{"id":214,"date":"2023-08-26T21:07:11","date_gmt":"2023-08-27T00:07:11","guid":{"rendered":"http:\/\/vigo.ime.unicamp.br\/ms650\/?p=214"},"modified":"2023-08-26T21:10:45","modified_gmt":"2023-08-27T00:10:45","slug":"derivadas-de-delta","status":"publish","type":"post","link":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/2023\/08\/26\/derivadas-de-delta\/","title":{"rendered":"Derivadas de delta"},"content":{"rendered":"<p>Como devemos interpretar\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\delta&#039;(x^2 -4) <\/span>? Aparentemente, h\u00e1 d\u00favidas sobre isso. Lembrando, as derivadas de delta s\u00e3o definidas como<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\delta&#039;(x) f(x) dx = -\\int \\delta(x) f&#039;(x) dx<\/span>\n<p>mas no nosso exemplo o argumento da delta \u00e9 uma fun\u00e7\u00e3o. O que fazer? Bem, devemos fazer uma mudan\u00e7a de vari\u00e1veis.\u00a0 Vamos introduzir a vari\u00e1vel <span class=\"katex-eq\" data-katex-display=\"false\"> \\displaystyle y = x^2 -4 <\/span> e lembrar que quando <span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span> percorre a reta real, <span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> vai de <span class=\"katex-eq\" data-katex-display=\"false\"> \\infty <\/span> a -4 e depois de -4 a <span class=\"katex-eq\" data-katex-display=\"false\"> \\infty <\/span>. Teremos<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^\\infty \\delta&#039;(x^2 - 4) f(x) dx = \\int_{-4}^\\infty \\delta&#039;(y) f(-\\sqrt{y +4}) \\frac{ dy}{2\\sqrt{y +4}}+\\int_{-4}^\\infty \\delta&#039;(y) f(\\sqrt{y +4}) \\frac{ dy}{2\\sqrt{y +4}}<\/span>\n<p>Pela defini\u00e7\u00e3o, teremos<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\u00a0 \\int_{-4}^\\infty \\delta&#039;(y) f(\\sqrt{y\u00a0 +4}) \\frac{ dy}{2\\sqrt{y +4}} = -\\frac{1}{2} \\int_{-4}^\\infty \\delta (y)\\frac{d}{dy}\\left(\\frac{f(\\sqrt{y +4})}{ \\sqrt{ y+4}}\\right)\u00a0 dy\u00a0 =\u00a0 -\\frac{1}{2} \\int_{-4}^\\infty \\delta (y)\u00a0 \u00a0\\left( \\frac{f&#039;(\\sqrt{y +4})}{2(y+4)} -\\frac{f(\\sqrt{y +4})}{2\\sqrt{y+4}^3} \u00a0 \\right) dy = -\\frac{f&#039;(2)}{16} + \\frac{f(2)}{32}<\/span>\n<p>de onde o <a href=\"https:\/\/www.wolframalpha.com\/input?i=DiracDelta%27%28x%5E2-4%29\">resultado<\/a> do Wolfram Alpha segue facilmente.<\/p>\n<p>(Post de contribui\u00e7\u00e3o do Mr. M).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Como devemos interpretar\u00a0 ? Aparentemente, h\u00e1 d\u00favidas sobre isso. Lembrando, as derivadas de delta s\u00e3o definidas como mas no nosso exemplo o argumento da delta \u00e9 uma fun\u00e7\u00e3o. O que fazer? Bem, devemos fazer uma mudan\u00e7a de vari\u00e1veis.\u00a0 Vamos introduzir a vari\u00e1vel e lembrar que quando percorre a reta real, vai de a -4 e &hellip; <a href=\"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/2023\/08\/26\/derivadas-de-delta\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Derivadas de delta&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-214","post","type-post","status-publish","format-standard","hentry","category-ms650"],"_links":{"self":[{"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/posts\/214","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/comments?post=214"}],"version-history":[{"count":18,"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/posts\/214\/revisions"}],"predecessor-version":[{"id":232,"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/posts\/214\/revisions\/232"}],"wp:attachment":[{"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/media?parent=214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/categories?post=214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vigo.ime.unicamp.br\/ms650\/index.php\/wp-json\/wp\/v2\/tags?post=214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}