .90925 m .48921 .9074 L .47547 .90152 L p F P 0 g s .761 .975 .707 r .25129 .41762 m .2895 .36163 L .32552 .33165 L p F P 0 g s 0 0 0 r .2895 .36163 m .25129 .41762 L .22272 .4653 L p F P 0 g s .796 .31 0 r .3267 .83737 m .28113 .80001 L .24977 .75374 L p F P 0 g s .62 .054 0 r .41422 .88876 m .36279 .86669 L .3267 .83737 L p F P 0 g s 0 .086 .553 r .28113 .80001 m .3267 .83737 L .36279 .86669 L p F P 0 g s .804 .495 0 r .83338 .49691 m .80961 .43409 L .82305 .48952 L p F P 0 g s .804 .495 0 r .82305 .48952 m .84742 .55321 L .83338 .49691 L p F P 0 g s .133 0 0 r .60481 .27402 m .54096 .26263 L .54577 .26374 L p F P 0 g s .133 0 0 r .54577 .26374 m .62825 .27845 L .60481 .27402 L p F P 0 g s .843 .793 .296 r .20301 .58521 m .20869 .52119 L .22272 .4653 L p F P 0 g s 0 0 .134 r .20869 .52119 m .20301 .58521 L .20513 .64399 L p F P 0 g s 0 0 0 r .60499 .89635 m .66194 .87875 L .62262 .8912 L p F P 0 g s 0 0 0 r .62262 .8912 m .58214 .9037 L .60499 .89635 L p F P 0 g s .803 .807 .32 r .8336 .67327 m .84517 .61333 L .82587 .67413 L p F P 0 g s .803 .807 .32 r .82587 .67413 m .81563 .7285 L .8336 .67327 L p F P 0 g s 0 0 0 r .38369 .29264 m .32552 .33165 L .2895 .36163 L p F P 0 g s .81 .461 0 r .24977 .75374 m .22393 .70192 L .20513 .64399 L p F P 0 g s 0 0 .5 r .22393 .70192 m .24977 .75374 L .28113 .80001 L p F P 0 g s .27 .771 .741 r .38369 .29264 m .46037 .26874 L .47495 .2665 L p F P 0 g s .003 0 0 r .54096 .26263 m .47495 .2665 L .46037 .26874 L p F P 0 g s .003 0 0 r .46037 .26874 m .54577 .26374 L .54096 .26263 L p F P 0 g s .243 0 0 r .48921 .9074 m .44545 .8983 L .41422 .88876 L p F P 0 g s 0 .398 .747 r .36279 .86669 m .41422 .88876 L .44545 .8983 L p F P 0 g s .679 .952 .699 r .74758 .81968 m .7818 .77839 L .73312 .81984 L p F P 0 g s .679 .952 .699 r .73312 .81984 m .70632 .85256 L .74758 .81968 L p F P 0 g s .803 .328 0 r .78225 .38866 m .72836 .33699 L .75065 .37815 L p F P 0 g s .803 .328 0 r .75065 .37815 m .80961 .43409 L .78225 .38866 L p F P 0 g s .512 .912 .824 r .66194 .87875 m .70632 .85256 L .65363 .87276 L p F P 0 g s .512 .912 .824 r .65363 .87276 m .62262 .8912 L .66194 .87875 L p F P 0 g s .596 .023 0 r .69682 .31049 m .62825 .27845 L .64761 .29947 L p F P 0 g s .596 .023 0 r .64761 .29947 m .72836 .33699 L .69682 .31049 L p F P 0 g s 0 0 0 r .22272 .4653 m .2641 .40487 L .2895 .36163 L p F P 0 g s 0 0 .146 r .2641 .40487 m .22272 .4653 L .20869 .52119 L p F P 0 g s 0 0 .134 r .20513 .64399 m .21084 .58265 L .20869 .52119 L p F P 0 g s 0 0 .489 r .21084 .58265 m .20513 .64399 L .22393 .70192 L p F P 0 g s .978 .747 .333 r .84742 .55321 m .82305 .48952 L .8209 .55232 L p F P 0 g s .978 .747 .333 r .8209 .55232 m .84517 .61333 L .84742 .55321 L p F P 0 g s .91 .962 .641 r .81563 .7285 m .82587 .67413 L .78976 .73208 L p F P 0 g s .91 .962 .641 r .78976 .73208 m .7818 .77839 L .81563 .7285 L p F P 0 g s 0 .086 .553 r .36279 .86669 m .3265 .83693 L .28113 .80001 L p F P 0 g s .028 .465 .877 r .3265 .83693 m .36279 .86669 L .40916 .88268 L p F P 0 g s 0 .398 .747 r .44545 .8983 m .40916 .88268 L .36279 .86669 L p F P 0 g s .164 0 0 r .46037 .26874 m .38369 .29264 L .35758 .31611 L p F P 0 g s 0 0 0 r .2895 .36163 m .35758 .31611 L .38369 .29264 L p F P 0 g s 0 0 .5 r .28113 .80001 m .25929 .7557 L .22393 .70192 L p F P 0 g s 0 .279 .782 r .25929 .7557 m .28113 .80001 L .3265 .83693 L p F P 0 g s .311 .765 .954 r .48921 .9074 m .53623 .90925 L .53057 .89751 L p F P 0 g s .395 .826 .959 r .53623 .90925 m .58214 .9037 L .55481 .89456 L p F P 0 g s .395 .826 .959 r .55481 .89456 m .53057 .89751 L .53623 .90925 L p F P 0 g s .08 0 .06 r .35758 .31611 m .2895 .36163 L .2641 .40487 L p F P 0 g s .281 .718 .965 r .44545 .8983 m .48921 .9074 L .50571 .89652 L p F P 0 g s .311 .765 .954 r .53057 .89751 m .50571 .89652 L .48921 .9074 L p F P 0 g s .559 .012 0 r .62825 .27845 m .54577 .26374 L .54975 .28217 L p F P 0 g s .559 .012 0 r .54975 .28217 m .64761 .29947 L .62825 .27845 L p F P 0 g s .514 .886 .972 r .58214 .9037 m .62262 .8912 L .57596 .88794 L p F P 0 g s .514 .886 .972 r .57596 .88794 m .55481 .89456 L .58214 .9037 L p F P 0 g s .741 .984 .904 r .70632 .85256 m .73312 .81984 L .67168 .84999 L p F P 0 g s .741 .984 .904 r .67168 .84999 m .65363 .87276 L .70632 .85256 L p F P 0 g s .305 .688 .977 r .40916 .88268 m .44545 .8983 L .4827 .89169 L p F P 0 g s .281 .718 .965 r .50571 .89652 m .4827 .89169 L .44545 .8983 L p F P 0 g s .418 0 0 r .54577 .26374 m .46037 .26874 L .44829 .28806 L p F P 0 g s .164 0 0 r .35758 .31611 m .44829 .28806 