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1210kine.fodt
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<?xml version="1.0" encoding="UTF-8"?>
<office:document xmlns:office="urn:oasis:names:tc:opendocument:xmlns:office:1.0" xmlns:ooo="http://openoffice.org/2004/office" xmlns:fo="urn:oasis:names:tc:opendocument:xmlns:xsl-fo-compatible:1.0" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:config="urn:oasis:names:tc:opendocument:xmlns:config:1.0" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:meta="urn:oasis:names:tc:opendocument:xmlns:meta:1.0" xmlns:style="urn:oasis:names:tc:opendocument:xmlns:style:1.0" xmlns:text="urn:oasis:names:tc:opendocument:xmlns:text:1.0" xmlns:rpt="http://openoffice.org/2005/report" xmlns:draw="urn:oasis:names:tc:opendocument:xmlns:drawing:1.0" xmlns:dr3d="urn:oasis:names:tc:opendocument:xmlns:dr3d:1.0" xmlns:svg="urn:oasis:names:tc:opendocument:xmlns:svg-compatible:1.0" xmlns:chart="urn:oasis:names:tc:opendocument:xmlns:chart:1.0" xmlns:table="urn:oasis:names:tc:opendocument:xmlns:table:1.0" xmlns:number="urn:oasis:names:tc:opendocument:xmlns:datastyle:1.0" xmlns:ooow="http://openoffice.org/2004/writer" xmlns:oooc="http://openoffice.org/2004/calc" xmlns:of="urn:oasis:names:tc:opendocument:xmlns:of:1.2" xmlns:xforms="http://www.w3.org/2002/xforms" xmlns:tableooo="http://openoffice.org/2009/table" xmlns:calcext="urn:org:documentfoundation:names:experimental:calc:xmlns:calcext:1.0" xmlns:drawooo="http://openoffice.org/2010/draw" xmlns:loext="urn:org:documentfoundation:names:experimental:office:xmlns:loext:1.0" xmlns:field="urn:openoffice:names:experimental:ooo-ms-interop:xmlns:field:1.0" xmlns:math="http://www.w3.org/1998/Math/MathML" xmlns:form="urn:oasis:names:tc:opendocument:xmlns:form:1.0" xmlns:script="urn:oasis:names:tc:opendocument:xmlns:script:1.0" xmlns:formx="urn:openoffice:names:experimental:ooxml-odf-interop:xmlns:form:1.0" xmlns:dom="http://www.w3.org/2001/xml-events" xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:grddl="http://www.w3.org/2003/g/data-view#" xmlns:css3t="http://www.w3.org/TR/css3-text/" xmlns:officeooo="http://openoffice.org/2009/office" office:version="1.3" office:mimetype="application/vnd.oasis.opendocument.text">
<office:meta><meta:generator>LibreOffice/7.4.6.2$Windows_X86_64 LibreOffice_project/5b1f5509c2decdade7fda905e3e1429a67acd63d</meta:generator><meta:initial-creator>A-S Koh</meta:initial-creator><meta:creation-date>2002-09-04T10:57:28</meta:creation-date><dc:creator>Aik-Siong Koh</dc:creator><dc:date>2006-10-14T11:59:15</dc:date><dc:language>en-US</dc:language><meta:editing-cycles>10</meta:editing-cycles><meta:editing-duration>PT34M45S</meta:editing-duration><meta:document-statistic meta:table-count="0" meta:image-count="0" meta:object-count="73" meta:page-count="1" meta:paragraph-count="53" meta:word-count="144" meta:character-count="896"/><meta:user-defined meta:name="Info 1"/><meta:user-defined meta:name="Info 2"/><meta:user-defined meta:name="Info 3"/><meta:user-defined meta:name="Info 4"/></office:meta>
<office:settings>
<config:config-item-set config:name="ooo:view-settings">
<config:config-item config:name="ViewAreaTop" config:type="long">501</config:config-item>
<config:config-item config:name="ViewAreaLeft" config:type="long">501</config:config-item>
<config:config-item config:name="ViewAreaWidth" config:type="long">41753</config:config-item>
<config:config-item config:name="ViewAreaHeight" config:type="long">22756</config:config-item>
<config:config-item config:name="ShowRedlineChanges" config:type="boolean">true</config:config-item>
<config:config-item config:name="InBrowseMode" config:type="boolean">true</config:config-item>
<config:config-item-map-indexed config:name="Views">
<config:config-item-map-entry>
<config:config-item config:name="ViewId" config:type="string">view2</config:config-item>
<config:config-item config:name="ViewLeft" config:type="long">702</config:config-item>
<config:config-item config:name="ViewTop" config:type="long">803</config:config-item>
<config:config-item config:name="VisibleLeft" config:type="long">501</config:config-item>
<config:config-item config:name="VisibleTop" config:type="long">501</config:config-item>
<config:config-item config:name="VisibleRight" config:type="long">42252</config:config-item>
