{"id":30262,"date":"2021-03-14T12:22:21","date_gmt":"2021-03-14T11:22:21","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=30262"},"modified":"2022-03-18T10:04:37","modified_gmt":"2022-03-18T09:04:37","slug":"separating-the-variables","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/integration-of-functions-of-several-real-variables\/substitution-in-integrals\/separating-the-variables\/","title":{"rendered":"Separating the variables"},"content":{"rendered":"<p>This page is a sub-page of our page on <a title=\"at Wikipedia\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/integration-of-functions-of-several-real-variables\/substitution-in-integrals\/\" target=\"_blank\" rel=\"noopener\">Substitution in Integrals<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>The sub-pages of this page are<\/strong>:<\/p>\n<p>\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/integration-of-functions-of-several-real-variables\/substitution-in-integrals\/separating-the-variables\/examples\/\" target=\"_blank\" rel=\"noopener\">Integration examples<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Related KMR-pages<\/strong>:<\/p>\n<p>\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/partial-differential-equations\/\" target=\"_blank\" rel=\"noopener\">Partial Differential Equations<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Other relevant sources of information<\/strong>:<\/p>\n<p>\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Separation_of_variables\" target=\"_blank\" rel=\"noopener\">Separation of variables in Differential Equations<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Orthogonal_coordinates\" target=\"_blank\" rel=\"noopener\">Orthogonal coordinates<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Curvilinear_coordinates\" target=\"_blank\" rel=\"noopener\">Curvilinear coordinates<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Confocal_conic_sections\" target=\"_blank\" rel=\"noopener\">Confocal conics sections<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Elliptic_coordinate_system\" target=\"_blank\" rel=\"noopener\">Elliptic coordinate system<\/a><br \/>\n\u2022 <a title=\"at Wolfram MathWorld\" href=\"http:\/\/mathworld.wolfram.com\/ParabolicCoordinates.html\" target=\"_blank\" rel=\"noopener\">Parabolic coordinates<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Elliptic_cylindrical_coordinates\" target=\"_blank\" rel=\"noopener\">Elliptic cylindrical coordinates<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Ellipsoidal_coordinates\" target=\"_blank\" rel=\"noopener\">Ellipsoidal coordinates<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Confocal_conic_sections#Confocal_quadrics\" target=\"_blank\" rel=\"noopener\">Confocal quadrics<\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Paraboloidal_coordinates\" target=\"_blank\" rel=\"noopener\">Paraboloidal coordinates<\/a><br \/>\n\u2022 <a title=\"at Wolfram MathWorld\" href=\"http:\/\/mathworld.wolfram.com\/ConfocalEllipsoidalCoordinates.html\" target=\"_blank\" rel=\"noopener\">Confocal Ellipsoidal coordinates<\/a><br \/>\n\u2022 <a title=\"at Wolfram MathWorld\" href=\"http:\/\/mathworld.wolfram.com\/ConfocalParaboloidalCoordinates.html\" target=\"_blank\" rel=\"noopener\">Confocal Paraboloidal coordinates<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>The interactive simulations on this page can be navigated with the <a href=\"http:\/\/www.pacifict.com\/FreeStuff.html\">Free Viewer<\/a><br \/>\nof the <a href=\"http:\/\/www.pacifict.com\/Home.html\">Graphing Calculator<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Polar_coordinate_system\" target=\"_blank\" rel=\"noopener\"><strong>Polar coordinates<\/strong><\/a>:<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/pi9irZntEog\" allowfullscreen=\"allowfullscreen\" width=\"457\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/pt2vyvjs?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/57ZpibDLNH0\" allowfullscreen=\"allowfullscreen\" width=\"449\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/2748c5mj?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Cylindrical_coordinate_system\" target=\"_blank\" rel=\"noopener\">Cylindrical coordinates<\/a><\/strong>:<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/xlgBJ1V4tMM\" allowfullscreen=\"allowfullscreen\" width=\"462\" height=\"454\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/yht3jpsa?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/5GlONer1rLA\" allowfullscreen=\"allowfullscreen\" width=\"428\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/2u833s8f?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Spherical_coordinate_system\" target=\"_blank\" rel=\"noopener\">Spherical coordinates<\/a><\/strong>:<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/i4uO_g2C13M\" allowfullscreen=\"allowfullscreen\" width=\"747\" height=\"454\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/3fa6bt9h?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/yYAtjOjfZ3o\" allowfullscreen=\"allowfullscreen\" width=\"462\" height=\"360\" frameborder=\"0\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/nyGQ8w4cKsU\" allowfullscreen=\"allowfullscreen\" width=\"436\" height=\"360\" frameborder=\"0\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/qJOWIEKIBZM\" allowfullscreen=\"allowfullscreen\" width=\"462\" height=\"360\" frameborder=\"0\"><\/iframe><\/p>\n<p><a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/5pakr6k9?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created the thee movies above<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Elliptic_coordinate_system\" target=\"_blank\" rel=\"noopener\">Elliptic coordinates<\/a><\/strong>:<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/bBMQMMy-zSw\" allowfullscreen=\"allowfullscreen\" width=\"380\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/2tv2j5mj?