L .46037 .26874 L p F P 0 g s .936 .569 .282 r .80961 .43409 m .75065 .37815 L .76165 .43264 L p F P 0 g s .936 .569 .282 r .76165 .43264 m .82305 .48952 L .80961 .43409 L p F P 0 g s 0 0 .489 r .22393 .70192 m .22997 .64639 L .21084 .58265 L p F P 0 g s .067 .204 .698 r .22997 .64639 m .22393 .70192 L .25929 .7557 L p F P 0 g s .418 0 0 r .44829 .28806 m .54975 .28217 L .54577 .26374 L p F P 0 g s .996 .848 .575 r .84517 .61333 m .8209 .55232 L .80243 .61903 L p F P 0 g s .996 .848 .575 r .80243 .61903 m .82587 .67413 L .84517 .61333 L p F P 0 g s 0 0 .146 r .20869 .52119 m .25155 .45982 L .2641 .40487 L p F P 0 g s .087 0 .439 r .25155 .45982 m .20869 .52119 L .21084 .58265 L p F P 0 g s .893 .979 .814 r .7818 .77839 m .78976 .73208 L .7383 .78355 L p F P 0 g s .893 .979 .814 r .7383 .78355 m .73312 .81984 L .7818 .77839 L p F P 0 g s .629 .92 .977 r .62262 .8912 m .65363 .87276 L .59184 .87828 L p F P 0 g s .629 .92 .977 r .59184 .87828 m .57596 .88794 L .62262 .8912 L p F P 0 g s .802 .348 .161 r .72836 .33699 m .64761 .29947 L .66135 .33734 L p F P 0 g s .802 .348 .161 r .66135 .33734 m .75065 .37815 L .72836 .33699 L p F P 0 g s 0 .279 .782 r .3265 .83693 m .30995 .802 L .25929 .7557 L p F P 0 g s .235 .511 .908 r .30995 .802 m .3265 .83693 L .38415 .86184 L p F P 0 g s .028 .465 .877 r .40916 .88268 m .38415 .86184 L .3265 .83693 L p F P 0 g s .363 .667 .973 r .38415 .86184 m .40916 .88268 L .46389 .88346 L p F P 0 g s .305 .688 .977 r .4827 .89169 m .46389 .88346 L .40916 .88268 L p F P 0 g s .08 0 .06 r .2641 .40487 m .33904 .35547 L .35758 .31611 L p F P 0 g s .287 .041 .356 r .33904 .35547 m .2641 .40487 L .25155 .45982 L p F P 0 g s .403 .034 .219 r .44829 .28806 m .35758 .31611 L .33904 .35547 L p F P 0 g s .712 .915 .964 r .65363 .87276 m .67168 .84999 L .60067 .86652 L p F P 0 g s .712 .915 .964 r .60067 .86652 m .59184 .87828 L .65363 .87276 L p F P 0 g s .067 .204 .698 r .25929 .7557 m .26569 .70861 L .22997 .64639 L p F P 0 g s .256 .398 .814 r .26569 .70861 m .25929 .7557 L .30995 .802 L p F P 0 g s .957 .876 .715 r .82587 .67413 m .80243 .61903 L .76811 .68554 L p F P 0 g s .957 .876 .715 r .76811 .68554 m .78976 .73208 L .82587 .67413 L p F P 0 g s .83 .941 .898 r .73312 .81984 m .7383 .78355 L .67423 .82509 L p F P 0 g s .83 .941 .898 r .67423 .82509 m .67168 .84999 L .73312 .81984 L p F P 0 g s .947 .678 .494 r .82305 .48952 m .76165 .43264 L .75988 .49786 L p F P 0 g s .947 .678 .494 r .75988 .49786 m .8209 .55232 L .82305 .48952 L p F P 0 g s .087 0 .439 r .21084 .58265 m .25348 .52388 L .25155 .45982 L p F P 0 g s .25 .203 .608 r .25348 .52388 m .21084 .58265 L .22997 .64639 L p F P 0 g s .569 .802 .987 r .52423 .86795 m .50571 .89652 L .53057 .89751 L closepath p F P 0 g s .589 .816 .986 r .52423 .86795 m .53057 .89751 L .55481 .89456 L closepath p F P 0 g s .557 .786 .985 r .52423 .86795 m .4827 .89169 L .50571 .89652 L closepath p F P 0 g s .706 .289 .253 r .64761 .29947 m .54975 .28217 L .55257 .31846 L p F P 0 g s .706 .289 .253 r .55257 .31846 m .66135 .33734 L .64761 .29947 L p F P 0 g s .612 .825 .982 r .52423 .86795 m .55481 .89456 L .57596 .88794 L closepath p F P 0 g s .235 .511 .908 r .38415 .86184 m .37351 .83771 L .30995 .802 L p F P 0 g s .432 .652 .952 r .37351 .83771 m .38415 .86184 L .4513 .87262 L p F P 0 g s .363 .667 .973 r .46389 .88346 m .4513 .87262 L .38415 .86184 L p F P 0 g s .554 .77 .979 r .52423 .86795 m .46389 .88346 L .4827 .89169 L closepath p F P 0 g s .636 .829 .975 r .52423 .86795 m .57596 .88794 L .59184 .87828 L closepath p F P 0 g s .849 .504 .401 r .75065 .37815 m .66135 .33734 L .66814 .39103 L p F P 0 g s .849 .504 .401 r .66814 .39103 m .76165 .43264 L .75065 .37815 L p F P 0 g s .582 .192 .268 r .54975 .28217 m .44829 .28806 L .43968 .32489 L p F P 0 g s .403 .034 .219 r .33904 .35547 m .43968 .32489 L .44829 .28806 L p F P 0 g s .256 .398 .814 r .30995 .802 m .31638 .76531 L .26569 .70861 L p F P 0 g s .391 .537 .881 r .31638 .76531 m .30995 .802 L .37351 .83771 L p F P 0 g s .558 .756 .97 r .52423 .86795 m .4513 .87262 L .46389 .88346 L closepath p F P 0 g s .582 .192 .268 r .43968 .32489 m .55257 .31846 L .54975 .28217 L p F P 0 g s .901 .876 .803 r .78976 .73208 m .76811 .68554 L .71969 .7475 L p F P 0 g s .901 .876 .803 r .71969 .7475 m .7383 .78355 L .78976 .73208 L p F P 0 g s .754 .883 .938 r .67168 .84999 m .67423 .82509 L .60134 .85387 L p F P 0 g s .754 .883 .938 r .60134 .85387 m .60067 .86652 L .67168 .84999 L p F P 0 g s .656 .826 .965 r .52423 .86795 m .59184 .87828 