<config:config-item config:name="VisibleBottom" config:type="long">23255</config:config-item>
<config:config-item config:name="ZoomType" config:type="short">0</config:config-item>
<config:config-item config:name="ViewLayoutColumns" config:type="short">1</config:config-item>
<config:config-item config:name="ViewLayoutBookMode" config:type="boolean">false</config:config-item>
<config:config-item config:name="ZoomFactor" config:type="short">100</config:config-item>
<config:config-item config:name="IsSelectedFrame" config:type="boolean">false</config:config-item>
<config:config-item config:name="KeepRatio" config:type="boolean">false</config:config-item>
<config:config-item config:name="HideWhitespace" config:type="boolean">false</config:config-item>
<config:config-item config:name="AnchoredTextOverflowLegacy" config:type="boolean">true</config:config-item>
</config:config-item-map-entry>
</config:config-item-map-indexed>
</config:config-item-set>
<config:config-item-set config:name="ooo:configuration-settings">
<config:config-item config:name="ProtectForm" config:type="boolean">false</config:config-item>
<config:config-item config:name="PrinterName" config:type="string">HP Officejet Pro 8600 (Network)</config:config-item>
<config:config-item config:name="EmbeddedDatabaseName" config:type="string"/>
<config:config-item config:name="CurrentDatabaseDataSource" config:type="string"/>
<config:config-item config:name="LinkUpdateMode" config:type="short">1</config:config-item>
<config:config-item config:name="AddParaTableSpacingAtStart" config:type="boolean">false</config:config-item>
<config:config-item config:name="FloattableNomargins" config:type="boolean">false</config:config-item>
<config:config-item config:name="UnbreakableNumberings" config:type="boolean">false</config:config-item>
<config:config-item config:name="FieldAutoUpdate" config:type="boolean">true</config:config-item>
<config:config-item config:name="AddVerticalFrameOffsets" config:type="boolean">false</config:config-item>
<config:config-item config:name="BackgroundParaOverDrawings" config:type="boolean">false</config:config-item>
<config:config-item config:name="AddParaTableSpacing" config:type="boolean">false</config:config-item>
<config:config-item config:name="ChartAutoUpdate" config:type="boolean">true</config:config-item>
<config:config-item config:name="CurrentDatabaseCommand" config:type="string"/>
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<config:config-item config:name="AlignTabStopPosition" config:type="boolean">false</config:config-item>
<config:config-item config:name="PrinterPaperFromSetup" config:type="boolean">false</config:config-item>
<config:config-item config:name="IsKernAsianPunctuation" config:type="boolean">false</config:config-item>
<config:config-item config:name="CharacterCompressionType" config:type="short">0</config:config-item>
<config:config-item config:name="ApplyUserData" config:type="boolean">true</config:config-item>
<config:config-item config:name="DoNotJustifyLinesWithManualBreak" config:type="boolean">true</config:config-item>
<config:config-item config:name="SaveThumbnail" config:type="boolean">true</config:config-item>
<config:config-item config:name="SaveGlobalDocumentLinks" config:type="boolean">false</config:config-item>
<config:config-item config:name="SmallCapsPercentage66" config:type="boolean">true</config:config-item>
<config:config-item config:name="CurrentDatabaseCommandType" config:type="int">0</config:config-item>
<config:config-item config:name="SaveVersionOnClose" config:type="boolean">false</config:config-item>
<config:config-item config:name="UpdateFromTemplate" config:type="boolean">false</config:config-item>
<config:config-item config:name="DoNotCaptureDrawObjsOnPage" config:type="boolean">false</config:config-item>
<config:config-item config:name="UseFormerObjectPositioning" config:type="boolean">true</config:config-item>
<config:config-item config:name="PrintSingleJobs" config:type="boolean">false</config:config-item>
<config:config-item config:name="EmbedSystemFonts" config:type="boolean">false</config:config-item>
<config:config-item config:name="PrinterIndependentLayout" config:type="string">disabled</config:config-item>
<config:config-item config:name="IsLabelDocument" config:type="boolean">false</config:config-item>
