{"id":13765,"date":"2022-08-11T11:50:50","date_gmt":"2022-08-11T03:50:50","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=13765"},"modified":"2026-03-23T11:32:22","modified_gmt":"2026-03-23T03:32:22","slug":"ca962_970320","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320","title":{"rendered":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"13765\" class=\"elementor elementor-13765\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-27f425e5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"27f425e5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-100 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5ae3c2c7\" data-id=\"5ae3c2c7\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7c72f094 elementor-widget elementor-widget-breadcrumbs\" data-id=\"7c72f094\" data-element_type=\"widget\" data-widget_type=\"breadcrumbs.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<link rel=\"stylesheet\" href=\"https:\/\/ocw.nycu.edu.tw\/wp-content\/plugins\/elementor-pro\/assets\/css\/widget-theme-elements.min.css\"><p id=\"breadcrumbs\"><span><span><a href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-43d3b5df elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"43d3b5df\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-2e31d00\" data-id=\"2e31d00\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62afa6e2 elementor-widget elementor-widget-penci-info-box\" data-id=\"62afa6e2\" data-element_type=\"widget\" data-widget_type=\"penci-info-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"penci_info_box_91679\" class=\"penci-block-vc penci-info-box penci-ibox-float-left penci-view-default penci-shape-circle\">\r\n\t\t\t<div class=\"penci-ibox-inner\">\r\n\t\t\t\t\t\t\t\t<div class=\"penci-ibox-icon penci-ibox-icon--icon penci-icon penci-tibox-text\"><span class=\"penci-ibox-icon-fa\"><\/span><\/div>\t\t\t\t<div class=\"penci-ibox-content-wrap\">\r\n\t\t\t\t\t\t\t\t\t\t<h3 class=\"penci-ibox-title\">12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface\n<\/h3>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-53f374c4 elementor-widget elementor-widget-video\" data-id=\"53f374c4\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/rpEcWphh2F4&quot;,&quot;modestbranding&quot;:&quot;yes&quot;,&quot;lazy_load&quot;:&quot;yes&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-video .elementor-widget-container{overflow:hidden;transform:translateZ(0)}.elementor-widget-video .elementor-wrapper{aspect-ratio:var(--video-aspect-ratio)}.elementor-widget-video .elementor-wrapper iframe,.elementor-widget-video .elementor-wrapper video{height:100%;width:100%;display:flex;border:none;background-color:#000}@supports not (aspect-ratio:1\/1){.elementor-widget-video .elementor-wrapper{position:relative;overflow:hidden;height:0;padding-bottom:calc(100% \/ var(--video-aspect-ratio))}.elementor-widget-video .elementor-wrapper iframe,.elementor-widget-video .elementor-wrapper video{position:absolute;top:0;right:0;bottom:0;left:0}}.elementor-widget-video .elementor-open-inline .elementor-custom-embed-image-overlay{position:absolute;top:0;left:0;width:100%;height:100%;background-size:cover;background-position:50%}.elementor-widget-video .elementor-custom-embed-image-overlay{cursor:pointer;text-align:center}.elementor-widget-video .elementor-custom-embed-image-overlay:hover .elementor-custom-embed-play i{opacity:1}.elementor-widget-video .elementor-custom-embed-image-overlay img{display:block;width:100%}.elementor-widget-video .e-hosted-video .elementor-video{-o-object-fit:cover;object-fit:cover}.e-con-inner>.elementor-widget-video,.e-con>.elementor-widget-video{width:var(--container-widget-width);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-inner-section elementor-element elementor-element-59b2106f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"59b2106f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-52097d67\" data-id=\"52097d67\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7bb7256c elementor-widget elementor-widget-penci-button\" data-id=\"7bb7256c\" data-element_type=\"widget\" data-widget_type=\"penci-button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pcbtn-wrapper pcbtn-align-left\">\r\n\t\t\t<a class=\"pcbtn pcbtn-icon-left elementor-animation-grow\" href=\"\/?post_type=course_page&#038;p=13613\">\r\n\t\t\t\t<span class=\"pcbtn-wrapperin\">\r\n\t\t\t\t\t<span class=\"pcbtn-content\"><svg class=\"e-font-icon-svg e-fas-arrow-left\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M257.5 445.1l-22.2 22.2c-9.4 9.4-24.6 9.4-33.9 0L7 273c-9.4-9.4-9.4-24.6 0-33.9L201.4 44.7c9.4-9.4 24.6-9.4 33.9 0l22.2 22.2c9.5 9.5 9.3 25-.4 34.3L136.6 216H424c13.3 0 24 10.7 24 24v32c0 13.3-10.7 24-24 24H136.6l120.5 114.8c9.8 9.3 10 24.8.4 34.3z\"><\/path><\/svg>\u8fd4\u56de\u8ab2\u7a0b\u9996\u9801<\/span>\r\n\t\t\t\t\t\t\t\t\t<\/span>\r\n\t\t\t<\/a>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-2 