L .60067 .86652 L closepath p F P 0 g s .25 .203 .608 r .22997 .64639 m .27054 .59343 L .25348 .52388 L p F P 0 g s .358 .35 .714 r .27054 .59343 m .22997 .64639 L .26569 .70861 L p F P 0 g s .287 .041 .356 r .25155 .45982 m .32985 .40952 L .33904 .35547 L p F P 0 g s .399 .222 .522 r .32985 .40952 m .25155 .45982 L .25348 .52388 L p F P 0 g s .925 .731 .622 r .8209 .55232 m .75988 .49786 L .74476 .57004 L p F P 0 g s .925 .731 .622 r .74476 .57004 m .80243 .61903 L .8209 .55232 L p F P 0 g s .432 .652 .952 r .4513 .87262 m .44641 .86025 L .37351 .83771 L p F P 0 g s .57 .745 .959 r .52423 .86795 m .44641 .86025 L .4513 .87262 L closepath p F P 0 g s .508 .229 .425 r .43968 .32489 m .33904 .35547 L .32985 .40952 L p F P 0 g s .391 .537 .881 r .37351 .83771 m .37911 .81272 L .31638 .76531 L p F P 0 g s .497 .645 .923 r .37911 .81272 m .37351 .83771 L .44641 .86025 L p F P 0 g s .837 .863 .863 r .7383 .78355 m .71969 .7475 L .66019 .80072 L p F P 0 g s .837 .863 .863 r .66019 .80072 m .67423 .82509 L .7383 .78355 L p F P 0 g s .67 .817 .954 r .52423 .86795 m .60067 .86652 L .60134 .85387 L closepath p F P 0 g s .358 .35 .714 r .26569 .70861 m .30222 .66405 L .27054 .59343 L p F P 0 g s .437 .463 .789 r .30222 .66405 m .26569 .70861 L .31638 .76531 L p F P 0 g s .497 .645 .923 r .44641 .86025 m .44993 .84766 L .37911 .81272 L p F P 0 g s .585 .737 .948 r .52423 .86795 m .44993 .84766 L .44641 .86025 L closepath p F P 0 g s .748 .427 .44 r .66135 .33734 m .55257 .31846 L .55395 .37175 L p F P 0 g s .748 .427 .44 r .55395 .37175 m .66814 .39103 L .66135 .33734 L p F P 0 g s .765 .839 .908 r .67423 .82509 m .66019 .80072 L .59355 .84169 L p F P 0 g s .765 .839 .908 r .59355 .84169 m .60134 .85387 L .67423 .82509 L p F P 0 g s .679 .789 .933 r .52423 .86795 m .59355 .84169 L .57797 .83138 L closepath p F P 0 g s .674 .774 .925 r .52423 .86795 m .57797 .83138 L .55622 .82414 L closepath p F P 0 g s .665 .76 .921 r .52423 .86795 m .55622 .82414 L .53075 .82087 L closepath p F P 0 g s .652 .749 .92 r .52423 .86795 m .53075 .82087 L .50452 .82197 L closepath p F P 0 g s .637 .74 .923 r .52423 .86795 m .50452 .82197 L .48059 .82732 L closepath p F P 0 g s .62 .735 .929 r .52423 .86795 m .48059 .82732 L .46171 .83624 L closepath p F P 0 g s .602 .734 .937 r .52423 .86795 m .46171 .83624 L .44993 .84766 L closepath p F P 0 g s .678 .804 .943 r .52423 .86795 m .60134 .85387 L .59355 .84169 L closepath p F P 0 g s .85 .586 .538 r .76165 .43264 m .66814 .39103 L .66702 .45803 L p F P 0 g s .85 .586 .538 r .66702 .45803 m .75988 .49786 L .76165 .43264 L p F P 0 g s .893 .761 .709 r .80243 .61903 m .74476 .57004 L .71684 .64449 L p F P 0 g s .893 .761 .709 r .71684 .64449 m .76811 .68554 L .80243 .61903 L p F P 0 g s .399 .222 .522 r .25348 .52388 m .33125 .47573 L .32985 .40952 L p F P 0 g s .467 .342 .626 r .33125 .47573 m .25348 .52388 L .27054 .59343 L p F P 0 g s .437 .463 .789 r .31638 .76531 m .34675 .731 L .30222 .66405 L p F P 0 g s .499 .557 .846 r .34675 .731 m .31638 .76531 L .37911 .81272 L p F P 0 g s .643 .342 .446 r .55257 .31846 m .43968 .32489 L .43538 .37832 L p F P 0 g s .508 .229 .425 r .32985 .40952 m .43538 .37832 L .43968 .32489 L p F P 0 g s .553 .645 .895 r .40117 .78968 m .37911 .81272 L .44993 .84766 L p F P 0 g s .553 .645 .895 r .44993 .84766 m .46171 .83624 L .40117 .78968 L p F P 0 g s .499 .557 .846 r .37911 .81272 m .40117 .78968 L .34675 .731 L p F P 0 g s .643 .342 .446 r .43538 .37832 m .55395 .37175 L .55257 .31846 L p F P 0 g s .757 .794 .883 r .57797 .83138 m .59355 .84169 L .66019 .80072 L p F P 0 g s .856 .779 .775 r .76811 .68554 m .71684 .64449 L .6778 .71606 L p F P 0 g s .856 .779 .775 r .6778 .71606 m .71969 .7475 L .76811 .68554 L p F P 0 g s .561 .346 .546 r .43538 .37832 m .32985 .40952 L .33125 .47573 L p F P 0 g s .601 .653 .874 r .43781 .77143 m .40117 .78968 L .46171 .83624 L p F P 0 g s .601 .653 .874 r .46171 .83624 m .48059 .82732 L .43781 .77143 L p F P 0 g s .812 .789 .831 r .63038 .77975 m .66019 .80072 L .71969 .7475 L p F P 0 g s .757 .794 .883 r .66019 .80072 m .63038 .77975 L .57797 .83138 L p F P 0 g s .812 .789 .831 r .71969 .7475 m .6778 .71606 L .63038 .77975 L p F P 0 g s .467 .342 .626 r .27054 .59343 m .34372 .5502 L .33125 .47573 L p F P 0 g s .513 .433 .702 r .34372 .5502 m .27054 .59343 L .30222 .66405 L p F P 0 g s .737 .753 .866 r .55622 .82414 m .57797 .83138 L .63038 .77975 L p F P 0 g s .84 .636 .628 r .75988 .49786 m .66702 .45803 L .65766 .53435 L p F P 0 g s .84 .636 .628 r .65766 .53435 m .74476 .57004 L .75988 .49786 L p F P 0 g s .643 .669 .86 r .48506 .76039 m .43781 .77143 L .48059 .82732 L p F P 0 g s .643 .669 .86 r .48059 .82732 m .50452 .82197 L .48506 .76039 L p F P 0 g s .76 .507 .551 r .66814 .39103 m .55395 .37175 L .55368 .43959 L p F P 0 g s .76 .507 .551 r .55368 .43959 m .66702 .45803 L .66814 .39103 L p F P 0 g s .737 .753 .866 r .63038 .77975 m .58774 .76486 L .55622 .82414 L p F P 0 g s .71 .719 .856 r .53075 .82087 m .55622 .82414 L .58774 .76486 L p F P 0 g s .577 .58 .819 r .39891 .70347 m .34675 .731 L .40117 .78968 L p F P 0 g s .577 .58 .819 r .40117 .78968 m .43781 .77143 L .39891 .70347 L p F P 0 g s .679 .69 .854 r .53722 .75809 m .48506 .76039 L .50452 .82197 L p F P 0 g s .679 .69 .854 r .50452 .82197 m .53075 .82087 L .53722 .75809 L p F P 0 g s .71 .719 .856 r .58774 .76486 m .53722 .75809 L .53075 .82087 L p F P 0 g s .513 .433 .702 r .30222 .66405 m .36676 .62795 L .34372 .5502 L p F P 0 g s .548 .509 .764 r .36676 .62795 m .30222 .66405 L .34675 .731 L p F P 0 g s .548 .509 .764 r .34675 .731 m .39891 .70347 L .36676 .62795 L p F P 0 g s .67 .432 .553 r .55395 .37175 m .43538 .37832 L .43601 .44587 L p F P 0 g s .561 .346 .546 r .33125 .47573 m .43601 .44587 L .43538 .37832 L p F P 0 g s .592 .428 .628 r .43601 .44587 m .33125 .47573 L .34372 .5502 L p F P 0 g s .775 .729 .81 r .58774 .76486 m .63038 .77975 L .6778 .71606 L p F P 0 g s .823 .673 .697 r .74476 .57004 m .65766 .53435 L .64044 .61477 L p F P 0 g s .823 .673 .697 r .64044 .61477 m .71684 .64449 L .74476 .57004 L p F P 0 g s .67 .432 .553 r .43601 .44587 m .55368 .43959 L .55395 .37175 L p F P 0 g s .638 .607 .803 r .46724 .68667 m .39891 .70347 L .43781 .77143 L p F P 0 g s .638 .607 .803 r .43781 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For example, what if we wanted to plot the graph of \ x = ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " ? You should know by looking at this equation that this is a paraboloid, \ just like our picture above, except this one opens in the direction of the \ positive x-axis. Suppose we tried to plot this with ", StyleBox["Plot3D", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(Plot3D[y^2\ + \ z^2, \ {y, \(-1\), 1}, {z, \(-1\), 1}]\)], "Input"], Cell[TextData[{ "We get exactly the same picture as before! The paraboloid opens upwards, \ which is incorrect. The reason is that ", StyleBox["Plot3D", FontWeight->"Bold"], " expects a function of two variables, and it interprets the values of the \ function as the height. Normally we give it a function of x and y, and since \ the z-axis represents height, this fits in to our view of the world. \n\nHow \ can we get a true graph of x = ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " from ", StyleBox["Mathematica", FontSlant->"Italic"], "? Using parametric equations is one possibility. Think back to last week \ when we did a \"trivial parametrization\" of the graph of y=g(x) by setting \ f(x) equal to (x, g(x)). Define f(y,z)=(", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], ", y, z):" }], "Text"], Cell[BoxData[ \(f[y_, z_] = {y^2\ + \ z^2, \ y, \ z}\)], "Input"], Cell[TextData[{ "Essentially we're saying that y and z are our parameters. Now we can plot \ the graph of f using ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(ParametricPlot3D[f[y, z], {y, \(-1\), 1}, {z, \(-1\), 1}, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"], Cell[TextData[{ "And you can see that this is a paraboloid opening in the direction of the \ positive x-axis, as desired.\n\nNote that we're using ", StyleBox["two", FontSlant->"Italic"], " parameters now, which is different than when we plotted curves. This \ makes sense, if you think about it. Curves are like a line, a \ one-dimensional object, so they require one parameter. Surfaces are \ two-dimensional creatures, and so they require two parameters." }], "Text"], Cell[TextData[{ "Note: if a ", StyleBox["curve", FontSlant->"Italic"], " is in the xy-plane, you can use ", StyleBox["ParametricPlot", FontWeight->"Bold"], ", or you can use ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " by letting z(t)=0. If you want to plot a surface, however, you ", StyleBox["must", FontSlant->"Italic"], " use ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ". The command ", StyleBox["ParametricPlot", FontWeight->"Bold"], " will not accept more than one parameter! (So you can't give it ranges \ for, say, y ", StyleBox["and", FontSlant->"Italic"], " z.)