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Parabolic_coordinates\" target=\"_blank\" rel=\"noopener\">Parabolic coordinates<\/a><\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/sHWru2guQM8\" allowfullscreen=\"allowfullscreen\" width=\"417\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/bc884vhd?download\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/geometry-2\/metric-geometry\/euclidean-geometry\/geometry\/surfaces\/quadrics\/confocal-quadrics\/\" target=\"_blank\" rel=\"noopener\">Confocal Quadrics<\/a><\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Confocal_conic_sections#Confocal_quadrics\" target=\"_blank\" rel=\"noopener\"><strong>Confocal quadric coordinates<\/strong><\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/awIOHokQXdU\" allowfullscreen=\"allowfullscreen\" width=\"369\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/xywyrt84\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/3tXwX46Jtik\" allowfullscreen=\"allowfullscreen\" width=\"632\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/spshs8w\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/zsUjX2zaZPE\" allowfullscreen=\"allowfullscreen\" width=\"616\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/v2cvarrc\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/Hi5lbCy-GgM\" allowfullscreen=\"allowfullscreen\" width=\"640\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/5h26jtrd\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/TbRjm7chFk0\" allowfullscreen=\"allowfullscreen\" width=\"558\" height=\"360\" frameborder=\"0\"><\/iframe><br \/>\n<a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/5cnxd5r4\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a><\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Parabolic_cylindrical_coordinates\" target=\"_blank\" rel=\"noopener\">Parabolic cylindrical coordinates<\/a><\/strong><\/p>\n<p>\u2022 <a title=\"at Wolfram MathWorld\" href=\"https:\/\/mathworld.wolfram.com\/ParabolicCoordinates.html\" target=\"_blank\" rel=\"noopener\">Parabolic coordinates<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Paraboloidal_coordinates\" target=\"_blank\" rel=\"noopener\">Paraboloidal coordinates<\/a><\/strong><\/p>\n<p><strong><a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Paraboloidal_coordinates\" target=\"_blank\" rel=\"noopener\">Confocal paraboloids<\/a> acting as mirrors to a planar wavefront<\/strong><\/p>\n<p><a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/8145?download\" target=\"_blank\" rel=\"noopener\">Confocal paraboloids reflecting a planar wavefront into parallel Dupin cyclides<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Reflection of a planar wavefront in a saddle surface<br \/>\nturns the wavefront into a Dupin cyclide<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/fmojic1LhvY?list=PLnDRHgzCOosTX5pKNKM4h4mBlbJDbxZkn\" allowfullscreen=\"allowfullscreen\" width=\"640\" height=\"360\" frameborder=\"0\"><\/iframe><\/p>\n<p><a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/8141?download\" target=\"_blank\" rel=\"noopener\">Confocal-paraboloids-planar-wavefront-into-cyclide-reflection.gcf<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>A set of (horizontally positioned) <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Paraboloidal_coordinates\" target=\"_blank\" rel=\"noopener\">confocal paraboloids<\/a> give the same (Dupine cyclide) reflection of a planar wave front that is incident along the vertical axis:<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/r5ZPHQg69ks\" allowfullscreen=\"allowfullscreen\" width=\"640\" height=\"360\" frameborder=\"0\"><\/iframe><\/p>\n<p><a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"https:\/\/tinyurl.com\/y785ccvz\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>The same variation of the <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Paraboloidal_coordinates\" target=\"_blank\" rel=\"noopener\">confocal paraboloids<\/a> mirrors is shown here without the reflected wavefront (which is the same for all the paraboloids of the confocal family:<br \/>\n<iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/UakF5Rnxs4Y\" allowfullscreen=\"allowfullscreen\" width=\"640\" height=\"360\" frameborder=\"0\"><\/iframe><\/p>\n<p><a title=\"in Ambj\u00f6rn's Confolio archive\" href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/8143?download\" target=\"_blank\" rel=\"noopener\">Confocal Paraboloids &#8211; reflect vertical rays into Dupin cyclide.gcf<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This page is a sub-page of our page on Substitution in Integrals \/\/\/\/\/\/\/ The sub-pages of this page are: \u2022 Integration examples \/\/\/\/\/\/\/ Related KMR-pages: \u2022 Partial Differential Equations \/\/\/\/\/\/\/ Other relevant sources of information: \u2022 Separation of variables in Differential Equations \u2022 Orthogonal coordinates \u2022 Curvilinear coordinates \u2022 Confocal conics sections \u2022 Elliptic coordinate &hellip; <a href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/integration-of-functions-of-several-real-variables\/substitution-in-integrals\/separating-the-variables\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Separating the variables<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":10427,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-30262","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/30262","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/comments?post=30262"}],"version-history":[{"count":6,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/30262\/revisions"}],"predecessor-version":[{"id":33939,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/30262\/revisions\/33939"}],"up":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/10427"}],"wp:attachment":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/media?parent=30262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}