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-723e657b\" data-id=\"723e657b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-336203b elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"336203b\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-divider{--divider-border-style:none;--divider-border-width:1px;--divider-color:#0c0d0e;--divider-icon-size:20px;--divider-element-spacing:10px;--divider-pattern-height:24px;--divider-pattern-size:20px;--divider-pattern-url:none;--divider-pattern-repeat:repeat-x}.elementor-widget-divider .elementor-divider{display:flex}.elementor-widget-divider .elementor-divider__text{font-size:15px;line-height:1;max-width:95%}.elementor-widget-divider .elementor-divider__element{margin:0 var(--divider-element-spacing);flex-shrink:0}.elementor-widget-divider .elementor-icon{font-size:var(--divider-icon-size)}.elementor-widget-divider .elementor-divider-separator{display:flex;margin:0;direction:ltr}.elementor-widget-divider--view-line_icon .elementor-divider-separator,.elementor-widget-divider--view-line_text .elementor-divider-separator{align-items:center}.elementor-widget-divider--view-line_icon .elementor-divider-separator:after,.elementor-widget-divider--view-line_icon .elementor-divider-separator:before,.elementor-widget-divider--view-line_text .elementor-divider-separator:after,.elementor-widget-divider--view-line_text .elementor-divider-separator:before{display:block;content:\"\";border-bottom:0;flex-grow:1;border-top:var(--divider-border-width) var(--divider-border-style) var(--divider-color)}.elementor-widget-divider--element-align-left .elementor-divider .elementor-divider-separator>.elementor-divider__svg:first-of-type{flex-grow:0;flex-shrink:100}.elementor-widget-divider--element-align-left .elementor-divider-separator:before{content:none}.elementor-widget-divider--element-align-left .elementor-divider__element{margin-left:0}.elementor-widget-divider--element-align-right .elementor-divider .elementor-divider-separator>.elementor-divider__svg:last-of-type{flex-grow:0;flex-shrink:100}.elementor-widget-divider--element-align-right .elementor-divider-separator:after{content:none}.elementor-widget-divider--element-align-right .elementor-divider__element{margin-right:0}.elementor-widget-divider:not(.elementor-widget-divider--view-line_text):not(.elementor-widget-divider--view-line_icon) .elementor-divider-separator{border-top:var(--divider-border-width) var(--divider-border-style) var(--divider-color)}.elementor-widget-divider--separator-type-pattern{--divider-border-style:none}.elementor-widget-divider--separator-type-pattern.elementor-widget-divider--view-line .elementor-divider-separator,.elementor-widget-divider--separator-type-pattern:not(.elementor-widget-divider--view-line) .elementor-divider-separator:after,.elementor-widget-divider--separator-type-pattern:not(.elementor-widget-divider--view-line) .elementor-divider-separator:before,.elementor-widget-divider--separator-type-pattern:not([class*=elementor-widget-divider--view]) .elementor-divider-separator{width:100%;min-height:var(--divider-pattern-height);-webkit-mask-size:var(--divider-pattern-size) 100%;mask-size:var(--divider-pattern-size) 100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-73059f3c elementor-widget elementor-widget-shortcode\" data-id=\"73059f3c\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-21\" class=\"tablepress tablepress-id-21\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">\u9031\u6b21<\/th><th class=\"column-2\">\u8ab2\u7a0b\u5167\u5bb9<\/th><th class=\"column-3\">\u8ab2\u7a0b\u5f71\u97f3<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter11 Functions and Model<br \/>\n11-1 Sequences<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13690\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-2 Series <br \/>\n11-3 The Integral Test and Estimates of Sums<br \/>\n11-4 The Comparison Tests<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13699\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-4 The Comparison Tests<br \/>\n11-5 Alternating Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13706\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-6 Absolute Convergence and the Ratio and Root Tests <br \/>\n11-8 Power Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13715\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-9 Representations of Functions as Power Series<br \/>\n11-10 Taylor and Maclaurin Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13725\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-10 Taylor and Maclaurin Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13731\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-11 The Binomial Series <br \/>\n11-12 Applications of Taylor Polynomials<br \/>\nChapter12 Vectors and the Geometry of Space<br \/>\n12-1 Three-Dimensional Coordinate Systems <br \/>\n12-2 Vectors<br \/>\n12-3 The Dot