\n\nAlso, most of this lab uses ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ", so if you're using ", StyleBox["Mathematica", FontSlant->"Italic"], " version 4.x it would be a good idea to execute the command ", StyleBox["Off[ParametricPlot3D::ppcom]", FontWeight->"Bold"], " to get rid of those blue error messages about compiling functions. \ People using ", StyleBox["Mathematica", FontSlant->"Italic"], " 5.0 needn't worry about this." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["More Examples", FontSize->16]], "Section"], Cell[TextData[{ "Let's look at more examples of how to parametrize surfaces. As in last \ week's lab, circular regions will be very important, and there is one \ technique in particular that you should learn. Many of our surfaces will be \ relatively easy to describe using cylindrical coordinates (r,\[Theta],z), \ which have been covered in class. A parametrization has to be in ", StyleBox["rectangular", FontSlant->"Italic"], " coordinates (x,y,z), so we can't just describe the surface in cylindrical \ coordinates and say we're done. But if we've described the surface in \ cylindrical coordinates, we can use the following formulas to convert our \ description into rectangular coordinates:" }], "Text"], Cell["\<\ x = r Cos[\[Theta]] y = r Sin[\[Theta]] z = z\ \>", "Text", TextAlignment->Center, FontWeight->"Bold"], Cell[TextData[{ StyleBox["Example 1(a)", FontSize->14, FontWeight->"Bold"], StyleBox[". Disks of Radius R in the Plane z=h\n", FontSize->14], "\nIn cylindrical coordinates, such a disk is described by (r,\[Theta],h) \ where 0\[LessEqual]r\[LessEqual] R, 0\[LessEqual]\[Theta]\[LessEqual]2\[Pi], \ and h is some constant number. Converting to rectangular coordinates, we \ have: ", Cell[BoxData[ \(TraditionalForm\`f(r, \[Theta]) = \((r\ Cos[\[Theta]], \ r\ Sin[\[Theta]], \ h)\)\)]], ", where r and \[Theta] have the same bounds. (r and \[Theta] are our \ parameters now.) For example, if R=1 and h=0," }], "Text"], Cell[BoxData[{ \(f[r_, theta_] = {r*Cos[theta], \ r*Sin[theta], \ 0}\), "\[IndentingNewLine]", \(disk0 = ParametricPlot3D[f[r, theta], {r, 0, 1}, {theta, 0, 2 Pi}]\)}], "Input"], Cell["Or, with other values of h:", "Text"], Cell[BoxData[{ \(disk1 = ParametricPlot3D[{r*Cos[theta], r*Sin[theta], 1}, {r, 0, 1}, {theta, 0, 2 Pi}]\), "\[IndentingNewLine]", \(disk2 = ParametricPlot3D[{r*Cos[theta], r*Sin[theta], 2}, {r, 0, 1}, {theta, 0, 2 Pi}]\)}], "Input"], Cell["Or, all together:", "Text"], Cell[BoxData[ \(Show[disk0, \ disk1, \ disk2]\)], "Input"], Cell[TextData[{ StyleBox["FYI", FontWeight->"Bold"], ": Normally when we use one parameter we choose the letter t. When we use \ two parameters, the standard letters are s and t. So often we would write \ these parametrizations as (for example):\n\nf(s,t) = (s\[CenterDot]Cos[t], s\ \[CenterDot]Sin[t], 0),\t\t0\[LessEqual]s\[LessEqual]1, 0\[LessEqual]t\ \[LessEqual]2\[Pi].\n\nWhen we do a parametrization related to cylindrical \ coordinates, it might be more intuitive to use r (=radius) and t (=theta), \ but you're not required to do so. You should also be aware that some \ textbooks use u and v as parameters instead of s and t (or r and t)." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ StyleBox["Example 1(b)", FontSize->14, FontWeight->"Bold"], StyleBox[". An Annulus\n", FontSize->14], "\nWe won't actually use this much, but it's an interesting example to \ point out. Suppose we parametrize the disk of radius 4 in the plane z=1:" }], "Text"], Cell[BoxData[{ \(f[s_, t_] = {s*Cos[t], \ s*Sin[t], \ 1}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[s, t], {s, 0, 4}, {t, 0, 2 Pi}]\)}], "Input"], Cell["\<\ Now, instead of letting s range from 0 to 4, let's use values of s \ from 2 to 4 instead:\ \>", "Text"], Cell[BoxData[ \(ParametricPlot3D[f[s, t], {s, 2, 4}, {t, 0, 2 Pi}]\)], "Input"], Cell[TextData[{ "This type of surface is called an ", StyleBox["annulus", FontSlant->"Italic"], ". More informally, we might refer to this surface as a \"washer,\" as in \ the little thing you can buy at the local hardware store." }], "Text"], Cell[TextData[{ StyleBox["Example 2", FontSize->14, FontWeight->"Bold"], StyleBox[". A Filled in Ellipse.\n", FontSize->14], "\nLast week we parametrized an ellipse:" }], "Text"], Cell[BoxData[ \(ParametricPlot3D[{4 Cos[t], 3 Sin[t], 0}, {t, 0, 2 Pi}]\)], "Input"], Cell["\<\ We can fill in the ellipse by introducing a second parameter, s. \ Notice the similarity to the disk above!