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href="/de/docs/Learn/CSS/CSS_layout/Supporting_Older_Browsers">Unterstützung älterer Browser</a></li><li><a href="/de/docs/Learn/CSS/CSS_layout/Fundamental_Layout_Comprehension">Grundlegendes Verständnis von Layouts</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/JavaScript">JavaScript — Dynamisches clientseitiges Skripting</a></li><li><details><summary>JavaScript erste Schritte</summary><ol><li><a href="/de/docs/Learn/JavaScript/First_steps">JavaScript erste Schritte</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/What_is_JavaScript">Was ist JavaScript?</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/A_first_splash">Ein erster Sprung in JavaScript</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/What_went_wrong">Was ist schiefgelaufen? JavaScript-Fehlerbehebung</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/Variables">Speichern der benötigten Informationen — Variablen</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/Math">Grundlegende Mathematik in JavaScript – Zahlen und Operatoren</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/Strings">Umgang mit Text — Strings in JavaScript</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/Useful_string_methods">Nützliche String-Methoden</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/Arrays">Arrays</a></li><li><a href="/de/docs/Learn/JavaScript/First_steps/Silly_story_generator">Silly Story Generator</a></li></ol></details></li><li><details><summary>JavaScript-Bausteine</summary><ol><li><a href="/de/docs/Learn/JavaScript/Building_blocks">JavaScript-Bausteine</a></li><li><a href="/de/docs/Learn/JavaScript/Building_blocks/conditionals">Entscheidungen in Ihrem Code treffen — Bedingte Anweisungen</a></li><li><a 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href="/de/docs/Learn/JavaScript/Objects/Object_prototypes">Objektprototypen</a></li><li><a href="/de/docs/Learn/JavaScript/Objects/Object-oriented_programming">Objektorientierte Programmierung</a></li><li><a href="/de/docs/Learn/JavaScript/Objects/Classes_in_JavaScript">Klassen in JavaScript</a></li><li><a href="/de/docs/Learn/JavaScript/Objects/JSON">Arbeiten mit JSON</a></li><li><a href="/de/docs/Learn/JavaScript/Objects/Object_building_practice">Objekt-Baupraxis</a></li><li><a href="/de/docs/Learn/JavaScript/Objects/Adding_bouncing_balls_features">Hinzufügen von Funktionen zu unserem hüpfenden Kugeln-Demo</a></li></ol></details></li><li><details><summary>Asynchrones JavaScript</summary><ol><li><a href="/de/docs/Learn/JavaScript/Asynchronous">Asynchrones JavaScript</a></li><li><a href="/de/docs/Learn/JavaScript/Asynchronous/Introducing">Einführung in asynchrones JavaScript</a></li><li><a href="/de/docs/Learn/JavaScript/Asynchronous/Promises">Anleitung zur Verwendung von Promises</a></li><li><a href="/de/docs/Learn/JavaScript/Asynchronous/Implementing_a_promise-based_API">Anleitung zur Implementierung einer Promise-basierten API</a></li><li><a href="/de/docs/Learn/JavaScript/Asynchronous/Introducing_workers">Einführung in Workers</a></li><li><a href="/de/docs/Learn/JavaScript/Asynchronous/Sequencing_animations">Sequenzierung von Animationen</a></li></ol></details></li><li><details><summary>Client-seitige Web-APIs</summary><ol><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs">Client-Side-Web-APIs</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Introduction">Einführung in Web-APIs</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Manipulating_documents">Manipulating documents</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Fetching_data">Abrufen von Daten vom Server</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Third_party_APIs">Third-party APIs</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Drawing_graphics">Grafiken zeichnen</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Video_and_audio_APIs">Video- und Audio-APIs</a></li><li><a href="/de/docs/Learn/JavaScript/Client-side_web_APIs/Client-side_storage">Client-side storage</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/Forms">Webformulare — Arbeiten mit Benutzerdaten</a></li><li><details><summary>Grundlagen der Webformulare</summary><ol><li><a href="/de/docs/Learn/Forms">Bausteine für Webformulare</a></li><li><a href="/de/docs/Learn/Forms/Your_first_form">Ihr erstes Formular</a></li><li><a href="/de/docs/Learn/Forms/How_to_structure_a_web_form">Wie Sie ein Webformular strukturieren</a></li><li><a href="/de/docs/Learn/Forms/Basic_native_form_controls">Basis-Native Formularelemente</a></li><li><a href="/de/docs/Learn/Forms/HTML5_input_types">Die HTML5 input Typen</a></li><li><a href="/de/docs/Learn/Forms/Other_form_controls">Andere Formularelemente</a></li><li><a href="/de/docs/Learn/Forms/Styling_web_forms">Styling von Webformularen</a></li><li><a href="/de/docs/Learn/Forms/Advanced_form_styling">Erweiterte Formular-Stilgestaltung</a></li><li><a href="/de/docs/Learn/Forms/UI_pseudo-classes">UI-Pseudo-Klassen</a></li><li><a href="/de/docs/Learn/Forms/Form_validation">Client-seitige Formularvalidierung</a></li><li><a href="/de/docs/Learn/Forms/Sending_and_retrieving_form_data">Senden von Formulardaten</a></li></ol></details></li><li><details><summary>Erweiterte Techniken für Webformulare</summary><ol><li><a href="/de/docs/Learn/Forms/How_to_build_custom_form_controls">Anleitung zur Erstellung benutzerdefinierter Formularelemente</a></li><li><a href="/de/docs/Learn/Forms/Sending_forms_through_JavaScript">Versenden von Formularen über JavaScript</a></li><li><a href="/de/docs/Learn/Forms/Property_compatibility_table_for_form_controls">CSS-Eigenschaftskompatibilitätstabelle für Formularelemente</a></li><li><a href="/de/docs/Learn/Forms/HTML_forms_in_legacy_browsers">HTML-Formulare in älteren Browsern</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/Accessibility">Barrierefreiheit — Das Web für alle nutzbar machen</a></li><li><details><summary>Barrierefreiheitsleitfäden</summary><ol><li><a href="/de/docs/Learn/Accessibility">Barrierefreiheit</a></li><li><a href="/de/docs/Learn/Accessibility/What_is_accessibility">Was ist Barrierefreiheit?</a></li><li><a href="/de/docs/Learn/Accessibility/HTML">HTML: Eine gute Grundlage für Barrierefreiheit</a></li><li><a href="/de/docs/Learn/Accessibility/CSS_and_JavaScript">CSS und JavaScript: Barrierefreiheits-Best Practices</a></li><li><a href="/de/docs/Learn/Accessibility/WAI-ARIA_basics">Grundlagen von WAI-ARIA</a></li><li><a href="/de/docs/Learn/Accessibility/Multimedia">Barrierefreie Multimedia</a></li><li><a href="/de/docs/Learn/Accessibility/Mobile">Barrierefreiheit auf Mobilgeräten</a></li><li><a href="/de/docs/Learn/Accessibility/Accessibility_troubleshooting">Bewertung: Barrierefreiheits-Troubleshooting</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/Performance">Leistung — Websites schnell und reaktionsschnell machen</a></li><li><details><summary>Leitfäden zur Leistung</summary><ol><li><a href="/de/docs/Learn/Performance">Webleistung</a></li><li><a href="/de/docs/Learn/Performance/why_web_performance">Der 'Warum' der Web-Performance</a></li><li><a href="/de/docs/Learn/Performance/What_is_web_performance">Was ist Web-Performance?