<config:config-item config:name="AddFrameOffsets" config:type="boolean">false</config:config-item>
<config:config-item config:name="AddExternalLeading" config:type="boolean">false</config:config-item>
<config:config-item config:name="MsWordCompMinLineHeightByFly" config:type="boolean">false</config:config-item>
<config:config-item config:name="UseOldNumbering" config:type="boolean">true</config:config-item>
<config:config-item config:name="OutlineLevelYieldsNumbering" config:type="boolean">false</config:config-item>
<config:config-item config:name="DoNotResetParaAttrsForNumFont" config:type="boolean">true</config:config-item>
<config:config-item config:name="IgnoreFirstLineIndentInNumbering" config:type="boolean">true</config:config-item>
<config:config-item config:name="AllowPrintJobCancel" config:type="boolean">true</config:config-item>
<config:config-item config:name="UseFormerLineSpacing" config:type="boolean">true</config:config-item>
<config:config-item config:name="AddParaSpacingToTableCells" config:type="boolean">false</config:config-item>
<config:config-item config:name="AddParaLineSpacingToTableCells" config:type="boolean">false</config:config-item>
<config:config-item config:name="UseFormerTextWrapping" config:type="boolean">true</config:config-item>
<config:config-item config:name="RedlineProtectionKey" config:type="base64Binary"/>
<config:config-item config:name="ConsiderTextWrapOnObjPos" config:type="boolean">false</config:config-item>
<config:config-item config:name="TableRowKeep" config:type="boolean">false</config:config-item>
<config:config-item config:name="TabsRelativeToIndent" config:type="boolean">true</config:config-item>
<config:config-item config:name="IgnoreTabsAndBlanksForLineCalculation" config:type="boolean">false</config:config-item>
<config:config-item config:name="RsidRoot" config:type="int">861244</config:config-item>
<config:config-item config:name="LoadReadonly" config:type="boolean">false</config:config-item>
<config:config-item config:name="ClipAsCharacterAnchoredWriterFlyFrames" config:type="boolean">true</config:config-item>
<config:config-item config:name="UnxForceZeroExtLeading" config:type="boolean">true</config:config-item>
<config:config-item config:name="UseOldPrinterMetrics" config:type="boolean">false</config:config-item>
<config:config-item config:name="TabAtLeftIndentForParagraphsInList" config:type="boolean">false</config:config-item>
<config:config-item config:name="Rsid" config:type="int">861244</config:config-item>
<config:config-item config:name="MsWordCompTrailingBlanks" config:type="boolean">false</config:config-item>
<config:config-item config:name="MathBaselineAlignment" config:type="boolean">false</config:config-item>
<config:config-item config:name="InvertBorderSpacing" config:type="boolean">false</config:config-item>
<config:config-item config:name="CollapseEmptyCellPara" config:type="boolean">false</config:config-item>
<config:config-item config:name="TabOverflow" config:type="boolean">false</config:config-item>
<config:config-item config:name="StylesNoDefault" config:type="boolean">false</config:config-item>
<config:config-item config:name="ClippedPictures" config:type="boolean">false</config:config-item>
<config:config-item config:name="EmbedFonts" config:type="boolean">false</config:config-item>
<config:config-item config:name="EmbedOnlyUsedFonts" config:type="boolean">false</config:config-item>
<config:config-item config:name="EmbedLatinScriptFonts" config:type="boolean">true</config:config-item>
<config:config-item config:name="EmbedAsianScriptFonts" config:type="boolean">true</config:config-item>
<config:config-item config:name="EmptyDbFieldHidesPara" config:type="boolean">false</config:config-item>
<config:config-item config:name="EmbedComplexScriptFonts" config:type="boolean">true</config:config-item>
<config:config-item config:name="TabOverMargin" config:type="boolean">false</config:config-item>
<config:config-item config:name="TabOverSpacing" config:type="boolean">false</config:config-item>
<config:config-item config:name="TreatSingleColumnBreakAsPageBreak" config:type="boolean">false</config:config-item>
<config:config-item config:name="SurroundTextWrapSmall" config:type="boolean">false</config:config-item>
