(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 134734, 5677] NotebookOptionsPosition[ 131966, 5597] NotebookOutlinePosition[ 132337, 5613] CellTagsIndexPosition[ 132294, 5610] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["Lab 07 - 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->14]], "Section", FontSize->12], Cell[TextData[{ "As in Lab 4A, there is no calculus in this notebook. In that lab 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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.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 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"[", RowBox[{"x_", ",", "y_"}], "]"}], "=", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}]}], "\[IndentingNewLine]", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]}], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can draw a great picture of graphs like this, but ", StyleBox["Plot3D", FontWeight->"Bold"], " has its limitations. For example, what if we wanted to plot the graph of \ x = ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], " ? 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[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"y", "^", "2"}], " ", "+", " ", RowBox[{"z", "^", "2"}]}], ",", " ", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "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[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], " 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[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], ", y, z):" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"y_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"y", "^", "2"}], " ", "+", " ", RowBox[{"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[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"y", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.412358532658198*^9, 3.412358532958613*^9}}], 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.)" }], "Text", CellFrame->True, CellChangeTimes->{{3.41235855445829*^9, 3.412358555335945*^9}}, Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["More Examples", FontSize->14]], "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[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"r", ",", "\[Theta]"}], ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", " ", "h"}], ")"}]}], TraditionalForm]]], ", 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[{ RowBox[{ RowBox[{"f", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"r", "*", RowBox[{"Cos", "[", "theta", "]"}]}], ",", " ", RowBox[{"r", "*", RowBox[{"Sin", "[", "theta", "]"}]}], ",", " ", "0"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"disk0", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell["Or, with other values of h:", "Text"], Cell[BoxData[{ RowBox[{"disk1", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"r", "*", RowBox[{"Cos", "[", "theta", "]"}]}], ",", RowBox[{"r", "*", RowBox[{"Sin", "[", "theta", "]"}]}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"disk2", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"r", "*", RowBox[{"Cos", "[", "theta", "]"}]}], ",", RowBox[{"r", "*", RowBox[{"Sin", "[", "theta", "]"}]}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell["Or, all together:", "Text"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"disk2", ",", " ", "disk1", ",", " ", "disk0"}], "]"}]], "Input", CellChangeTimes->{{3.412358569506587*^9, 3.412358586346044*^9}}], Cell[TextData[{ StyleBox["FYI", FontWeight->"Bold"], ": Normally when we use one parameter in this lab we'll choose the letter t. \ When we use two parameters, the standard letters choices are (s,t) or (u,v). \ We'll use 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, CellChangeTimes->{{3.412358607474095*^9, 3.412358619238554*^9}}, 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[{ RowBox[{ RowBox[{"f", "[", RowBox[{"s_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"s", "*", RowBox[{"Cos", "[", "t", "]"}]}], ",", " ", RowBox[{"s", "*", RowBox[{"Sin", "[", "t", "]"}]}], ",", " ", "1"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"s", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", "0", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"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[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"s", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", "2", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"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[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"4", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"3", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"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[{ RowBox[{ RowBox[{"f", "[", RowBox[{"s_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"4", "s", "*", RowBox[{"Cos", "[", "t", "]"}]}], ",", " ", RowBox[{"3", "s", "*", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"s", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"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[ FormBox[ RowBox[{"z", "=", "3"}], TraditionalForm]]], " centered at (1.5,1):" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", RowBox[{"s_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"1.5", " ", "+", " ", RowBox[{"s", "*", RowBox[{"Cos", "[", "t", "]"}]}]}], ",", RowBox[{"1", "+", " ", RowBox[{"s", "*", RowBox[{"Sin", "[", "t", "]"}]}]}], ",", " ", "3"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"s", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"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[{ RowBox[{ RowBox[{"f", "[", RowBox[{"s_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"4", " ", "+", " ", RowBox[{"4", "s", "*", RowBox[{"Cos", "[", "t", "]"}]}]}], ",", " ", RowBox[{"2", "+", RowBox[{"2", "s", "*", RowBox[{"Sin", "[", "t", "]"}]}]}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"s", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "Input"], Cell["\<\ Rotate the previous picture and see which plane the ellipse is in.\ \>", "Text", CellChangeTimes->{3.412358665582121*^9}], 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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " which is above the unit disk in the xy-plane. We could try the \ \"trivial\" parametrization, \n\nf(x,y) = (x, y, ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "), \t\twhere -1\[LessEqual]x\[LessEqual]1 and -", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"1", " ", "-", " ", FormBox[ SuperscriptBox["x", "2"], TraditionalForm]}]], TraditionalForm]]], " \[LessEqual] y \[LessEqual] ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"1", "-", FormBox[ SuperscriptBox["x", "2"], TraditionalForm]}]], TraditionalForm]]] }], "Text"], Cell[TextData[{ "The problem is that, until recently, neither ", StyleBox["Plot3D", FontWeight->"Bold"], " or ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " accepted bounds like that for y. In ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0 this has been fixed, but the edges of the graph are still a bit \ ragged. Try it! Both of these commands will produce pictures with edges that \ aren't quite smooth:" }], "Text", CellChangeTimes->{{3.412358692597516*^9, 3.412358730538976*^9}}], Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"x", "^", "2"}]}], "]"}]}], ",", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"x", "^", "2"}]}], "]"}]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "y", ",", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"x", "^", "2"}]}], "]"}]}], ",", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "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[ FormBox[ RowBox[{ SuperscriptBox["r", "2"], "."}], TraditionalForm]]], " So our parametrization is:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "theta", "]"}]}], ",", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", "theta", "]"}]}], ",", " ", RowBox[{"r", "^", "2"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "which is above the unit disk.\" Even more commonly we might call it \"the \ portion of the paraboloid z=", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " cut off by the cylinder ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "= 1;\" it's really the same thing if you think about it. If we draw the \ cylinder ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "= 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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "=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[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", "+", " ", RowBox[{"y", "^", "2"}]}], " ", "==", " ", "9"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "6"}], ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "6"}], ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.412358765221036*^9, 3.412358765700578*^9}}], 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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "=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[{ RowBox[{ RowBox[{"f", "[", RowBox[{"theta_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"3", RowBox[{"Cos", "[", "theta", "]"}]}], ",", RowBox[{"3", RowBox[{"Sin", "[", "theta", "]"}]}], ",", "z"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"theta", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"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[{ RowBox[{ RowBox[{"f", "[", RowBox[{"s_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"s", "*", RowBox[{"Cos", "[", RowBox[{"2", "Pi", " ", "t"}], "]"}]}], ",", " ", RowBox[{"s", "*", RowBox[{"Sin", "[", RowBox[{"2", "Pi", " ", "t"}], "]"}]}], ",", " ", "t"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"s", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"s", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "4"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", RowBox[{"{", RowBox[{"15", ",", "30"}], "}"}]}]}], "]"}]}], "Input"], Cell[TextData[{ "A filled-in helix such as this is called a ", StyleBox["helicoid", FontSlant->"Italic"], "." }], "Text"], 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[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], " which is cut off by the cylinder ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], "= 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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "=9. Now graph the hyperbolic paraboloid ", StyleBox["z = ", FontSize->16], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FormBox[ SuperscriptBox["x", "2"], TraditionalForm], "-", " ", FormBox[ SuperscriptBox["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[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], " = 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 } 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Cell[TextData[{ "This lab was written entirely from scratch in January 2002. I made up all \ of the examples here, but they're all standard examples, so I'm not making \ any claims of originality! Many of the examples correspond to examples in \ Lab 2A - i.e. disks instead of circles, a helicoid instead of a circle, etc. \ I made a few minor adjustments and added some exercises in January 2004. \ Extremely minor updates in Spring 2008 for ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0.\n\nThis lab is copyright 2002, 2004 by Jonathan Rogness \ (rogness@math.umn.edu) and is protected by the Creative Commons \ Attribution-NonCommercial-ShareAlike License. You can find more information \ on this license at http://creativecommons.org/licenses/by-nc-sa/1.0/\n\n\ Although it's not specifically required by the license, I'd appreciate it if \ you let me know if you use parts of our labs, just so I can keep track of it. \ Please send me any questions or comments!" }], "Text", CellChangeTimes->{{3.412358849042299*^9, 3.412358857057184*^9}}] }, Closed]] }, Closed]] }, ScreenStyleEnvironment->"Working", WindowSize->{583, 710}, WindowMargins->{{258, Automatic}, {142, Automatic}}, FrontEndVersion->"6.0 for Linux x86 (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 362, 11, 113, "Text"], Cell[CellGroupData[{ Cell[955, 36, 81, 2, 65, "Section"], Cell[1039, 40, 871, 18, 191, "Text"], Cell[1913, 60, 44466, 2165, 120, 39812, 2086, "GraphicsData", "PostScript", \ "Graphics"] }, Closed]], Cell[CellGroupData[{ Cell[46416, 2230, 81, 2, 35, "Section"], Cell[46500, 2234, 121, 3, 47, "Text"], Cell[46624, 2239, 483, 16, 52, "Input"], Cell[47110, 2257, 679, 21, 101, "Text"], Cell[47792, 2280, 341, 11, 31, "Input"], Cell[48136, 2293, 1051, 30, 173, "Text"], Cell[49190, 2325, 261, 9, 31, "Input"], Cell[49454, 2336, 188, 6, 47, "Text"], Cell[49645, 2344, 382, 11, 31, "Input"], Cell[50030, 2357, 478, 9, 137, "Text"], Cell[50511, 2368, 753, 27, 99, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[51301, 2400, 67, 1, 35, "Section"], Cell[51371, 2403, 719, 12, 155, "Text"], Cell[52093, 2417, 114, 6, 65, "Text"], Cell[52210, 2425, 875, 24, 122, "Text"], Cell[53088, 2451, 644, 20, 52, "Input"], Cell[53735, 2473, 43, 0, 29, "Text"], Cell[53781, 2475, 947, 28, 92, "Input"], Cell[54731, 2505, 33, 0, 29, "Text"], Cell[54767, 2507, 178, 3, 31, "Input"], Cell[54948, 2512, 838, 15, 207, "Text"], Cell[55789, 2529, 287, 8, 86, "Text"], Cell[56079, 2539, 589, 19, 52, "Input"], Cell[56671, 2560, 113, 3, 29, "Text"], Cell[56787, 2565, 296, 9, 31, "Input"], Cell[57086, 2576, 248, 6, 47, "Text"], Cell[57337, 2584, 188, 7, 68, "Text"], Cell[57528, 2593, 342, 11, 31, "Input"], Cell[57873, 2606, 132, 3, 47, "Text"], Cell[58008, 2611, 594, 19, 52, "Input"], Cell[58605, 2632, 487, 14, 104, "Text"], Cell[59095, 2648, 656, 21, 52, "Input"], Cell[59754, 2671, 116, 3, 29, "Text"], Cell[59873, 2676, 659, 20, 52, "Input"], Cell[60535, 2698, 132, 3, 29, "Text"], Cell[60670, 2703, 1052, 39, 126, "Text"], Cell[61725, 2744, 516, 14, 83, "Text"], Cell[62244, 2760, 507, 17, 31, "Input"], Cell[62754, 2779, 582, 19, 52, "Input"], Cell[63339, 2800, 795, 21, 101, "Text"], Cell[64137, 2823, 630, 19, 52, "Input"], Cell[64770, 2844, 407, 11, 65, "Text"], Cell[65180, 2857, 1289, 43, 153, "Text"], Cell[66472, 2902, 921, 26, 158, "Text"], Cell[67396, 2930, 522, 15, 52, "Input"], Cell[67921, 2947, 992, 21, 191, "Text"], Cell[68916, 2970, 590, 19, 52, "Input"], Cell[69509, 2991, 624, 15, 176, "Text"], Cell[70136, 3008, 735, 23, 72, "Input"], Cell[70874, 3033, 126, 5, 29, "Text"], Cell[71003, 3040, 61, 0, 29, "Text"], Cell[71067, 3042, 59749, 2528, 1803, "Text"], Cell[CellGroupData[{ Cell[130841, 5574, 29, 0, 38, "Subsection"], Cell[130873, 5576, 1065, 17, 281, "Text"] }, Closed]] }, Closed]] } ] *) (* End of internal cache information *)