</a></li><li><a href="/de/docs/Learn/Performance/Perceived_performance">Wahrgenommene Leistung</a></li><li><a href="/de/docs/Learn/Performance/Measuring_performance">Messung der Performance</a></li><li><a href="/de/docs/Learn/Performance/Multimedia">Multimedia: Bilder</a></li><li><a href="/de/docs/Learn/Performance/video">Multimedia: Video</a></li><li><a href="/de/docs/Learn/Performance/JavaScript">JavaScript-Leistungsoptimierung</a></li><li><a href="/de/docs/Learn/Performance/HTML">HTML-Leistungsoptimierung</a></li><li><a href="/de/docs/Learn/Performance/CSS">CSS-Leistungsoptimierung</a></li><li><a href="/de/docs/Learn/Performance/business_case_for_performance">Der geschäftliche Nutzen von Web-Performance</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/MathML">MathML — Schreiben von Mathematik mit MathML</a></li><li><details open=""><summary>MathML erste Schritte</summary><ol><li><a href="/de/docs/Learn/MathML/First_steps">Erste Schritte mit MathML</a></li><li><a href="/de/docs/Learn/MathML/First_steps/Getting_started">Erste Schritte mit MathML</a></li><li><a href="/de/docs/Learn/MathML/First_steps/Text_containers">MathML Text-Container</a></li><li><a href="/de/docs/Learn/MathML/First_steps/Fractions_and_roots">MathML-Brüche und -Wurzeln</a></li><li><a href="/de/docs/Learn/MathML/First_steps/Scripts">MathML gescriptete Elemente</a></li><li><a href="/de/docs/Learn/MathML/First_steps/Tables">MathML-Tabellen</a></li><li><em><a href="/de/docs/Learn/MathML/First_steps/Three_famous_mathematical_formulas" aria-current="page">Drei berühmte mathematische Formeln</a></em></li></ol></details></li><li class="section"><a href="/de/docs/Learn/../Games">Spiele — Entwicklung von Spielen für das Web</a></li><li><details><summary>Anleitungen und Tutorials</summary><ol><li><a href="/de/docs/Games/Introduction">Einführung in die Spieleentwicklung für das Web</a></li><li><a href="/de/docs/Games/Techniques">Techniken für die Spieleentwicklung</a></li><li><a href="/de/docs/Games/Tutorials">Tutorials</a></li><li><a href="/de/docs/Games/Publishing_games">Publishing Games</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/Tools_and_testing">Werkzeuge und Tests</a></li><li><details><summary>Client-seitige Webentwicklungstools</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools">Verständnis von Client-seitigen Webentwicklungstools</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Overview">Übersicht über clientseitige Werkzeuge</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Command_line">Crashkurs zur Kommandozeile</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Package_management">Grundlagen des Paketmanagements</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Introducing_complete_toolchain">Einführung in eine vollständige Toolchain</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Deployment">Bereitstellung unserer App</a></li></ol></details></li><li><details><summary>Einführung in client-seitige Frameworks</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Introduction">Einführung in client-seitige Frameworks</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Main_features">Hauptmerkmale von Frameworks</a></li></ol></details></li><li><details><summary>React</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_getting_started">Erste Schritte mit React</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_todo_list_beginning">Beginn unserer React-Task-Liste</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_components">Komponentisieren unserer React-App</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_interactivity_events_state">React-Interaktivität: Ereignisse und Status</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_interactivity_filtering_conditional_rendering">React Interaktivität: Bearbeiten, Filtern, bedingtes Rendering</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_accessibility">Accessibility in React</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_resources">React-Ressourcen</a></li></ol></details></li><li><details><summary>Ember</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_getting_started">Einstieg in Ember</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_structure_componentization">Ember App-Struktur und Komponentisierung</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_interactivity_events_state">Ember-Interaktivität: Events, Klassen und Zustand</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_conditional_footer">Ember Interaktivität: Footer-Funktionalität, bedingtes Rendering</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_routing">Routing in Ember</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_resources">Ember-Ressourcen und Fehlerbehebung</a></li></ol></details></li><li><details><summary>Vue</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_getting_started">Einstieg in Vue</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_first_component">Erstellen unserer ersten Vue-Komponente</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_rendering_lists">Rendering einer Liste von Vue-Komponenten</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_methods_events_models">Hinzufügen eines neuen Todo-Formulars: Vue-Ereignisse, Methoden und Modelle</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_styling">Styling von Vue-Komponenten mit CSS</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_computed_properties">Verwendung von Vue computed properties</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_conditional_rendering">Vue bedingte Darstellung: Bearbeitung bestehender Todos</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_refs_focus_management">Vue-Refs und Lifecycle-Methoden zur Fokusverwaltung</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_resources">Vue-Ressourcen</a></li></ol></details></li><li><details><summary>Svelte</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_getting_started">Erste Schritte mit Svelte</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_Todo_list_beginning">Starten unserer Svelte-Tasklisten-App</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_variables_props">Dynamisches Verhalten in Svelte: Arbeiten mit Variablen und Props</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_components">Komponentisierung unserer Svelte-App</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_reactivity_lifecycle_accessibility">Fortgeschrittenes Svelte: Reaktivität, Lebenszyklus, Barrierefreiheit</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_stores">Arbeiten mit Svelte Stores</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_TypeScript">TypeScript-Unterstützung in Svelte</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_deployment_next">Bereitstellung und nächste Schritte</a></li></ol></details></li><li><details><summary>Angular</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_getting_started">Erste Schritte mit Angular</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_todo_list_beginning">Anfang unserer Angular-To-Do-Liste-App</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_styling">Styling unserer Angular-Anwendung</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_item_component">Erstellen einer Item-Komponente</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_filtering">Filtern unserer To-Do-Elemente</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_building">Building Angular applications and further resources</a></li></ol></details></li><li><details><summary>Git und GitHub</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/GitHub">Git und GitHub</a></li></ol></details></li><li><details><summary>Cross-Browser-Tests</summary><ol><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing">Cross-Browser-Testing</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Introduction">Einführung in das Cross-Browser-Testing</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Testing_strategies">Strategien zur Durchführung von Tests</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/HTML_and_CSS">Umgang mit häufigen HTML- und CSS-Problemen</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/JavaScript">Umgang mit häufigen JavaScript-Problemen</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Accessibility">Umgang mit häufigen Problemen der Barrierefreiheit</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Feature_detection">Implementierung von