Product<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13737\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">12-4 The Cross Product <br \/>\n12-5 Equations of Lines and Planes<br \/>\n12-6 Cylinders and Quadric Surface<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13765\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">12-6 Cylinders and Quadric Surface <br \/>\n12-7 Cylindrical and Spherical Coordinates  <br \/>\nChapter13 Vector Functions<br \/>\n13-1 Vector Functions and Space Curves<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13771\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">13-1 Vector Functions and Space Curves<br \/>\n13-2 Derivatives and Integrals of Vector Functions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13777\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">13-2 Derivatives and Integrals of Vector Functions<br \/>\n13-3 Arc Length and Curvature <br \/>\n13-4 Motion in Space:Velocity and Acceleration<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13783\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter14 Partial Derivatives<br \/>\n14-1 Functions of Several Variables <br \/>\n14-2 Limits and Continuity<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13789\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-3 Partial Derivatives<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13798\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-4 Tangent Lanes and Linear Approximations<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13804\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-5 The Chain Rule <br \/>\n14-6 Directional Derivatives and the Gradient Vector<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13810\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-6 Directional Derivatives and the Gradient Vector  <br \/>\n14-7 Maximum and Minimum Values<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13816\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-7 Maximum and Minimum Values<br \/>\n14-8 Lagrange Multipliers<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13822\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-8 Lagrange Multipliers<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13828\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-20 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter15 Multiple Integrals<br \/>\n15-1 Double Integrals over Rectangles <br \/>\n15-2 Iterated Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13834\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-21 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-3 Double Integrals over General Regions <br \/>\n15-4 Double Integrals in Polar Coordinates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13840\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-22 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-6 Surface Area <br \/>\n15-7 Triple Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13846\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-23 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-7 Triple Integrals<br \/>\n15-8 Triple Integrals in Cylindrical and Spherical Coordinates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13852\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-24 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-9 Change of Variables in Multiple Integrals <br \/>\nChapter16 Vector Calculus<br \/>\n16-1 Vector Fields<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13858\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-25 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">16-2 Line Integrals <br \/>\n16-3 The Fundanental Theorem for Line Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13864\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-26 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">16-3 The Fundanental Theorem for Line Integrals  <br \/>\n16-5 Curl and Divergence<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13871\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-27 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">16-6 Parametric Surface and Their Areas <br \/>\n16-7 Surface Area<br \/>\n16-8 Stokes Theorem<br \/>\n16-9 The Divergence Theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13877\" rel=\"noopener\" target=\"_blank\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-21 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-48a0b44\" data-id=\"48a0b44\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62a7c21 elementor-widget-divider--view-line_text elementor-widget-divider--element-align-center elementor-widget elementor-widget-divider\" data-id=\"62a7c21\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t\t<span class=\"elementor-divider__text elementor-divider__element\">\n\t\t\t\t\u8ab2\u7a0b\u5c08\u5340\t\t\t\t<\/span>\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-68371dbc elementor-nav-menu__text-align-center