\ \>", "Text"], Cell[BoxData[{ \(f[s_, t_] = {4 s*Cos[t], \ 3 s*Sin[t], 0}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[s, t], {s, 0, 1}, {t, 0, 2 Pi}]\)}], "Input"], Cell[TextData[{ StyleBox["Example 3", FontSize->14, FontWeight->"Bold"], StyleBox[". Shifted Circles and Ellipses.\n", FontSize->14], "\nAs with our circles last week, we can shift a circle so that it's \ centered at another point by adding the appropriate numbers to the components \ of the parametrization. For example, here's a disk of radius 2 in the plane \ ", Cell[BoxData[ \(TraditionalForm\`z = 3\)]], " centered at (1.5,1):" }], "Text"], Cell[BoxData[{ \(f[s_, t_] = {1.5\ + \ s*Cos[t], 1 + \ \ s*Sin[t], \ 3}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[s, t], {s, 0, 2}, {t, 0, 2 Pi}]\)}], "Input"], Cell["\<\ It's also easy to graph a circle or ellipse in, say, the yz-plane \ instead of the xy-plane:\ \>", "Text"], Cell[BoxData[{ \(f[s_, t_] = {0, \ 4\ + \ 4 s*Cos[t], \ 2 + 2 s*Sin[t]}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[s, t], {s, 0, 1}, {t, 0, 2 Pi}, AxesLabel \[Rule] {"\", "\", "\"}]\)}], "Input"], Cell["\<\ Use the following command to rotate the previous picture and see \ which plane the ellipse is in.\ \>", "Text"], Cell[BoxData[ \(ShowLive[%]\)], "Input"], Cell[TextData[{ StyleBox["Example 4", FontSize->14, FontWeight->"Bold"], StyleBox[". A Piece of a Paraboloid\n", FontSize->14], "\nSuppose we want to work with the portion of the paraboloid z=", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " which is above the unit disk in the xy-plane. We could try the \"trivial\ \" parametrization, \n\nf(x,y) = (x, y, ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "), \t\twhere -1\[LessEqual]x\[LessEqual]1 and -", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"1", " ", "-", " ", FormBox[\(x\^2\), "TraditionalForm"]}]], TraditionalForm]]], " \[LessEqual] y \[LessEqual] ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"1", "-", FormBox[\(x\^2\), "TraditionalForm"]}]], TraditionalForm]]] }], "Text"], Cell[TextData[{ "The problem is that neither ", StyleBox["Plot3D", FontWeight->"Bold"], " or ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " accept bounds like that for y. Try it! Both of these commands will \ produce errors:" }], "Text"], Cell[BoxData[ \(Plot3D[ x^2 + y^2, {x, \(-1\), 1}, {y, \(-Sqrt[1 - x^2]\), Sqrt[1 - x^2]}]\)], "Input"], Cell[BoxData[ \(ParametricPlot3D[{x, y, x^2 + y^2}, {x, \(-1\), 1}, {y, \(-Sqrt[1 - x^2]\), Sqrt[1 - x^2]}]\)], "Input"], Cell[TextData[{ "We can solve this problem, though. It turns out to be very easy to \ parametrize the surface if we describe it in cylindrical coordinates first. \ Our x and y values are in the unit disk; this corresponds to 0\[LessEqual]r\ \[LessEqual]1 and 0\[LessEqual]\[Theta]\[LessEqual]2\[Pi]. All we need to do \ is describe z=", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "in terms of r and \[Theta]; but we know that x=rCos[\[Theta]] and y=rSin[\ \[Theta]]! On paper, square these and add them together; you should get z = \ ", Cell[BoxData[ \(TraditionalForm\`\(\(r\^2\)\(.\)\)\)]], " So our parametrization is:" }], "Text"], Cell[BoxData[{ \(f[r_, theta_] = {r\ Cos[theta], \ r\ Sin[theta], \ r^2}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[r, theta], {r, 0, 1}, {theta, 0, 2 Pi}]\)}], "Input"], Cell[TextData[{ "Scroll up and compare this graph to the very first graph in this lab, \ which is of the very same paraboloid. Notice how much nicer it looks when we \ use a circular domain and ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " instead of ", StyleBox["Plot3D", FontWeight->"Bold"], " and the rectangle -1\[LessEqual]x\[LessEqual]1, -1\[LessEqual]y\ \[LessEqual]1." }], "Text"], Cell[TextData[{ "FYI: Sometimes this graph is called \"the portion of the paraboloid z=", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "which is above the unit disk.\" Even more commonly we might call it \"the \ portion of the paraboloid z=", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " cut off by the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "= 1;\" it's really the same thing if you think about it. If we draw the \ cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "= 1 and only take the points of the paraboloid which are ", StyleBox["inside", FontSlant->"Italic"], " the cylinder, we get those points which, in cylindrical coordinates, have \ a radius between 0 and 1. Those are precisely the points we used while \ creating this graph!" }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ StyleBox["Example 5", FontSize->14, FontWeight->"Bold"], StyleBox[". A