Feature-Erkennung</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Automated_testing">Einführung in automatisiertes Testen</a></li><li><a href="/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Your_own_automation_environment">Einrichten Ihrer eigenen Testautomatisierungsumgebung</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/Server-side">Server-seitige Webprogrammierung</a></li><li><details><summary>Erste Schritte</summary><ol><li><a href="/de/docs/Learn/Server-side/First_steps">Server-seitige Website-Programmierung: Erste Schritte</a></li><li><a href="/de/docs/Learn/Server-side/First_steps/Introduction">Einführung in die serverseitige Programmierung</a></li><li><a href="/de/docs/Learn/Server-side/First_steps/Client-Server_overview">Überblick über Client-Server</a></li><li><a href="/de/docs/Learn/Server-side/First_steps/Web_frameworks">Serverseitige Web-Frameworks</a></li><li><a href="/de/docs/Learn/Server-side/First_steps/Website_security">Website-Sicherheit</a></li></ol></details></li><li><details><summary>Django Web-Framework (Python)</summary><ol><li><a href="/de/docs/Learn/Server-side/Django">Django Web Framework (Python)</a></li><li><a href="/de/docs/Learn/Server-side/Django/Introduction">Einführung in Django</a></li><li><a href="/de/docs/Learn/Server-side/Django/development_environment">Einrichten einer Django-Entwicklungsumgebung</a></li><li><a href="/de/docs/Learn/Server-side/Django/Tutorial_local_library_website">Django-Tutorial: Die Local Library Website</a></li><li><a href="/de/docs/Learn/Server-side/Django/skeleton_website">Django-Tutorial Teil 2: Erstellung einer Skelett-Website</a></li><li><a href="/de/docs/Learn/Server-side/Django/Models">Django-Tutorial Teil 3: Verwenden von Modellen</a></li><li><a href="/de/docs/Learn/Server-side/Django/Admin_site">Django Tutorial Teil 4: Django Admin-Seite</a></li><li><a href="/de/docs/Learn/Server-side/Django/Home_page">Django-Tutorial Teil 5: Erstellen unserer Startseite</a></li><li><a href="/de/docs/Learn/Server-side/Django/Generic_views">Django Tutorial Teil 6: Generische Listen- und Detailansichten</a></li><li><a href="/de/docs/Learn/Server-side/Django/Sessions">Django-Tutorial Teil 7: Sessions-Framework</a></li><li><a href="/de/docs/Learn/Server-side/Django/Authentication">Django-Tutorial Teil 8: Benutzer-Authentifizierung und Berechtigungen</a></li><li><a href="/de/docs/Learn/Server-side/Django/Forms">Django Tutorial Teil 9: Arbeiten mit Formularen</a></li><li><a href="/de/docs/Learn/Server-side/Django/Testing">Django Tutorial Teil 10: Testen einer Django-Webanwendung</a></li><li><a href="/de/docs/Learn/Server-side/Django/Deployment">Django-Tutorial Teil 11: Django in Produktion bereitstellen</a></li><li><a href="/de/docs/Learn/Server-side/Django/web_application_security">Django-Webanwendungssicherheit</a></li><li><a href="/de/docs/Learn/Server-side/Django/django_assessment_blog">Bewertung: DIY Django Mini-Blog</a></li></ol></details></li><li><details><summary>Express Web-Framework (Node.js/JavaScript)</summary><ol><li><a href="/de/docs/Learn/Server-side/Express_Nodejs">Express-Webframework (Node.js/JavaScript)</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/Introduction">Einführung in Express/Node</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/development_environment">Einrichtung einer Node-Entwicklungsumgebung</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/Tutorial_local_library_website">Express-Tutorial: Die Website der lokalen Bibliothek</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/skeleton_website">Express Tutorial Teil 2: Erstellung einer Grundstruktur für eine Website</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/mongoose">Express Tutorial Teil 3: Verwendung einer Datenbank (mit Mongoose)</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/routes">Express Tutorial Teil 4: Routen und Controller</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/Displaying_data">Express Tutorial Teil 5: Bibliotheksdaten anzeigen</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/forms">Express Tutorial Teil 6: Arbeiten mit Formularen</a></li><li><a href="/de/docs/Learn/Server-side/Express_Nodejs/deployment">Express-Tutorial Teil 7: Bereitstellung für die Produktion</a></li></ol></details></li><li class="section"><a href="/de/docs/Learn/Common_questions">Weitere Ressourcen</a></li><li><details><summary>Häufige Fragen</summary><ol><li><a href="/de/docs/Learn/Common_questions">Häufige Fragen</a></li><li><a href="/de/docs/Learn/HTML/Howto">HTML verwenden, um häufige Probleme zu lösen</a></li><li><a href="/de/docs/Learn/CSS/Howto">CSS verwenden, um häufige Probleme zu lösen</a></li><li><a href="/de/docs/Learn/JavaScript/Howto">Lösen Sie häufige Probleme in Ihrem JavaScript-Code</a></li><li><a href="/de/docs/Learn/Common_questions/Web_mechanics">Web-Mechanik</a></li><li><a href="/de/docs/Learn/Common_questions/Tools_and_setup">Tools und Einrichtung</a></li><li><a href="/de/docs/Learn/Common_questions/Design_and_accessibility">Design und Barrierefreiheit</a></li></ol></details></li></ol></div></div><section class="place side"></section></nav></aside><div class="toc-container"><aside class="toc"><nav><div class="document-toc-container"><section class="document-toc"><header><h2 class="document-toc-heading">In diesem Artikel</h2></header><ul class="document-toc-list"><li class="document-toc-item "><a class="document-toc-link" href="#ein_kleiner_mathematikartikel">Ein kleiner Mathematikartikel</a></li><li class="document-toc-item "><a class="document-toc-link" href="#ausgangspunkt">Ausgangspunkt</a></li><li class="document-toc-item "><a class="document-toc-link" href="#hinweise_und_tipps">Hinweise und Tipps</a></li></ul></section></div></nav></aside><section class="place side"></section></div></div><main id="content" class="main-content "><article class="main-page-content" lang="de"><header><h1>Drei berühmte mathematische Formeln</h1></header><div class="section-content"><ul class="prev-next"> <li><a class="button secondary" href="/de/docs/Learn/MathML/First_steps/Tables"><span class="button-wrap"> Zurück </span></a></li> <li><a class="button secondary" href="/de/docs/Learn/MathML/First_steps"><span class="button-wrap"> Übersicht: Erste Schritte mit MathML</span></a></li> </ul> <figure class="table-container"><table> <tbody> <tr> <th scope="row">Voraussetzungen:</th> <td> Bevor Sie diese Bewertung versuchen, sollten Sie bereits alle Artikel in diesem Modul durchgearbeitet haben und auch die Grundlagen von HTML verstehen (lesen Sie <a href="/de/docs/Learn/HTML/Introduction_to_HTML">Einführung in HTML</a>). </td> </tr> <tr> <th scope="row">Ziel:</th> <td>Ein wenig mit MathML zu experimentieren und Ihr neu erworbenes Wissen zu testen.