elementor-nav-menu__align-left elementor-nav-menu--dropdown-tablet elementor-nav-menu--toggle elementor-nav-menu--burger elementor-widget elementor-widget-nav-menu\" data-id=\"68371dbc\" data-element_type=\"widget\" data-settings=\"{&quot;layout&quot;:&quot;vertical&quot;,&quot;submenu_icon&quot;:{&quot;value&quot;:&quot;&lt;svg class=\\&quot;e-font-icon-svg e-fas-caret-down\\&quot; viewBox=\\&quot;0 0 320 512\\&quot; xmlns=\\&quot;http:\\\/\\\/www.w3.org\\\/2000\\\/svg\\&quot;&gt;&lt;path d=\\&quot;M31.3 192h257.3c17.8 0 26.7 21.5 14.1 34.1L174.1 354.8c-7.8 7.8-20.5 7.8-28.3 0L17.2 226.1C4.6 213.5 13.5 192 31.3 192z\\&quot;&gt;&lt;\\\/path&gt;&lt;\\\/svg&gt;&quot;,&quot;library&quot;:&quot;fa-solid&quot;},&quot;toggle&quot;:&quot;burger&quot;}\" data-widget_type=\"nav-menu.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<link rel=\"stylesheet\" href=\"https:\/\/ocw.nycu.edu.tw\/wp-content\/uploads\/elementor\/css\/custom-pro-widget-nav-menu.min.css?ver=1776064852\">\t\t\t<nav class=\"elementor-nav-menu--main elementor-nav-menu__container elementor-nav-menu--layout-vertical e--pointer-double-line e--animation-slide\">\n\t\t\t\t<ul id=\"menu-1-68371dbc\" class=\"elementor-nav-menu sm-vertical\"><li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-121575\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/self-learning-courses-for-high-school-students\" class=\"elementor-item\">\u9ad8\u4e2d\u751f\u81ea\u4e3b\u5b78\u7fd2\u8ab2\u7a0b<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-121576\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/pre-university-courses\" class=\"elementor-item\">\u5927\u5b78\u5148\u4fee\u8ab2\u7a0b<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118057\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\" class=\"elementor-item\">\u7406\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118058\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-engineering\" class=\"elementor-item\">\u5de5\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118059\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-electrical-and-computer-engineering\" class=\"elementor-item\">\u96fb\u6a5f\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118056\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-management\" class=\"elementor-item\">\u7ba1\u7406\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118060\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-computer-science\" class=\"elementor-item\">\u8cc7\u8a0a\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118061\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-biological-science-and-technology\" class=\"elementor-item\">\u5de5\u7a0b\u751f\u7269\u79d1\u5b78\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118062\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-humanities-and-social-sciences\" class=\"elementor-item\">\u4eba\u6587\u85dd\u8853\u8207\u793e\u6703\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118063\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-hakka-affair-culture\" class=\"elementor-item\">\u5ba2\u5bb6\u6587\u5316\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118064\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/school-of-law\" class=\"elementor-item\">\u79d1\u6280\u6cd5\u5f8b\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118055\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/school-of-nursing\" class=\"elementor-item\">\u8b77\u7406\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118065\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-photonics\" class=\"elementor-item\">\u5149\u96fb\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-125873\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-biomedical-science-and-engineering\" class=\"elementor-item\">\u751f\u7269\u91ab\u5b78\u66a8\u5de5\u7a0b\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-132737\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-life-sciences\" class=\"elementor-item\">\u751f\u547d\u79d1\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-133529\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-medicine-ch\" class=\"elementor-item\">\u91ab\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118068\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/course-of-chinese-center\" class=\"elementor-item\">\u83ef\u8a9e\u4e2d\u5fc3<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120428\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/forum-of-liberal-arts\" class=\"elementor-item\">\u535a\u96c5\u8b1b\u5ea7<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123757\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/speech-of-general-education-center\" class=\"elementor-item\">\u7d93\u5178\u901a\u8b58\u6559\u80b2\u8b1b\u5ea7<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-128238\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/reading-of-classic-masterpiece\" class=\"elementor-item\">\u7d93\u5178\u540d\u8457\u9078\u8b80<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120395\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/general-education\" class=\"elementor-item\">\u901a\u8b58\u8ab2\u7a0b<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"\u9078\u55ae\u5207\u63db\" aria-expanded=\"false\">\n\t\t\t<svg aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open e-font-icon-svg e-eicon-menu-bar\" viewBox=\"0 0 1000 1000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M104 333H896C929 333 958 304 958 271S929 208 896 208H104C71 208 42 237 42 271S71 333 104 333ZM104 583H896C929 583 958 554 958 521S929 458 896 458H104C71 458 42 487 42 521S71 583 104 583ZM104 833H896C929 833 958 804 958 771S929 708 896 708H104C71 708 42 737 42 771S71 833 104 833Z\"><\/path><\/svg><svg aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close e-font-icon-svg e-eicon-close\" viewBox=\"0 0 1000 1000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M742 167L500 408 258 167C246 154 233 150 217 150 196 150 179 158 167 167 154 179 150 196 150 212 150 229 154 242 171 254L408 500 167 742C138 771 138 800 167 829 196 858 225 858 254 829L496 587 738 829C750 842 767 846 783 846 800 846 817 842 829 829 842 817 846 804 846 783 846 767 842 750 829 737L588 500 833 258C863 229 863 200 833 171 804 137 775 137 742 167Z\"><\/path><\/svg>\t\t\t<span class=\"elementor-screen-only\">\u9078\u55ae<\/span>\n\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-68371dbc\" class=\"elementor-nav-menu sm-vertical\"><li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-121575\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/self-learning-courses-for-high-school-students\" class=\"elementor-item\" tabindex=\"-1\">\u9ad8\u4e2d\u751f\u81ea\u4e3b\u5b78\u7fd2\u8ab2\u7a0b<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-121576\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/pre-university-courses\" class=\"elementor-item\" tabindex=\"-1\">\u5927\u5b78\u5148\u4fee\u8ab2\u7a0b<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118057\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\" class=\"elementor-item\" tabindex=\"-1\">\u7406\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118058\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-engineering\" class=\"elementor-item\" tabindex=\"-1\">\u5de5\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118059\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-electrical-and-computer-engineering\" class=\"elementor-item\" tabindex=\"-1\">\u96fb\u6a5f\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118056\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-management\" class=\"elementor-item\" tabindex=\"-1\">\u7ba1\u7406\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118060\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-computer-science\" class=\"elementor-item\" tabindex=\"-1\">\u8cc7\u8a0a\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118061\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-biological-science-and-technology\" class=\"elementor-item\" tabindex=\"-1\">\u5de5\u7a0b\u751f\u7269\u79d1\u5b78\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118062\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-humanities-and-social-sciences\" class=\"elementor-item\" tabindex=\"-1\">\u4eba\u6587\u85dd\u8853\u8207\u793e\u6703\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118063\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-hakka-affair-culture\" class=\"elementor-item\" tabindex=\"-1\">\u5ba2\u5bb6\u6587\u5316\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118064\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/school-of-law\" class=\"elementor-item\" tabindex=\"-1\">\u79d1\u6280\u6cd5\u5f8b\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118055\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/school-of-nursing\" class=\"elementor-item\" tabindex=\"-1\">\u8b77\u7406\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118065\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-photonics\" class=\"elementor-item\" tabindex=\"-1\">\u5149\u96fb\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-125873\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-biomedical-science-and-engineering\" class=\"elementor-item\" tabindex=\"-1\">\u751f\u7269\u91ab\u5b78\u66a8\u5de5\u7a0b\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-132737\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-life-sciences\" class=\"elementor-item\" tabindex=\"-1\">\u751f\u547d\u79d1\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-133529\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-medicine-ch\" class=\"elementor-item\" tabindex=\"-1\">\u91ab\u5b78\u9662<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118068\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/course-of-chinese-center\" class