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<config:config-item config:name="PrintHiddenText" config:type="boolean">false</config:config-item>
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<config:config-item config:name="PrintTables" config:type="boolean">true</config:config-item>
<config:config-item config:name="PrintProspectRTL" config:type="boolean">false</config:config-item>
<config:config-item config:name="PrintReversed" config:type="boolean">false</config:config-item>
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<config:config-item config:name="PrintFaxName" config:type="string"/>
<config:config-item config:name="PrintPaperFromSetup" config:type="boolean">false</config:config-item>
<config:config-item config:name="PrintEmptyPages" config:type="boolean">true</config:config-item>
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</office:settings>
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</draw:frame>is a time varying surface wrt a marker.</text:p>
<text:h text:style-name="Heading_20_3" text:outline-level="3">Surface with Constant Profile (Prism)</text:h>
<text:p text:style-name="Standard">Profile is defined in the marker<draw:frame draw:style-name="fr1" draw:name="Object123" text:anchor-type="as-char" svg:width="0.2402in" svg:height="0.1902in" draw:z-index="3"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
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<mi mathvariant="italic">xy</mi>
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</draw:frame>plane. Therefore the parametric equations are</text:p>
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<mi>x</mi>
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<mrow>
<msub>
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<mn>1</mn>
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<mrow>
<mi>y</mi>
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<mi>f</mi>
<mi>y</mi>
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<mrow>
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<mrow>
<mi>z</mi>
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<mi>s</mi>
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<annotation encoding="StarMath 5.0">x=f_x(s_1)~~y=f_y(s_1)~~z=s_2</annotation>
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</draw:frame></text:p>
<text:p text:style-name="equation">It is possible to simplify to use<draw:frame draw:style-name="fr1" draw:name="Object121" text:anchor-type="as-char" svg:width="0.2047in" svg:height="0.2138in" draw:z-index="5"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
<annotation encoding="StarMath 5.0">s_1</annotation>
</semantics>
</math>
</draw:object><draw:image>
<office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQgCAAAf
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</draw:image>
</draw:frame>only.</text:p>
<text:h text:style-name="Heading_20_3" text:outline-level="3">Plane Surface</text:h>
<text:p text:style-name="Standard">The surface is the marker<draw:frame draw:style-name="fr1" draw:name="Object124" text:anchor-type="as-char" svg:width="0.2402in" svg:height="0.1902in" draw:z-index="6"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<mi mathvariant="italic">xy</mi>
<annotation encoding="StarMath 5.0">xy</annotation>
</semantics>
</math>
</draw:object><draw:image>
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</office:binary-data>
</draw:image>
</draw:frame>plane.</text:p>
<text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object122" text:anchor-type="as-char" svg:width="1.4508in" svg:height="0.2138in" draw:z-index="7"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
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<mrow>
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<mi>x</mi>
<mo stretchy="false">=</mo>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
</mrow>
<mspace width="4em"/>
<mrow>
<mi>y</mi>
<mo stretchy="false">=</mo>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
</mrow>
<mspace width="4em"/>
<mrow>
<mi>z</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</mrow>
<annotation encoding="StarMath 5.0">x=s_1~~y=s_2~~z=0</annotation>
</semantics>
</math>
</draw:object><draw:image>
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</draw:image>
</draw:frame></text:p>