</td> </tr> </tbody> </table></figure></div><section aria-labelledby="ein_kleiner_mathematikartikel"><h2 id="ein_kleiner_mathematikartikel"><a href="#ein_kleiner_mathematikartikel">Ein kleiner Mathematikartikel</a></h2><div class="section-content"><p>Das Ziel ist es, den folgenden Mathematikartikel mit HTML und MathML neu zu schreiben:</p> <p> <img src="/de/docs/Learn/MathML/First_steps/Three_famous_mathematical_formulas/xelatex-output.png" alt="Screenshot der durch XeLaTeX erzeugten PDF-Ausgabe" width="1013" height="697" loading="lazy"> </p> <p>Auch wenn Sie mit <a href="https://en.wikipedia.org/wiki/LaTeX" class="external" target="_blank">LaTeX</a> nicht vertraut sein müssen, könnte es nützlich sein, den LaTeX-Quellcode zu kennen, aus dem er erzeugt wurde:</p> <div class="code-example"><div class="example-header"><span class="language-name">latex</span></div><pre class="brush: latex notranslate"><code>\documentclass{article} \usepackage{amsmath} \usepackage{amssymb} \begin{document} To solve the cubic equation $t^3 + pt + q = 0$ (where the real numbers $p, q$ satisfy ${4p^3 + 27q^2} > 0$) one can use Cardano's formula: \[ \sqrt[{3}]{ -\frac{q}{2} +\sqrt{\frac{q^2}{4} + {\frac{p^{3}}{27}}} }+ \sqrt[{3}]{ -\frac{q}{2} -\sqrt{\frac{q^2}{4} + {\frac{p^{3}}{27}}} } \] For any $u_1, \dots, u_n \in \mathbb{C}$ and $v_1, \dots, v_n \in \mathbb{C}$, the Cauchy–Bunyakovsky–Schwarz inequality can be written as follows: \[ \left| \sum_{k=1}^n {u_k \bar{v_k}} \right|^2 \leq { \left( \sum_{k=1}^n {|u_k|} \right)^2 \left( \sum_{k=1}^n {|v_k|} \right)^2 } \] Finally, the determinant of a Vandermonde matrix can be calculated using the following expression: \[ \begin{vmatrix} 1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \dots & x_2^{n-1} \\ 1 & x_3 & x_3^2 & \dots & x_3^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \dots & x_n^{n-1} \\ \end{vmatrix} = {\prod_{1 \leq {i,j} \leq n} {(x_i - x_j)}} \] \end{document} </code></pre></div></div></section><section aria-labelledby="ausgangspunkt"><h2 id="ausgangspunkt"><a href="#ausgangspunkt">Ausgangspunkt</a></h2><div class="section-content"><p>Um mit dieser Bewertung zu beginnen, können Sie auf unsere übliche HTML-Vorlage zurückgreifen. Standardmäßig wird UTF-8-Kodierung verwendet, spezielle Webfonts für die <code><body></code> und <code><math></code> Tags (mit ähnlichem Look & Feel wie die LaTeX-Ausgabe). Das Ziel ist es, die Fragezeichen <code>???</code> durch tatsächlichen MathML-Inhalt zu ersetzen.</p> <div class="code-example"><div class="example-header"><span class="language-name">html</span></div><pre class="brush: html notranslate"><code><!doctype html> <html lang="en-US"> <head> <meta charset="utf-8" /> <title>Three famous mathematical formulas</title> <link rel="stylesheet" href="https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css" /> </head> <body class="htmlmathparagraph"> <p> To solve the cubic equation ??? (where the real numbers ??? satisfy ???) one can use Cardano's formula: ??? </p> <p> For any ??? and ???, the Cauchy–Bunyakovsky–Schwarz inequality can be written as follows: ??? </p> <p> Finally, the determinant of a Vandermonde matrix can be calculated using the following expression: ??? </p> </body> </html> </code></pre></div></div></section><section aria-labelledby="hinweise_und_tipps"><h2 id="hinweise_und_tipps"><a href="#hinweise_und_tipps">Hinweise und Tipps</a></h2><div class="section-content"><ul> <li>Beginnen Sie mit dem Einfügen leerer <code><math></code> Tags und entscheiden Sie, ob sie ein <code>display="block"</code> Attribut haben sollen oder nicht.</li> <li>Überprüfen Sie den verwendeten Text und finden Sie deren <a href="https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode" class="external" target="_blank">Unicode-Zeichen</a> ("−", "ℂ", "∑", ...).</li> <li>Analysieren Sie die Semantik jedes Textabschnitts (Variable? Operator? Zahl?) und bestimmen Sie das richtige Tokenelement, das Sie verwenden sollen.</li> <li>Suchen Sie nach erweiterten Konstruktionen (Brüche? Wurzeln? Indizes? 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Metadaten in HTML</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/HTML_text_fundamentals\">Grundlagen des HTML-Textes</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/Creating_hyperlinks\">Erstellen von Hyperlinks</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/Advanced_text_formatting\">Erweiterte Textformatierung</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/Document_and_website_structure\">Struktur eines Dokuments und einer Website</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/Debugging_HTML\">Debugging HTML</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/Marking_up_a_letter\">Markierung eines Briefes</a></li><li><a href=\"/de/docs/Learn/HTML/Introduction_to_HTML/Structuring_a_page_of_content\">Die Strukturierung einer Seite mit Inhalt</a></li></ol></details></li><li><details><summary>Multimedia und Einbettung</summary><ol><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding\">Multimedia und Einbettung</a></li><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding/Images_in_HTML\">Bilder in HTML</a></li><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding/Video_and_audio_content\">Video- und Audioinhalte</a></li><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding/Other_embedding_technologies\">Von Objekt zu iframe — andere Einbettungstechnologien</a></li><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding/Adding_vector_graphics_to_the_Web\">Vektorgrafiken zum Web hinzufügen</a></li><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding/Responsive_images\">Responsive Images</a></li><li><a href=\"/de/docs/Learn/HTML/Multimedia_and_embedding/Mozilla_splash_page\">Mozilla Splash-Seite</a></li></ol></details></li><li><details><summary>HTML-Tabellen</summary><ol><li><a href=\"/de/docs/Learn/HTML/Tables\">HTML-Tabellen</a></li><li><a href=\"/de/docs/Learn/HTML/Tables/Basics\">HTML Table Grundlagen</a></li><li><a href=\"/de/docs/Learn/HTML/Tables/Advanced\">Erweiterte Funktionen und Barrierefreiheit von HTML-Tabellen</a></li><li><a href=\"/de/docs/Learn/HTML/Tables/Structuring_planet_data\">Strukturierung von Planeten-Daten</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/CSS\">CSS — Gestaltung des Webs</a></li><li><details><summary>CSS erste Schritte</summary><ol><li><a href=\"/de/docs/Learn/CSS/First_steps\">Erste Schritte mit CSS</a></li><li><a href=\"/de/docs/Learn/CSS/First_steps/What_is_CSS\">Was ist CSS?</a></li><li><a href=\"/de/docs/Learn/CSS/First_steps/Getting_started\">Einstieg in CSS</a></li><li><a href=\"/de/docs/Learn/CSS/First_steps/How_CSS_is_structured\">Wie CSS strukturiert ist</a></li><li><a href=\"/de/docs/Learn/CSS/First_steps/How_CSS_works\">Wie CSS funktioniert</a></li><li><a href=\"/de/docs/Learn/CSS/First_steps/Styling_a_biography_page\">Eine Biografieseite stylen</a></li></ol></details></li><li><details><summary>CSS-Bausteine</summary><ol><li><a href=\"/de/docs/Learn/CSS/Building_blocks\">CSS-Grundbausteine</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Selectors\">CSS-Selektoren</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Selectors/Type_Class_and_ID_Selectors\">Typ-, Klassen- und ID-Selektoren</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Selectors/Attribute_selectors\">Attributselektoren</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Selectors/Pseudo-classes_and_pseudo-elements\">Pseudo-Klassen und Pseudo-Elemente</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Selectors/Combinators\">Kombinatoren</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Cascade_and_inheritance\">Kaskade, Spezifität und Vererbung</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Cascade_layers\">Kaskadenschichten</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/The_box_model\">Das Boxmodell</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Backgrounds_and_borders\">Hintergründe und Rahmen</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Handling_different_text_directions\">Umgang mit unterschiedlichen Textausrichtungen</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Overflowing_content\">Überlaufender Inhalt</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Values_and_units\">CSS-Werte und Einheiten</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Sizing_items_in_CSS\">Größe von Elementen in CSS</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Images_media_form_elements\">Bilder, Medien und Formularelemente</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Styling_tables\">Tabellen stylen</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Advanced_styling_effects\">Erweiterte Styling-Effekte</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Debugging_CSS\">Debugging CSS</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Organizing\">Organisieren Ihres CSS</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Fundamental_CSS_comprehension\">Grundlegendes