=\"elementor-item\" tabindex=\"-1\">\u83ef\u8a9e\u4e2d\u5fc3<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120428\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/forum-of-liberal-arts\" class=\"elementor-item\" tabindex=\"-1\">\u535a\u96c5\u8b1b\u5ea7<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123757\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/speech-of-general-education-center\" class=\"elementor-item\" tabindex=\"-1\">\u7d93\u5178\u901a\u8b58\u6559\u80b2\u8b1b\u5ea7<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-128238\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/reading-of-classic-masterpiece\" class=\"elementor-item\" tabindex=\"-1\">\u7d93\u5178\u540d\u8457\u9078\u8b80<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120395\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/general-education\" class=\"elementor-item\" tabindex=\"-1\">\u901a\u8b58\u8ab2\u7a0b<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>12-4 The Cross&hellip;<\/p>\n","protected":false},"featured_media":0,"parent":13613,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","format":"standard","meta":{"footnotes":""},"categories":[892],"tags":[],"class_list":["post-13765","course_page","type-course_page","status-publish","format-standard","hentry","category-calculus-ii-english-96"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.8 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface - NYCU OCW<\/title>\n<meta name=\"description\" content=\"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface https:\/\/youtu.be\/rpEcWphh2F4 \u8fd4\u56de\u8ab2\u7a0b\u9996\u9801\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/\u5fae\u7a4d\u5206\u4e8c-calculus-ii-english-\u61c9\u7528\u6578\u5b78\u7cfb-\u7b26\u9ea5\u514b\u8001\u5e2b\/ca962_970320\" \/>\n<meta property=\"og:locale\" content=\"zh_TW\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface - NYCU OCW\" \/>\n<meta property=\"og:description\" content=\"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface https:\/\/youtu.be\/rpEcWphh2F4 \u8fd4\u56de\u8ab2\u7a0b\u9996\u9801\" \/>\n<meta property=\"og:url\" content=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/\u5fae\u7a4d\u5206\u4e8c-calculus-ii-english-\u61c9\u7528\u6578\u5b78\u7cfb-\u7b26\u9ea5\u514b\u8001\u5e2b\/ca962_970320\" \/>\n<meta property=\"og:site_name\" content=\"NYCU OCW\" \/>\n<meta property=\"article:modified_time\" content=\"2026-03-23T03:32:22+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u9810\u4f30\u95b1\u8b80\u6642\u9593\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 \u5206\u9418\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320\",\"url\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320\",\"name\":\"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface - NYCU OCW\",\"isPartOf\":{\"@id\":\"https:\/\/ocw.nycu.edu.tw\/#website\"},\"datePublished\":\"2022-08-11T03:50:50+00:00\",\"dateModified\":\"2026-03-23T03:32:22+00:00\",\"description\":\"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface https:\/\/youtu.be\/rpEcWphh2F4 \u8fd4\u56de\u8ab2\u7a0b\u9996\u9801\",\"breadcrumb\":{\"@id\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320#breadcrumb\"},\"inLanguage\":\"zh-TW\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Homepage\",\"item\":\"https:\/\/ocw.nycu.edu.tw\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"\u8ab2\u7a0b\u5c08\u5340\",\"item\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"\u7406\u5b78\u9662\",\"item\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\"},{\"@type\":\"ListItem\",\"position\":4,\"name\":\"\u61c9\u7528\u6578\u5b78\u7cfb\",\"item\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\"},{\"@type\":\"ListItem\",\"position\":5,\"name\":\"\u5fae\u7a4d\u5206(\u4e8c) Calculus II (English) | \u61c9\u7528\u6578\u5b78\u7cfb \u7b26\u9ea5\u514b\u8001\u5e2b\",\"item\":\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\"},{\"@type\":\"ListItem\",\"position\":6,\"name\":\"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/ocw.nycu.edu.tw\/#website\",\"url\":\"https:\/\/ocw.nycu.edu.tw\/\",\"name\":\"NYCU OCW\",\"description\":\"\u958b\u653e\u6559\u80b2\u662f\u73fe\u4ee3\u6559\u80b2\u601d\u6f6e\u7684\u4e00\u80a1\u4e3b\u6d41\uff0c\u570b\u5167\u5916\u5927\u5b78\u8fd1\u5e7e\u5e74\u9678\u7e8c\u958b\u653e\u6559\u80b2\u8cc7\u6e90\uff0c\u900f\u904e\u958b\u653e\u5f0f\u8ab2\u7a0b (OpenCourseWare, OCW)\u66a8\u5176\u61c9\u7528\u4e4b\u5efa\u7f6e\u5c07\u77e5\u8b58\u5206\u4eab\u65bc\u793e\u6703\u3002\",\"publisher\":{\"@id\":\"https:\/\/ocw.nycu.edu.tw\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/ocw.nycu.edu.tw\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"zh-TW\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/ocw.nycu.edu.tw\/#organization\",\"name\":\"NYCU OCW\",\"url\":\"https:\/\/ocw.nycu.edu.tw\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-TW\",\"@id\":\"https:\/\/ocw.nycu.edu.tw\/#\/schema\/logo\/image\/\",\"url\":\"\",\"contentUrl\":\"\",\"caption\":\"NYCU