<text:p text:style-name="equation">It is possible to avoid using<draw:frame draw:style-name="fr1" draw:name="Object125" text:anchor-type="as-char" svg:width="0.1453in" svg:height="0.1929in" draw:z-index="8"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<mstyle mathvariant="bold">
<mi mathvariant="normal">s</mi>
</mstyle>
<annotation encoding="StarMath 5.0">{bold nitalic s}</annotation>
</semantics>
</math>
</draw:object><draw:image>
<office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAXABAADr
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</office:binary-data>
</draw:image>
</draw:frame>.</text:p>
<text:h text:style-name="Heading_20_3" text:outline-level="3">Surface tangent<draw:frame draw:style-name="fr1" draw:name="Object85" text:anchor-type="as-char" svg:width="0.902in" svg:height="0.2165in" draw:z-index="9"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<mrow>
<msub>
<mstyle mathvariant="bold">
<mi mathvariant="normal">t</mi>
</mstyle>
<mrow>
<mn>1</mn>
<mi mathvariant="italic">em</mi>
</mrow>
</msub>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mi>t</mi>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
<annotation encoding="StarMath 5.0">{bold nitalic t}_1em(t,s_1,s_2)</annotation>
</semantics>
</math>
</draw:object><draw:image>
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</office:binary-data>
</draw:image>
</draw:frame></text:h>
<text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object86" text:anchor-type="as-char" svg:width="1.452in" svg:height="0.4583in" draw:z-index="10"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<mrow>
<msub>
<mstyle mathvariant="bold">
<mi mathvariant="normal">t</mi>
</mstyle>
<mrow>
<mn>1</mn>
<mi mathvariant="italic">em</mi>
</mrow>
</msub>
<mo stretchy="false">=</mo>
<mfrac>
<mrow>
<mo stretchy="false">∂</mo>
<msub>
<mstyle mathvariant="bold">
<mi mathvariant="normal">r</mi>
</mstyle>
<mi mathvariant="italic">mem</mi>
</msub>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mi>t</mi>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
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</mrow>
</mrow>
<mrow>
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<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mrow>
<annotation encoding="StarMath 5.0">{bold nitalic t}_1em
=
{{partial {bold nitalic r}_mem(t,s_1,s_2)} over {partial s_1}}</annotation>
</semantics>
</math>
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<text:p text:style-name="equation">Similar to curve tangent.</text:p>
<text:h text:style-name="Heading_20_3" text:outline-level="3">Surface tangent<draw:frame draw:style-name="fr1" draw:name="Object87" text:anchor-type="as-char" svg:width="0.9047in" svg:height="0.2165in" draw:z-index="11"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<mrow>
<msub>
<mstyle mathvariant="bold">
<mi mathvariant="normal">t</mi>
</mstyle>
<mrow>
<mn>2</mn>
<mi mathvariant="italic">em</mi>
</mrow>
</msub>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mi>t</mi>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
<annotation encoding="StarMath 5.0">{bold nitalic t}_2em(t,s_1,s_2)</annotation>
</semantics>
</math>
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</draw:image>
</draw:frame></text:h>
<text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object88" text:anchor-type="as-char" svg:width="1.4547in" svg:height="0.4583in" draw:z-index="12"><draw:object>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<semantics>
<mrow>
<msub>
<mstyle mathvariant="bold">
<mi mathvariant="normal">t</mi>
</mstyle>
<mrow>
<mn>2</mn>
<mi mathvariant="italic">em</mi>
</mrow>
</msub>
<mo stretchy="false">=</mo>
<mfrac>
<mrow>
<mo stretchy="false">∂</mo>
<msub>
<mstyle mathvariant="bold">
<mi mathvariant="normal">r</mi>
</mstyle>
<mi mathvariant="italic">mem</mi>
</msub>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mi>t</mi>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>1</mn>
</msub>
<mi>,</mi>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">∂</mo>
<msub>
<mi>s</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mrow>
<annotation encoding="StarMath 5.0">{bold nitalic t}_2em
=
{{partial {bold nitalic r}_mem(t,s_1,s_2)} over {partial s_2}}</annotation>
</semantics>
</math>
</draw:object><draw:image>
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