CSS-Verständnis</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/Creating_fancy_letterheaded_paper\">Erstellen von stilvollem Briefpapier</a></li><li><a href=\"/de/docs/Learn/CSS/Building_blocks/A_cool_looking_box\">Eine cool aussehende Box</a></li></ol></details></li><li><details><summary>Textgestaltung</summary><ol><li><a href=\"/de/docs/Learn/CSS/Styling_text\">CSS Textgestaltung</a></li><li><a href=\"/de/docs/Learn/CSS/Styling_text/Fundamentals\">Grundlegende Text- und Schriftgestaltung</a></li><li><a href=\"/de/docs/Learn/CSS/Styling_text/Styling_lists\">Listen stilisieren</a></li><li><a href=\"/de/docs/Learn/CSS/Styling_text/Styling_links\">Styling von Links</a></li><li><a href=\"/de/docs/Learn/CSS/Styling_text/Web_fonts\">Web-Fonts</a></li><li><a href=\"/de/docs/Learn/CSS/Styling_text/Typesetting_a_homepage\">Setzen einer Community-Schul-Startseite</a></li></ol></details></li><li><details><summary>CSS-Layout</summary><ol><li><a href=\"/de/docs/Learn/CSS/CSS_layout\">CSS-Layout</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Introduction\">Einführung in CSS Layout</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Normal_Flow\">Normaler Fluss</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Flexbox\">Flexbox</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Grids\">Raster</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Floats\">Floats</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Positioning\">Positioning</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Multiple-column_Layout\">Mehrspaltiges Layout</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Responsive_Design\">Responsives Design</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Media_queries\">Einsteigerleitfaden für Media Queries</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Legacy_Layout_Methods\">Legacy-Layout-Methoden</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Supporting_Older_Browsers\">Unterstützung älterer Browser</a></li><li><a href=\"/de/docs/Learn/CSS/CSS_layout/Fundamental_Layout_Comprehension\">Grundlegendes Verständnis von Layouts</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/JavaScript\">JavaScript — Dynamisches clientseitiges Skripting</a></li><li><details><summary>JavaScript erste Schritte</summary><ol><li><a href=\"/de/docs/Learn/JavaScript/First_steps\">JavaScript erste Schritte</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/What_is_JavaScript\">Was ist JavaScript?</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/A_first_splash\">Ein erster Sprung in JavaScript</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/What_went_wrong\">Was ist schiefgelaufen? JavaScript-Fehlerbehebung</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/Variables\">Speichern der benötigten Informationen — Variablen</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/Math\">Grundlegende Mathematik in JavaScript – Zahlen und Operatoren</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/Strings\">Umgang mit Text — Strings in JavaScript</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/Useful_string_methods\">Nützliche String-Methoden</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/Arrays\">Arrays</a></li><li><a href=\"/de/docs/Learn/JavaScript/First_steps/Silly_story_generator\">Silly Story Generator</a></li></ol></details></li><li><details><summary>JavaScript-Bausteine</summary><ol><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks\">JavaScript-Bausteine</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/conditionals\">Entscheidungen in Ihrem Code treffen — Bedingte Anweisungen</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Looping_code\">Schleifen-Code</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Functions\">Funktionen — wiederverwendbare Codeblöcke</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Build_your_own_function\">Erstellen Sie Ihre eigene Funktion</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Return_values\">Funktionsrückgabewerte</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Events\">Einführung in Ereignisse</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Event_bubbling\">Event bubbling</a></li><li><a href=\"/de/docs/Learn/JavaScript/Building_blocks/Image_gallery\">Bildgalerie</a></li></ol></details></li><li><details><summary>Einführung in JavaScript-Objekte</summary><ol><li><a href=\"/de/docs/Learn/JavaScript/Objects\">Einführung in JavaScript-Objekte</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/Basics\">JavaScript Objekt Grundlagen</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/Object_prototypes\">Objektprototypen</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/Object-oriented_programming\">Objektorientierte Programmierung</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/Classes_in_JavaScript\">Klassen in JavaScript</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/JSON\">Arbeiten mit JSON</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/Object_building_practice\">Objekt-Baupraxis</a></li><li><a href=\"/de/docs/Learn/JavaScript/Objects/Adding_bouncing_balls_features\">Hinzufügen von Funktionen zu unserem hüpfenden Kugeln-Demo</a></li></ol></details></li><li><details><summary>Asynchrones JavaScript</summary><ol><li><a href=\"/de/docs/Learn/JavaScript/Asynchronous\">Asynchrones JavaScript</a></li><li><a href=\"/de/docs/Learn/JavaScript/Asynchronous/Introducing\">Einführung in asynchrones JavaScript</a></li><li><a href=\"/de/docs/Learn/JavaScript/Asynchronous/Promises\">Anleitung zur Verwendung von Promises</a></li><li><a href=\"/de/docs/Learn/JavaScript/Asynchronous/Implementing_a_promise-based_API\">Anleitung zur Implementierung einer Promise-basierten API</a></li><li><a href=\"/de/docs/Learn/JavaScript/Asynchronous/Introducing_workers\">Einführung in Workers</a></li><li><a href=\"/de/docs/Learn/JavaScript/Asynchronous/Sequencing_animations\">Sequenzierung von Animationen</a></li></ol></details></li><li><details><summary>Client-seitige Web-APIs</summary><ol><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs\">Client-Side-Web-APIs</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Introduction\">Einführung in Web-APIs</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Manipulating_documents\">Manipulating documents</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Fetching_data\">Abrufen von Daten vom Server</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Third_party_APIs\">Third-party APIs</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Drawing_graphics\">Grafiken zeichnen</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Video_and_audio_APIs\">Video- und Audio-APIs</a></li><li><a href=\"/de/docs/Learn/JavaScript/Client-side_web_APIs/Client-side_storage\">Client-side storage</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/Forms\">Webformulare — Arbeiten mit Benutzerdaten</a></li><li><details><summary>Grundlagen der Webformulare</summary><ol><li><a href=\"/de/docs/Learn/Forms\">Bausteine für Webformulare</a></li><li><a href=\"/de/docs/Learn/Forms/Your_first_form\">Ihr erstes Formular</a></li><li><a href=\"/de/docs/Learn/Forms/How_to_structure_a_web_form\">Wie Sie ein Webformular strukturieren</a></li><li><a href=\"/de/docs/Learn/Forms/Basic_native_form_controls\">Basis-Native Formularelemente</a></li><li><a href=\"/de/docs/Learn/Forms/HTML5_input_types\">Die HTML5 input Typen</a></li><li><a href=\"/de/docs/Learn/Forms/Other_form_controls\">Andere Formularelemente</a></li><li><a href=\"/de/docs/Learn/Forms/Styling_web_forms\">Styling von Webformularen</a></li><li><a href=\"/de/docs/Learn/Forms/Advanced_form_styling\">Erweiterte Formular-Stilgestaltung</a></li><li><a href=\"/de/docs/Learn/Forms/UI_pseudo-classes\">UI-Pseudo-Klassen</a></li><li><a href=\"/de/docs/Learn/Forms/Form_validation\">Client-seitige Formularvalidierung</a></li><li><a href=\"/de/docs/Learn/Forms/Sending_and_retrieving_form_data\">Senden von Formulardaten</a></li></ol></details></li><li><details><summary>Erweiterte Techniken für Webformulare</summary><ol><li><a href=\"/de/docs/Learn/Forms/How_to_build_custom_form_controls\">Anleitung zur Erstellung benutzerdefinierter Formularelemente</a></li><li><a href=\"/de/docs/Learn/Forms/Sending_forms_through_JavaScript\">Versenden von Formularen über JavaScript</a></li><li><a href=\"/de/docs/Learn/Forms/Property_compatibility_table_for_form_controls\">CSS-Eigenschaftskompatibilitätstabelle für Formularelemente</a></li><li><a href=\"/de/docs/Learn/Forms/HTML_forms_in_legacy_browsers\">HTML-Formulare in älteren Browsern</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/Accessibility\">Barrierefreiheit — Das Web für alle nutzbar machen</a></li><li><details><summary>Barrierefreiheitsleitfäden</summary><ol><li><a href=\"/de/docs/Learn/Accessibility\">Barrierefreiheit</a></li><li><a href=\"/de/docs/Learn/Accessibility/What_is_accessibility\">Was ist Barrierefreiheit?