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"Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 207 "" 0 "" {TEXT 278 32 "MAT 272 Fall 2007 Sept 19, 20 07" }{TEXT 278 0 "" }}{PARA 207 "" 0 "" {TEXT 278 6 "Review" }{TEXT 278 0 "" }}{PARA 215 "" 0 "" {TEXT 284 8 "Dr Firoz" }}{PARA 245 "" 0 " " {TEXT 227 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 298 110 "We review on typing vectors and finding dot and cross product under with(LinearAlg ebra), with(VectorCalculus):" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetti ng:-mi(\"with\"), Typesetting:-mo(\"⁡\", form = \"infix \", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"i nfinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fal se\", accent = \"false\", font_style_name = \"2D Input\", size = \"12 \", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Types etting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"LinearAlgebra\"))) ), Typesetting:-mo(\":\", form = \"infix\", fence = \"false\", separat or = \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspace \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Input\", size = \"12\", fore ground = \"[255,0,0]\", background = \"[255,255,255]\"));" "-I%mrowG6# /I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6#Q %withF'-I#moGF$63Q0⁡F'/%%formGQ&infixF'/%&fenceGQ&falseF '/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)stretchyGF=/%*symmetri cGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.movablelim itsGF=/%'accentGF=/%0font_style_nameGQ)2D~InputF'/%%sizeGQ#12F'/%+fore groundGQ*[255,0,0]F'/%+backgroundGQ.[255,255,255]F'-I(mfencedGF$6#-F#6 #-F,6#Q.LinearAlgebraF'-F563Q\":F'F8F;F>/FAQ/thickmathspaceF'/FDFgoFEF GFIFLFOFQFSFUFXFenFhn" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Type setting:-mrow(Typesetting:-mo(\"?\", form = \"infix\", fence = \"false \", separator = \"false\", lspace = \"verythinmathspace\", rspace = \" verythinmathspace\", stretchy = \"false\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Input\", siz e = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mi(\"vector\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6$-I#moGF$63Q\"?F'/%%formGQ&infixF'/%&fenceGQ&falseF '/%*separatorGF4/%'lspaceGQ2verythinmathspaceF'/%'rspaceGF9/%)stretchy GF4/%*symmetricGF4/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF 4/%.movablelimitsGF4/%'accentGF4/%0font_style_nameGQ)2D~InputF'/%%size GQ#12F'/%+foregroundGQ*[255,0,0]F'/%+backgroundGQ.[255,255,255]F'-I#mi GF$6#Q'vectorF'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Typesettin g:-mrow(Typesetting:-mo(\"?\", form = \"infix\", fence = \"false\", se parator = \"false\", lspace = \"verythinmathspace\", rspace = \"veryth inmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Input\", size = \" 12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mi(\"dotproduct\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6$-I#moGF$63Q\"?F'/%%formGQ&infixF'/%&fenceGQ&falseF'/ %*separatorGF4/%'lspaceGQ2verythinmathspaceF'/%'rspaceGF9/%)stretchyGF 4/%*symmetricGF4/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF4/ %.movablelimitsGF4/%'accentGF4/%0font_style_nameGQ)2D~InputF'/%%sizeGQ #12F'/%+foregroundGQ*[255,0,0]F'/%+backgroundGQ.[255,255,255]F'-I#miGF $6#Q+dotproductF'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Typesett ing:-mrow(Typesetting:-mo(\"?\", form = \"infix\", fence = \"false\", \+ separator = \"false\", lspace = \"verythinmathspace\", rspace = \"very thinmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize \+ = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \+ \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), T ypesetting:-mi(\"crossproduct\"));" "-I%mrowG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6$-I#moGF$63Q\"?F'/%%formGQ&infixF'/%&fenceGQ&fals eF'/%*separatorGF4/%'lspaceGQ2verythinmathspaceF'/%'rspaceGF9/%)stretc hyGF4/%*symmetricGF4/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeop GF4/%.movablelimitsGF4/%'accentGF4/%0font_style_nameGQ)2D~InputF'/%%si zeGQ#12F'/%+foregroundGQ*[255,0,0]F'/%+backgroundGQ.[255,255,255]F'-I# miGF$6#Q-crossproductF'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }{MPLTEXT 1 0 23 "with(VectorCalculus):\n" }{MPLTEXT 1 0 35 "C rossProduct(, );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 73 "For two vectors a =2i+3j-4k and b = 3i-4k find their dot and c rossproduct" }}}{PARA 215 "" 0 "" {TEXT 284 0 "" }}}{SECT 0 {PARA 229 "" 0 "" {TEXT 231 8 "Examples" }{TEXT 231 0 "" }}{SECT 1 {PARA 263 "" 0 "" {TEXT 248 41 "I. Graphs of Functions with two variables" }{TEXT 248 0 "" }}{PARA 245 "" 0 "" {TEXT 227 5 "The " }{TEXT 225 6 "plot3d" }{TEXT 227 99 " command plots the graph of a function of two variabl es. For instance, the graph of the function " }{XPPEDIT 18 0 "Typeset ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"&Appl yFunction;\", form = \"infix\", fence = \"false\", separator = \"false \", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \+ \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinma thspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesett ing:-mo(\",\", form = \"infix\", fence = \"false\", separator = \"true \", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"fa lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", f ont_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mi(\"y\")), Typesett ing:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"fal se\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stre tchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")) , Typesetting:-mo(\"=\", form = \"infix\", fence = \"false\", separato r = \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspace \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), T ypesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"2\"), super scriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"+\", f orm = \"infix\", fence = \"false\", separator = \"false\", lspace = \" mediummathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", large op = \"false\", movablelimits = \"false\", accent = \"false\", font_st yle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", ba ckground = \"[255,255,255]\"), Typesetting:-mi(\"y\")), Typesetting:-m i(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-F#6&-F,6#Q\"fF'-I#moGF$63Q0 ⁡F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF?/%' lspaceGQ$0emF'/%'rspaceGFD/%)stretchyGF?/%*symmetricGF?/%(maxsizeGQ)in finityF'/%(minsizeGQ\"1F'/%(largeopGF?/%.movablelimitsGF?/%'accentGF?/ %0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F '/%+backgroundGQ.[255,255,255]F'-F#6%-F763Q\"(F'/F;Q'prefixF'/F>Q%true F'F@/FCQ.thinmathspaceF'/FFFgo/FHFeoFIFKFNFQFSFUFWFZFgnFjn-F#6%-F,6#Q \"xF'-F763Q\",F'F:F=/FAFeoFB/FFQ3verythickmathspaceF'FGFIFKFNFQFSFUFWF ZFgnFjn-F,6#Q\"yF'-F763Q\")F'/F;Q(postfixF'FdoF@Ffo/FFQ2verythinmathsp aceF'FioFIFKFNFQFSFUFWFZFgnFjnF+-F763Q\"=F'F:F=F@/FCQ/thickmathspaceF' /FFFcqFGFIFKFNFQFSFUFWFZFgnFjn-F#6&F+-F#6%F+-I%msupGF$6%F\\p-I#mnGF$6# Q\"2F'/%1superscriptshiftGQ\"0F'F+-F763Q\"+F'F:F=F@/FCQ0mediummathspac eF'/FFFgrFGFIFKFNFQFSFUFWFZFgnFjnFepF+F+" }{TEXT 227 26 " is plotte d as follows:" }{TEXT 227 0 "" 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