Cylinder\n", FontSize->14], "\nSo far nearly everything we've graphed has actually been a function of x \ and y -- for example, the circles were really just plots of z=0 (or z=1, z=2) \ where we happened to restrict our domain of x- and y- values to get a circle. \ Here's an example which is ", StyleBox["not", FontSlant->"Italic"], " a function of x and y! Let's graph the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "=9, 0\[LessEqual]z\[LessEqual]10. You learned in an earlier lab that you \ can graph this cylinder with the following command. If you don't remember \ for sure how ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " works, you should look back at Lab 1B." }], "Text"], Cell[BoxData[ \(ContourPlot3D[ x^2\ + \ y^2\ - \ 9, \ {x, \(-6\), 6}, {y, \(-6\), 6}, {z, 0, 10}]\)], "Input"], Cell[TextData[{ "The problem with this approach is that we can see the cylinder, but we \ can't really work with it because we don't have a parametrization. So let's \ find a way to plot it using ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ".\n\nBecause we're graphing a cylinder, it's very natural to describe it \ in cylindrical coordinates. With our earlier disks and other surfaces, r and \ \[Theta] varied while z stayed constant (or at least depended on r and \ \[Theta]). So r and \[Theta] were our two parameters. Now our situation is \ a little different. In the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "=9, r is constantly equal to three, \[Theta] ranges from 0 to 2\[Pi], and \ z ranges from 0 to 5. So now \[Theta] and z are our two parameters, and when \ we convert from cylindrical to rectangular coordinates we get:" }], "Text"], Cell[BoxData[{ \(f[theta_, z_] = {3 Cos[theta], 3 Sin[theta], z}\), "\[IndentingNewLine]", \(ParametricPlot3D[ f[theta, z], {theta, 0, 2 Pi}, {z, 0, 10}]\)}], "Input"], Cell[TextData[{ "Compare this to the plot from ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " and decide which one you think looks better!\n\n", StyleBox[" Example 6", FontSize->14, FontWeight->"Bold"], StyleBox[". A Helicoid.\n", FontSize->14], "\nHere's another example which is not a function of x and y. Using the \ same sort of technique as with the circle and the ellipse -- multiplying x(t) \ and y(t) by a new parameter that ranges from 0 to 1 -- we can \"fill in\" the \ helix from last week's lab, producing sort of a spiral staircase. (Well, at \ least a spiral ramp, anyway...)" }], "Text"], Cell[BoxData[{ \(f[s_, t_] = {s*Cos[2 Pi\ t], \ s*Sin[2 Pi\ t], \ t}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[s, t], {s, 0, 1}, {t, 0, 4}, PlotPoints \[Rule] {15, 30}]\)}], "Input"], Cell[TextData[{ "This is somewhat choppy, but if we use the ", StyleBox["PlotPoints", FontWeight->"Bold"], " option with", " ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ", we can make it look better:" }], "Text"], Cell[BoxData[{ \(f[s_, t_] = {s\ Cos[2 Pi\ t], \ s\ Sin[2 Pi\ t], \ t}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[s, t], {s, 0, 1}, {t, 0, 4}, PlotPoints \[Rule] {15, 80}]\)}], "Input"], Cell[TextData[{ "A filled-in helix such as this is called a ", StyleBox["helicoid", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ StyleBox["FYI:", FontWeight->"Bold"], " Note that the ", StyleBox["PlotPoints", FontWeight->"Bold"], " option is used a little differently if we're plotting surfaces instead of \ curves. Last week you learned that with just one parameter, ", StyleBox["PlotPoints", FontWeight->"Bold"], " tells ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " how many different values of that one parameter to use. When you have ", StyleBox["two", FontSlant->"Italic"], " parameters, however, you must tell ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " how many different values of ", StyleBox["each", FontSlant->"Italic"], " parameter to use. That's why in the command above, we used a ", StyleBox["list", FontSlant->"Italic"], " of numbers after ", StyleBox["PlotPoints", FontWeight->"Bold"], " instead of just one number. In the command above we told ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " to use 15 different values for s, and 50 different values for t. Because \ ", StyleBox["Mathematica", FontSlant->"Italic"], " plots a point of the surface for every possible combination of s and t, \ that means we drew the nicer looking helix with 15*80=1200 points, instead of \ 15*15=225 points with the default values. No wonder it looks better!