</a></li><li><a href=\"/de/docs/Learn/Accessibility/HTML\">HTML: Eine gute Grundlage für Barrierefreiheit</a></li><li><a href=\"/de/docs/Learn/Accessibility/CSS_and_JavaScript\">CSS und JavaScript: Barrierefreiheits-Best Practices</a></li><li><a href=\"/de/docs/Learn/Accessibility/WAI-ARIA_basics\">Grundlagen von WAI-ARIA</a></li><li><a href=\"/de/docs/Learn/Accessibility/Multimedia\">Barrierefreie Multimedia</a></li><li><a href=\"/de/docs/Learn/Accessibility/Mobile\">Barrierefreiheit auf Mobilgeräten</a></li><li><a href=\"/de/docs/Learn/Accessibility/Accessibility_troubleshooting\">Bewertung: Barrierefreiheits-Troubleshooting</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/Performance\">Leistung — Websites schnell und reaktionsschnell machen</a></li><li><details><summary>Leitfäden zur Leistung</summary><ol><li><a href=\"/de/docs/Learn/Performance\">Webleistung</a></li><li><a href=\"/de/docs/Learn/Performance/why_web_performance\">Der 'Warum' der Web-Performance</a></li><li><a href=\"/de/docs/Learn/Performance/What_is_web_performance\">Was ist Web-Performance?</a></li><li><a href=\"/de/docs/Learn/Performance/Perceived_performance\">Wahrgenommene Leistung</a></li><li><a href=\"/de/docs/Learn/Performance/Measuring_performance\">Messung der Performance</a></li><li><a href=\"/de/docs/Learn/Performance/Multimedia\">Multimedia: Bilder</a></li><li><a href=\"/de/docs/Learn/Performance/video\">Multimedia: Video</a></li><li><a href=\"/de/docs/Learn/Performance/JavaScript\">JavaScript-Leistungsoptimierung</a></li><li><a href=\"/de/docs/Learn/Performance/HTML\">HTML-Leistungsoptimierung</a></li><li><a href=\"/de/docs/Learn/Performance/CSS\">CSS-Leistungsoptimierung</a></li><li><a href=\"/de/docs/Learn/Performance/business_case_for_performance\">Der geschäftliche Nutzen von Web-Performance</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/MathML\">MathML — Schreiben von Mathematik mit MathML</a></li><li><details open=\"\"><summary>MathML erste Schritte</summary><ol><li><a href=\"/de/docs/Learn/MathML/First_steps\">Erste Schritte mit MathML</a></li><li><a href=\"/de/docs/Learn/MathML/First_steps/Getting_started\">Erste Schritte mit MathML</a></li><li><a href=\"/de/docs/Learn/MathML/First_steps/Text_containers\">MathML Text-Container</a></li><li><a href=\"/de/docs/Learn/MathML/First_steps/Fractions_and_roots\">MathML-Brüche und -Wurzeln</a></li><li><a href=\"/de/docs/Learn/MathML/First_steps/Scripts\">MathML gescriptete Elemente</a></li><li><a href=\"/de/docs/Learn/MathML/First_steps/Tables\">MathML-Tabellen</a></li><li><em><a href=\"/de/docs/Learn/MathML/First_steps/Three_famous_mathematical_formulas\" aria-current=\"page\">Drei berühmte mathematische Formeln</a></em></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/../Games\">Spiele — Entwicklung von Spielen für das Web</a></li><li><details><summary>Anleitungen und Tutorials</summary><ol><li><a href=\"/de/docs/Games/Introduction\">Einführung in die Spieleentwicklung für das Web</a></li><li><a href=\"/de/docs/Games/Techniques\">Techniken für die Spieleentwicklung</a></li><li><a href=\"/de/docs/Games/Tutorials\">Tutorials</a></li><li><a href=\"/de/docs/Games/Publishing_games\">Publishing Games</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/Tools_and_testing\">Werkzeuge und Tests</a></li><li><details><summary>Client-seitige Webentwicklungstools</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools\">Verständnis von Client-seitigen Webentwicklungstools</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Overview\">Übersicht über clientseitige Werkzeuge</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Command_line\">Crashkurs zur Kommandozeile</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Package_management\">Grundlagen des Paketmanagements</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Introducing_complete_toolchain\">Einführung in eine vollständige Toolchain</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Understanding_client-side_tools/Deployment\">Bereitstellung unserer App</a></li></ol></details></li><li><details><summary>Einführung in client-seitige Frameworks</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Introduction\">Einführung in client-seitige Frameworks</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Main_features\">Hauptmerkmale von Frameworks</a></li></ol></details></li><li><details><summary>React</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_getting_started\">Erste Schritte mit React</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_todo_list_beginning\">Beginn unserer React-Task-Liste</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_components\">Komponentisieren unserer React-App</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_interactivity_events_state\">React-Interaktivität: Ereignisse und Status</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_interactivity_filtering_conditional_rendering\">React Interaktivität: Bearbeiten, Filtern, bedingtes Rendering</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_accessibility\">Accessibility in React</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/React_resources\">React-Ressourcen</a></li></ol></details></li><li><details><summary>Ember</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_getting_started\">Einstieg in Ember</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_structure_componentization\">Ember App-Struktur und Komponentisierung</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_interactivity_events_state\">Ember-Interaktivität: Events, Klassen und Zustand</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_conditional_footer\">Ember Interaktivität: Footer-Funktionalität, bedingtes Rendering</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_routing\">Routing in Ember</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Ember_resources\">Ember-Ressourcen und Fehlerbehebung</a></li></ol></details></li><li><details><summary>Vue</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_getting_started\">Einstieg in Vue</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_first_component\">Erstellen unserer ersten Vue-Komponente</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_rendering_lists\">Rendering einer Liste von Vue-Komponenten</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_methods_events_models\">Hinzufügen eines neuen Todo-Formulars: Vue-Ereignisse, Methoden und Modelle</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_styling\">Styling von Vue-Komponenten mit CSS</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_computed_properties\">Verwendung von Vue computed properties</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_conditional_rendering\">Vue