OCW\"},\"image\":{\"@id\":\"https:\/\/ocw.nycu.edu.tw\/#\/schema\/logo\/image\/\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface - NYCU OCW","description":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface https:\/\/youtu.be\/rpEcWphh2F4 \u8fd4\u56de\u8ab2\u7a0b\u9996\u9801","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/\u5fae\u7a4d\u5206\u4e8c-calculus-ii-english-\u61c9\u7528\u6578\u5b78\u7cfb-\u7b26\u9ea5\u514b\u8001\u5e2b\/ca962_970320","og_locale":"zh_TW","og_type":"article","og_title":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface - NYCU OCW","og_description":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface https:\/\/youtu.be\/rpEcWphh2F4 \u8fd4\u56de\u8ab2\u7a0b\u9996\u9801","og_url":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/\u5fae\u7a4d\u5206\u4e8c-calculus-ii-english-\u61c9\u7528\u6578\u5b78\u7cfb-\u7b26\u9ea5\u514b\u8001\u5e2b\/ca962_970320","og_site_name":"NYCU OCW","article_modified_time":"2026-03-23T03:32:22+00:00","twitter_card":"summary_large_image","twitter_misc":{"\u9810\u4f30\u95b1\u8b80\u6642\u9593":"1 \u5206\u9418"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320","url":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320","name":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface - NYCU OCW","isPartOf":{"@id":"https:\/\/ocw.nycu.edu.tw\/#website"},"datePublished":"2022-08-11T03:50:50+00:00","dateModified":"2026-03-23T03:32:22+00:00","description":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface https:\/\/youtu.be\/rpEcWphh2F4 \u8fd4\u56de\u8ab2\u7a0b\u9996\u9801","breadcrumb":{"@id":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320#breadcrumb"},"inLanguage":"zh-TW","potentialAction":[{"@type":"ReadAction","target":["https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab\/ca962_970320#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Homepage","item":"https:\/\/ocw.nycu.edu.tw\/"},{"@type":"ListItem","position":2,"name":"\u8ab2\u7a0b\u5c08\u5340","item":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course"},{"@type":"ListItem","position":3,"name":"\u7406\u5b78\u9662","item":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science"},{"@type":"ListItem","position":4,"name":"\u61c9\u7528\u6578\u5b78\u7cfb","item":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am"},{"@type":"ListItem","position":5,"name":"\u5fae\u7a4d\u5206(\u4e8c) Calculus II (English) | \u61c9\u7528\u6578\u5b78\u7cfb \u7b26\u9ea5\u514b\u8001\u5e2b","item":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e7%a9%8d%e5%88%86%e4%ba%8c-calculus-ii-english-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e7%ac%a6%e9%ba%a5%e5%85%8b%e8%80%81%e5%b8%ab"},{"@type":"ListItem","position":6,"name":"12-4 The Cross Product, 12-5 Equations of Lines and Planes, 12-6 Cylinders and Quadric Surface"}]},{"@type":"WebSite","@id":"https:\/\/ocw.nycu.edu.tw\/#website","url":"https:\/\/ocw.nycu.edu.tw\/","name":"NYCU OCW","description":"\u958b\u653e\u6559\u80b2\u662f\u73fe\u4ee3\u6559\u80b2\u601d\u6f6e\u7684\u4e00\u80a1\u4e3b\u6d41\uff0c\u570b\u5167\u5916\u5927\u5b78\u8fd1\u5e7e\u5e74\u9678\u7e8c\u958b\u653e\u6559\u80b2\u8cc7\u6e90\uff0c\u900f\u904e\u958b\u653e\u5f0f\u8ab2\u7a0b (OpenCourseWare, OCW)\u66a8\u5176\u61c9\u7528\u4e4b\u5efa\u7f6e\u5c07\u77e5\u8b58\u5206\u4eab\u65bc\u793e\u6703\u3002","publisher":{"@id":"https:\/\/ocw.nycu.edu.tw\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/ocw.nycu.edu.tw\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"zh-TW"},{"@type":"Organization","@id":"https:\/\/ocw.nycu.edu.tw\/#organization","name":"NYCU OCW","url":"https:\/\/ocw.nycu.edu.tw\/","logo":{"@type":"ImageObject","inLanguage":"zh-TW","@id":"https:\/\/ocw.nycu.edu.tw\/#\/schema\/logo\/image\/","url":"","contentUrl":"","caption":"NYCU OCW"},"image":{"@id":"https:\/\/ocw.nycu.edu.tw\/#\/schema\/logo\/image\/"}}]}},"_links":{"self":[{"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=\/wp\/v2\/page\/13765","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=\/wp\/v2\/page"}],"about":[{"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=\/wp\/v2\/types\/course_page"}],"replies":[{"embeddable":true,"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13765"}],"version-history":[{"count":11,"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=\/wp\/v2\/page\/13765\/revisions"}],"predecessor-version":[{"id":154943,"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=\/wp\/v2\/page\/13765\/revisions\/154943"}],"up":[{"embeddable":true,"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=\/wp\/v2\/page\/13613"}],"wp:attachment":[{"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13765"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13765"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ocw.nycu.edu.tw\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}