\n\n", StyleBox["Warning!", FontWeight->"Bold"], " With just one parameter, the default ", StyleBox["PlotPoints", FontWeight->"Bold"], " value is 75, and we often increased it to 400. With surfaces -- i.e. if \ you give ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " two parameters -- the default value is {15, 15}. (Sometimes ", StyleBox["Mathematica", FontSlant->"Italic"], " changes its defaults to make things look better, but it doesn't always do \ a good job.) If you try to increase these numbers as high you can with one \ parameter, you will run out of memory and ", StyleBox["Mathematica will crash, and you will lose your work", FontWeight->"Bold"], ". As a rule of thumb, when plotting surfaces you should never use values \ {a, b} with ", StyleBox["PlotPoints", FontWeight->"Bold"], " if a*b is bigger than, say, 1500 or so." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["And now for something completely different...", "Text"], Cell[TextData[{ StyleBox["Exercise 1", FontSize->14, FontWeight->"Bold"], "\n\nGraph the portion of the paraboloid x = ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " which is cut off by the cylinder ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], "= 1.\n\nHint: use the ideas of Example 4. In fact, you can start with the \ same parametrization, and then flip-flop some of the components to get the \ right answer!\n\nYou should hand in your parametrization (including the \ bounds for your parameters!), a picture of the graph, and an explanation of \ how you arrived at your parametrization. Stumbling across the correct graph \ without understanding why it works will result in very little credit.\n\n", StyleBox["Exercise 2", FontSize->14, FontWeight->"Bold"], "\n\nIn Example 5 we graphed the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "=9. Now graph the hyperbolic paraboloid ", StyleBox["z = ", FontSize->16], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], "-", " ", FormBox[\(y\^2\), "TraditionalForm"]}], "10"], TraditionalForm]], FontSize->14], StyleBox["+ 3", FontSize->14], ". (Suggestion: use ", StyleBox["Plot3D", FontWeight->"Bold"], " with x- and y-ranges of {x,-4,4} and {y,-4,4}.) Show the two of these \ graphs together so you can see what the intersection looks like.\n\n(a) The \ intersection of the cylinder and the hyperboloid is a curve. Find a \ parametrization for this curve. (Your answer should NOT be the \ parametrization of a surface!) \n\nHint: you did a problem similar to this \ in Exercise 3 in last week's lab. Now that you've used cylindrical \ coordinates, here's an easy way to find the parametrization if you were \ confused by that problem: you can actually describe this curve in cylindrical \ coordinates. r will be constant, \[Theta] will change, and z will depend on \ \[Theta], so \[Theta] is your parameter. Now express z in terms of \[Theta] \ (and the constant r) and change back to rectangular coordinates.\n\n(b) Find \ a parametrization for the portion of the hyperboloid which is cut off by the \ cylinder. In other words, fill in the part of the surface which is inside \ the curve you found in part (a). (Suggestion: when you graph the surface, \ use the option ", StyleBox["BoxRatios\[RightArrow]{1,1,1}", FontWeight->"Bold"], " in ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ".)\n\nYou should hand in your parametrizations, complete with bounds for \ your parameters, and well as pictures of their graphs and an explanation of \ how you arrived at your answers.\n\n", StyleBox["Exercise 3", FontSize->14, FontWeight->"Bold"], "\n\nThroughout this lab we have described many surfaces using cylindrical \ coordinates, and then converted back to rectangular coordinates to get a \ parametrization. Use this same idea to produce a graph of the unit sphere, \ ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " = 1, using ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ".\n\nHint: describe the sphere in ", StyleBox["spherical", FontSlant->"Italic"], " coordinates, and then convert this description to rectangular coordinates \ using the given change-of-coordinate formulas in your book.\n\nYou should \ hand in your parametrization and an explanation of what you did to find it.\n\ \n", StyleBox["Exercise 4", FontSize->14, FontWeight->"Bold"], "\n\nFind a parametrization for the following surface:\n\n", Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .59685 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics3D %%IncludeResource: font Courier %%IncludeFont: Courier 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