bedingte Darstellung: Bearbeitung bestehender Todos</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_refs_focus_management\">Vue-Refs und Lifecycle-Methoden zur Fokusverwaltung</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Vue_resources\">Vue-Ressourcen</a></li></ol></details></li><li><details><summary>Svelte</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_getting_started\">Erste Schritte mit Svelte</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_Todo_list_beginning\">Starten unserer Svelte-Tasklisten-App</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_variables_props\">Dynamisches Verhalten in Svelte: Arbeiten mit Variablen und Props</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_components\">Komponentisierung unserer Svelte-App</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_reactivity_lifecycle_accessibility\">Fortgeschrittenes Svelte: Reaktivität, Lebenszyklus, Barrierefreiheit</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_stores\">Arbeiten mit Svelte Stores</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_TypeScript\">TypeScript-Unterstützung in Svelte</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Svelte_deployment_next\">Bereitstellung und nächste Schritte</a></li></ol></details></li><li><details><summary>Angular</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_getting_started\">Erste Schritte mit Angular</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_todo_list_beginning\">Anfang unserer Angular-To-Do-Liste-App</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_styling\">Styling unserer Angular-Anwendung</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_item_component\">Erstellen einer Item-Komponente</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_filtering\">Filtern unserer To-Do-Elemente</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Client-side_JavaScript_frameworks/Angular_building\">Building Angular applications and further resources</a></li></ol></details></li><li><details><summary>Git und GitHub</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/GitHub\">Git und GitHub</a></li></ol></details></li><li><details><summary>Cross-Browser-Tests</summary><ol><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing\">Cross-Browser-Testing</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Introduction\">Einführung in das Cross-Browser-Testing</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Testing_strategies\">Strategien zur Durchführung von Tests</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/HTML_and_CSS\">Umgang mit häufigen HTML- und CSS-Problemen</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/JavaScript\">Umgang mit häufigen JavaScript-Problemen</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Accessibility\">Umgang mit häufigen Problemen der Barrierefreiheit</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Feature_detection\">Implementierung von Feature-Erkennung</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Automated_testing\">Einführung in automatisiertes Testen</a></li><li><a href=\"/de/docs/Learn/Tools_and_testing/Cross_browser_testing/Your_own_automation_environment\">Einrichten Ihrer eigenen Testautomatisierungsumgebung</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/Server-side\">Server-seitige Webprogrammierung</a></li><li><details><summary>Erste Schritte</summary><ol><li><a href=\"/de/docs/Learn/Server-side/First_steps\">Server-seitige Website-Programmierung: Erste Schritte</a></li><li><a href=\"/de/docs/Learn/Server-side/First_steps/Introduction\">Einführung in die serverseitige Programmierung</a></li><li><a href=\"/de/docs/Learn/Server-side/First_steps/Client-Server_overview\">Überblick über Client-Server</a></li><li><a href=\"/de/docs/Learn/Server-side/First_steps/Web_frameworks\">Serverseitige Web-Frameworks</a></li><li><a href=\"/de/docs/Learn/Server-side/First_steps/Website_security\">Website-Sicherheit</a></li></ol></details></li><li><details><summary>Django Web-Framework (Python)</summary><ol><li><a href=\"/de/docs/Learn/Server-side/Django\">Django Web Framework (Python)</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Introduction\">Einführung in Django</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/development_environment\">Einrichten einer Django-Entwicklungsumgebung</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Tutorial_local_library_website\">Django-Tutorial: Die Local Library Website</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/skeleton_website\">Django-Tutorial Teil 2: Erstellung einer Skelett-Website</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Models\">Django-Tutorial Teil 3: Verwenden von Modellen</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Admin_site\">Django Tutorial Teil 4: Django Admin-Seite</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Home_page\">Django-Tutorial Teil 5: Erstellen unserer Startseite</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Generic_views\">Django Tutorial Teil 6: Generische Listen- und Detailansichten</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Sessions\">Django-Tutorial Teil 7: Sessions-Framework</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Authentication\">Django-Tutorial Teil 8: Benutzer-Authentifizierung und Berechtigungen</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Forms\">Django Tutorial Teil 9: Arbeiten mit Formularen</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Testing\">Django Tutorial Teil 10: Testen einer Django-Webanwendung</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/Deployment\">Django-Tutorial Teil 11: Django in Produktion bereitstellen</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/web_application_security\">Django-Webanwendungssicherheit</a></li><li><a href=\"/de/docs/Learn/Server-side/Django/django_assessment_blog\">Bewertung: DIY Django Mini-Blog</a></li></ol></details></li><li><details><summary>Express Web-Framework (Node.js/JavaScript)</summary><ol><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs\">Express-Webframework (Node.js/JavaScript)</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/Introduction\">Einführung in Express/Node</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/development_environment\">Einrichtung einer Node-Entwicklungsumgebung</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/Tutorial_local_library_website\">Express-Tutorial: Die Website der lokalen Bibliothek</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/skeleton_website\">Express Tutorial Teil 2: Erstellung einer Grundstruktur für eine Website</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/mongoose\">Express Tutorial Teil 3: Verwendung einer Datenbank (mit Mongoose)</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/routes\">Express Tutorial Teil 4: Routen und Controller</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/Displaying_data\">Express Tutorial Teil 5: Bibliotheksdaten anzeigen</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/forms\">Express Tutorial Teil 6: Arbeiten mit Formularen</a></li><li><a href=\"/de/docs/Learn/Server-side/Express_Nodejs/deployment\">Express-Tutorial Teil 7: Bereitstellung für die Produktion</a></li></ol></details></li><li class=\"section\"><a href=\"/de/docs/Learn/Common_questions\">Weitere Ressourcen</a></li><li><details><summary>Häufige Fragen</summary><ol><li><a href=\"/de/docs/Learn/Common_questions\">Häufige Fragen</a></li><li><a href=\"/de/docs/Learn/HTML/Howto\">HTML verwenden, um häufige Probleme zu lösen</a></li><li><a href=\"/de/docs/Learn/CSS/Howto\">CSS verwenden, um häufige Probleme zu lösen</a></li><li><a href=\"/de/docs/Learn/JavaScript/Howto\">Lösen Sie häufige Probleme in Ihrem JavaScript-Code</a></li><li><a href=\"/de/docs/Learn/Common_questions/Web_mechanics\">Web-Mechanik</a></li><li><a href=\"/de/docs/Learn/Common_questions/Tools_and_setup\">Tools und Einrichtung</a></li><li><a href=\"/de/docs/Learn/Common_questions/Design_and_accessibility\">Design und Barrierefreiheit</a></li></ol></details></li></ol>","sidebarMacro":"LearnSidebar","body":[{"type":"prose","value":{"id":null,"title":null,"isH3":false,"content":"<ul class=\"prev-next\">\n <li><a class=\"button secondary\" href=\"/de/docs/Learn/MathML/First_steps/Tables\"><span class=\"button-wrap\"> Zurück </span></a></li>\n <li><a class=\"button secondary\" href=\"/de/docs/Learn/MathML/First_steps\"><span class=\"button-wrap\"> Übersicht: Erste Schritte mit MathML</span></a></li>\n \n</ul>\n<figure class=\"table-container\"><table>\n <tbody>\n <tr>\n <th scope=\"row\">Voraussetzungen:</th>\n <td>\n Bevor Sie diese Bewertung versuchen, sollten Sie bereits alle Artikel in diesem Modul durchgearbeitet haben und auch die Grundlagen von HTML verstehen (lesen Sie\n <a href=\"/de/docs/Learn/HTML/Introduction_to_HTML\">Einführung in HTML</a>).