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		<title>La diferencial de una aplicación</title>
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		<pubDate>Tue, 13 Oct 2020 09:02:08 +0000</pubDate>
				<category><![CDATA[Geometría de Riemann]]></category>
		<category><![CDATA[Lecciones]]></category>
		<category><![CDATA[cambio de coordenadas]]></category>
		<category><![CDATA[diferencial de una aplicación]]></category>
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					<description><![CDATA[<p>La diferencial de una aplicación entre variedades. La diferencial en coordenadas. Caso especial: el cambio de coordenadas. Ejemplos. Coordenadas polares y canónicas. La diferencial de la aplicación antípoda. La velocidad de una curva.</p>
<p>La entrada <a href="https://gravitacion.es/la-diferencial-de-una-aplicacion/">La diferencial de una aplicación</a> aparece primero en <a href="https://gravitacion.es">Gravitación</a>.</p>
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<legend>Biblio</legend>
<blockquote>
<p>Introduction to Smooth Manifolds.<br /><cite>John M. Lee. <em>Springer Science</em>. <strong>2013</strong>. Capítulo 3.</cite></p>
</blockquote>
</fieldset>
<h2>La diferencial de una función</h2>
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<figure><img fetchpriority="high" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/09/Page1-4.jpg" alt="" data-id="1997" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page1-4.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page1-4-2/" class="wp-image-1997" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page1-4.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-4-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-4-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-4-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-4-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-4-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-4-1080x1417.jpg 1080w" sizes="(max-width: 1905px) 100vw, 1905px" /></figure>
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<h2>Curvas diferenciables</h2>
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<figure><img decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/09/Page2-3.jpg" alt="" data-id="1998" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page2-3.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page2-3-2/" class="wp-image-1998" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page2-3.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-3-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-3-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-3-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-3-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-3-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-3-1080x1417.jpg 1080w" sizes="(max-width: 1905px) 100vw, 1905px" /></figure>
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<li class="blocks-gallery-item">
<figure><img decoding="async" width="1905" height="888" src="https://gravitacion.es/wp-content/uploads/2021/09/Page3-3.jpg" alt="" data-id="1999" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page3-3.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page3-3-2/" class="wp-image-1999" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page3-3.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-3-300x140.jpg 300w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-3-1024x477.jpg 1024w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-3-768x358.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-3-1536x716.jpg 1536w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-3-1080x503.jpg 1080w" sizes="(max-width: 1905px) 100vw, 1905px" /></figure>
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<hr>
<h2>La diferencial de una aplicación</h2>
<figure class="wp-block-gallery columns-1 is-cropped is-style-no-gutter">
<ul class="blocks-gallery-grid">
<li class="blocks-gallery-item">
<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/09/Page1-5.jpg" alt="" data-id="2004" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page1-5.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page1-5-2/" class="wp-image-2004" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page1-5.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-5-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-5-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-5-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-5-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-5-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page1-5-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
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<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/09/Page2-4.jpg" alt="" data-id="2005" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page2-4.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page2-4-2/" class="wp-image-2005" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page2-4.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-4-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-4-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-4-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-4-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-4-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page2-4-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
</li>
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<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/09/Page3-4.jpg" alt="" data-id="2006" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page3-4.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page3-4-2/" class="wp-image-2006" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page3-4.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-4-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-4-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-4-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-4-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-4-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page3-4-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
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<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/09/Page4-1.jpg" alt="" data-id="2007" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page4-1.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page4-1/" class="wp-image-2007" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page4-1.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/09/Page4-1-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page4-1-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page4-1-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page4-1-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page4-1-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page4-1-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
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<figure><img loading="lazy" decoding="async" width="780" height="1024" src="https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-780x1024.jpg" alt="" data-id="2008" data-full-url="https://gravitacion.es/wp-content/uploads/2021/09/Page5-1.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page5-1/" class="wp-image-2008" srcset="https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/09/Page5-1-1080x1417.jpg 1080w, https://gravitacion.es/wp-content/uploads/2021/09/Page5-1.jpg 1905w" sizes="auto, (max-width: 780px) 100vw, 780px" /></figure>
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<hr>
<h2>Matando moscas a cañonazos</h2>
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<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/10/Page1.jpg" alt="" data-id="2012" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page1-20/" class="wp-image-2012" srcset="https://gravitacion.es/wp-content/uploads/2021/10/Page1.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/10/Page1-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/10/Page1-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/10/Page1-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/10/Page1-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/10/Page1-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/10/Page1-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
</li>
<li class="blocks-gallery-item">
<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/10/Page2.jpg" alt="" data-id="2013" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page2-20/" class="wp-image-2013" srcset="https://gravitacion.es/wp-content/uploads/2021/10/Page2.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/10/Page2-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/10/Page2-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/10/Page2-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/10/Page2-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/10/Page2-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/10/Page2-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
</li>
<li class="blocks-gallery-item">
<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/10/Page3.jpg" alt="" data-id="2014" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page3-19/" class="wp-image-2014" srcset="https://gravitacion.es/wp-content/uploads/2021/10/Page3.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/10/Page3-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/10/Page3-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/10/Page3-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/10/Page3-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/10/Page3-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/10/Page3-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
</li>
<li class="blocks-gallery-item">
<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/10/Page4.jpg" alt="" data-id="2015" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page4-17/" class="wp-image-2015" srcset="https://gravitacion.es/wp-content/uploads/2021/10/Page4.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/10/Page4-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/10/Page4-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/10/Page4-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/10/Page4-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/10/Page4-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/10/Page4-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
</li>
<li class="blocks-gallery-item">
<figure><img loading="lazy" decoding="async" width="780" height="1024" src="https://gravitacion.es/wp-content/uploads/2021/10/Page5-780x1024.jpg" alt="" data-id="2016" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page5-17/" class="wp-image-2016" srcset="https://gravitacion.es/wp-content/uploads/2021/10/Page5-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/10/Page5-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/10/Page5-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/10/Page5-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/10/Page5-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/10/Page5-1080x1417.jpg 1080w, https://gravitacion.es/wp-content/uploads/2021/10/Page5.jpg 1905w" sizes="auto, (max-width: 780px) 100vw, 780px" /></figure>
</li>
<li class="blocks-gallery-item">
<figure><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2021/10/Page6.jpg" alt="" data-id="2018" data-full-url="https://gravitacion.es/wp-content/uploads/2021/10/Page6.jpg" data-link="https://gravitacion.es/la-diferencial-de-una-aplicacion/page6-12/" class="wp-image-2018" srcset="https://gravitacion.es/wp-content/uploads/2021/10/Page6.jpg 1905w, https://gravitacion.es/wp-content/uploads/2021/10/Page6-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2021/10/Page6-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2021/10/Page6-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2021/10/Page6-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2021/10/Page6-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2021/10/Page6-1080x1417.jpg 1080w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></figure>
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</ul>
</figure>
<hr>
<h2>Epílogo: más allá del carácter local de los vectores</h2>
<p><img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57.jpg" alt="" width="1904" height="2500" class="aligncenter size-full wp-image-2421" srcset="https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57.jpg 1904w, https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2020/10/Nota-5-nov-2022-19_50_57-1080x1418.jpg 1080w" sizes="auto, (max-width: 1904px) 100vw, 1904px" /></p>
<p>La entrada <a href="https://gravitacion.es/la-diferencial-de-una-aplicacion/">La diferencial de una aplicación</a> aparece primero en <a href="https://gravitacion.es">Gravitación</a>.</p>
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