\n </td>\n </tr>\n <tr>\n <th scope=\"row\">Ziel:</th>\n <td>Ein wenig mit MathML zu experimentieren und Ihr neu erworbenes Wissen zu testen.</td>\n </tr>\n </tbody>\n</table></figure>"}},{"type":"prose","value":{"id":"ein_kleiner_mathematikartikel","title":"Ein kleiner Mathematikartikel","isH3":false,"content":"<p>Das Ziel ist es, den folgenden Mathematikartikel mit HTML und MathML neu zu schreiben:</p>\n<p>\n <img src=\"/de/docs/Learn/MathML/First_steps/Three_famous_mathematical_formulas/xelatex-output.png\" alt=\"Screenshot der durch XeLaTeX erzeugten PDF-Ausgabe\" width=\"1013\" height=\"697\" loading=\"lazy\">\n</p>\n<p>Auch wenn Sie mit <a href=\"https://en.wikipedia.org/wiki/LaTeX\" class=\"external\" target=\"_blank\">LaTeX</a> nicht vertraut sein müssen, könnte es nützlich sein, den LaTeX-Quellcode zu kennen, aus dem er erzeugt wurde:</p>\n<div class=\"code-example\"><div class=\"example-header\"><span class=\"language-name\">latex</span></div><pre class=\"brush: latex notranslate\"><code>\\documentclass{article}\n\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\n\\begin{document}\n\nTo solve the cubic equation $t^3 + pt + q = 0$ (where the real numbers\n$p, q$ satisfy ${4p^3 + 27q^2} > 0$) one can use Cardano's formula:\n\n\\[\n \\sqrt[{3}]{\n -\\frac{q}{2}\n +\\sqrt{\\frac{q^2}{4} + {\\frac{p^{3}}{27}}}\n }+\n \\sqrt[{3}]{\n -\\frac{q}{2}\n -\\sqrt{\\frac{q^2}{4} + {\\frac{p^{3}}{27}}}\n }\n\\]\n\nFor any $u_1, \\dots, u_n \\in \\mathbb{C}$ and\n$v_1, \\dots, v_n \\in \\mathbb{C}$, the Cauchy–Bunyakovsky–Schwarz\ninequality can be written as follows:\n\n\\[\n \\left| \\sum_{k=1}^n {u_k \\bar{v_k}} \\right|^2\n \\leq\n {\n \\left( \\sum_{k=1}^n {|u_k|} \\right)^2\n \\left( \\sum_{k=1}^n {|v_k|} \\right)^2\n }\n\\]\n\nFinally, the determinant of a Vandermonde matrix can be calculated\nusing the following expression:\n\n\\[\n \\begin{vmatrix}\n 1 & x_1 & x_1^2 & \\dots & x_1^{n-1} \\\\\n 1 & x_2 & x_2^2 & \\dots & x_2^{n-1} \\\\\n 1 & x_3 & x_3^2 & \\dots & x_3^{n-1} \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 1 & x_n & x_n^2 & \\dots & x_n^{n-1} \\\\\n \\end{vmatrix}\n = {\\prod_{1 \\leq {i,j} \\leq n} {(x_i - x_j)}}\n\\]\n\n\\end{document}\n</code></pre></div>"}},{"type":"prose","value":{"id":"ausgangspunkt","title":"Ausgangspunkt","isH3":false,"content":"<p>Um mit dieser Bewertung zu beginnen, können Sie auf unsere übliche HTML-Vorlage zurückgreifen. Standardmäßig wird UTF-8-Kodierung verwendet, spezielle Webfonts für die <code><body></code> und <code><math></code> Tags (mit ähnlichem Look & Feel wie die LaTeX-Ausgabe). Das Ziel ist es, die Fragezeichen <code>???</code> durch tatsächlichen MathML-Inhalt zu ersetzen.</p>\n<div class=\"code-example\"><div class=\"example-header\"><span class=\"language-name\">html</span></div><pre class=\"brush: html notranslate\"><code><!doctype html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\" />\n <title>Three famous mathematical formulas</title>\n <link\n rel=\"stylesheet\"\n href=\"https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css\" />\n </head>\n <body class=\"htmlmathparagraph\">\n <p>\n To solve the cubic equation ??? (where the real numbers ??? satisfy ???)\n one can use Cardano's formula: ???\n </p>\n\n <p>\n For any ??? and ???, the Cauchy–Bunyakovsky–Schwarz inequality can be\n written as follows: ???\n </p>\n\n <p>\n Finally, the determinant of a Vandermonde matrix can be calculated using\n the following expression: ???\n </p>\n </body>\n</html>\n</code></pre></div>"}},{"type":"prose","value":{"id":"hinweise_und_tipps","title":"Hinweise und Tipps","isH3":false,"content":"<ul>\n <li>Beginnen Sie mit dem Einfügen leerer <code><math></code> Tags und entscheiden Sie, ob sie ein <code>display=\"block\"</code> Attribut haben sollen oder nicht.</li>\n <li>Überprüfen Sie den verwendeten Text und finden Sie deren <a href=\"https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode\" class=\"external\" target=\"_blank\">Unicode-Zeichen</a> (\"−\", \"ℂ\", \"∑\", ...).</li>\n <li>Analysieren Sie die Semantik jedes Textabschnitts (Variable? Operator? Zahl?) und bestimmen Sie das richtige Tokenelement, das Sie verwenden sollen.</li>\n <li>Suchen Sie nach erweiterten Konstruktionen (Brüche? Wurzeln? Indizes? Matrizen?) und bestimmen Sie das passende MathML-Element für jede davon.</li>\n <li>Vergessen Sie nicht, <code><mrow></code> zu verwenden, um Unterausdrücke zu gruppieren.</li>\n <li>Achten Sie auf dehnbare und große Operatoren!</li>\n <li>Verwenden Sie den <a href=\"https://validator.w3.org/nu/\" class=\"external\" target=\"_blank\">W3C-Validator</a>, um unbeabsichtigte Fehler in Ihrem HTML/MathML-Markup zu erkennen.</li>\n <li>Wenn Sie nicht weiterkommen oder feststellen, wie schmerzhaft es ist, MathML von Hand zu schreiben, zögern Sie nicht, Tools zu nutzen, um <a href=\"/de/docs/Web/MathML/Authoring\">MathML zu schreiben</a> wie <a href=\"https://fred-wang.github.io/TeXZilla/\" class=\"external\" target=\"_blank\">TeXZilla</a>.</li>\n</ul><ul class=\"prev-next\">\n <li><a class=\"button secondary\" href=\"/de/docs/Learn/MathML/First_steps/Tables\"><span class=\"button-wrap\"> Zurück </span></a></li>\n <li><a class=\"button secondary\" href=\"/de/docs/Learn/MathML/First_steps\"><span class=\"button-wrap\"> Übersicht: Erste Schritte mit MathML</span></a></li>\n \n</ul>"}}],"toc":[{"text":"Ein kleiner Mathematikartikel","id":"ein_kleiner_mathematikartikel"},{"text":"Ausgangspunkt","id":"ausgangspunkt"},{"text":"Hinweise und Tipps","id":"hinweise_und_tipps"}],"summary":"Das Ziel ist es, den folgenden Mathematikartikel mit HTML und MathML neu zu schreiben:","popularity":0,"modified":null,"other_translations":[{"locale":"en-US","title":"Three famous mathematical formulas","native":"English (US)"},{"locale":"ja","title":"三大数式","native":"日本語"},{"locale":"zh-CN","title":"三个著名的数学公式","native":"中文 (简体)"}],"pageType":"unknown","source":{"folder":"de/learn/mathml/first_steps/three_famous_mathematical_formulas","github_url":"https://github.com/mdn/translated-content/blob/main/files/de/learn/mathml/first_steps/three_famous_mathematical_formulas/index.md","last_commit_url":"https://github.com/mdn/translated-content/commit/null","filename":"index.md"},"short_title":"Drei berühmte mathematische Formeln","parents":[{"uri":"/de/docs/Learn","title":"Webentwicklung lernen"},{"uri":"/de/docs/Learn/MathML","title":"Mathematik mit MathML schreiben"},{"uri":"/de/docs/Learn/MathML/First_steps","title":"Erste Schritte mit MathML"},{"uri":"/de/docs/Learn/MathML/First_steps/Three_famous_mathematical_formulas","title":"Drei berühmte mathematische Formeln"}],"pageTitle":"Drei berühmte mathematische Formeln - Webentwicklung lernen | MDN","noIndexing":false}}</script></body></html>