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	<title>espacio tangente Archives | Gravitación</title>
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	<description>Geometría § Relatividad § Universo</description>
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	<title>espacio tangente Archives | Gravitación</title>
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		<title>Vectores tangentes</title>
		<link>https://gravitacion.es/vectores-tangentes/</link>
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		<dc:creator><![CDATA[admin]]></dc:creator>
		<pubDate>Tue, 06 Oct 2020 21:40:29 +0000</pubDate>
				<category><![CDATA[Geometría de Riemann]]></category>
		<category><![CDATA[Lecciones]]></category>
		<category><![CDATA[derivación]]></category>
		<category><![CDATA[espacio tangente]]></category>
		<category><![CDATA[vector tangente]]></category>
		<guid isPermaLink="false">https://gravitacion.es/?p=1394</guid>

					<description><![CDATA[<p>La dificultad para definir un vector tangente a una variedad. Vectores como derivaciones. Vectores tangentes a una variedad. El espacio tangente. La base de las parciales. Ejemplos. Vectores tangentes de una superficie regular. Sobre el abuso de notación. Coordenadas polares en el plano.</p>
<p>La entrada <a href="https://gravitacion.es/vectores-tangentes/">Vectores tangentes</a> aparece primero en <a href="https://gravitacion.es">Gravitación</a>.</p>
]]></description>
										<content:encoded><![CDATA[<fieldset>
<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>Vector tangente: definición</h2>

<a href='https://gravitacion.es/wp-content/uploads/2022/11/Page1-1.jpg'><img fetchpriority="high" decoding="async" width="1904" height="2500" src="https://gravitacion.es/wp-content/uploads/2022/11/Page1-1.jpg" class="attachment-full size-full" alt="" srcset="https://gravitacion.es/wp-content/uploads/2022/11/Page1-1.jpg 1904w, https://gravitacion.es/wp-content/uploads/2022/11/Page1-1-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2022/11/Page1-1-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2022/11/Page1-1-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2022/11/Page1-1-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2022/11/Page1-1-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2022/11/Page1-1-1080x1418.jpg 1080w" sizes="(max-width: 1904px) 100vw, 1904px" /></a>
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<a href='https://gravitacion.es/wp-content/uploads/2022/11/Page5-1.jpg'><img loading="lazy" decoding="async" width="1904" height="2500" src="https://gravitacion.es/wp-content/uploads/2022/11/Page5-1.jpg" class="attachment-full size-full" alt="" srcset="https://gravitacion.es/wp-content/uploads/2022/11/Page5-1.jpg 1904w, https://gravitacion.es/wp-content/uploads/2022/11/Page5-1-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2022/11/Page5-1-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2022/11/Page5-1-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2022/11/Page5-1-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2022/11/Page5-1-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2022/11/Page5-1-1080x1418.jpg 1080w" sizes="auto, (max-width: 1904px) 100vw, 1904px" /></a>
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<a href='https://gravitacion.es/wp-content/uploads/2022/11/Page7.jpg'><img loading="lazy" decoding="async" width="1904" height="2500" src="https://gravitacion.es/wp-content/uploads/2022/11/Page7.jpg" class="attachment-full size-full" alt="" srcset="https://gravitacion.es/wp-content/uploads/2022/11/Page7.jpg 1904w, https://gravitacion.es/wp-content/uploads/2022/11/Page7-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2022/11/Page7-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2022/11/Page7-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2022/11/Page7-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2022/11/Page7-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2022/11/Page7-1080x1418.jpg 1080w" sizes="auto, (max-width: 1904px) 100vw, 1904px" /></a>
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<a href='https://gravitacion.es/wp-content/uploads/2022/11/Page9.jpg'><img loading="lazy" decoding="async" width="1904" height="2500" src="https://gravitacion.es/wp-content/uploads/2022/11/Page9.jpg" class="attachment-full size-full" alt="" srcset="https://gravitacion.es/wp-content/uploads/2022/11/Page9.jpg 1904w, https://gravitacion.es/wp-content/uploads/2022/11/Page9-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2022/11/Page9-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2022/11/Page9-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2022/11/Page9-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2022/11/Page9-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2022/11/Page9-1080x1418.jpg 1080w" sizes="auto, (max-width: 1904px) 100vw, 1904px" /></a>
<a href='https://gravitacion.es/wp-content/uploads/2022/11/Page10.jpg'><img loading="lazy" decoding="async" width="1904" height="2500" src="https://gravitacion.es/wp-content/uploads/2022/11/Page10.jpg" class="attachment-full size-full" alt="" srcset="https://gravitacion.es/wp-content/uploads/2022/11/Page10.jpg 1904w, https://gravitacion.es/wp-content/uploads/2022/11/Page10-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2022/11/Page10-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2022/11/Page10-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2022/11/Page10-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2022/11/Page10-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2022/11/Page10-1080x1418.jpg 1080w" sizes="auto, (max-width: 1904px) 100vw, 1904px" /></a>
<a href='https://gravitacion.es/wp-content/uploads/2022/11/Page11.jpg'><img loading="lazy" decoding="async" width="1904" height="2500" src="https://gravitacion.es/wp-content/uploads/2022/11/Page11.jpg" class="attachment-full size-full" alt="" srcset="https://gravitacion.es/wp-content/uploads/2022/11/Page11.jpg 1904w, https://gravitacion.es/wp-content/uploads/2022/11/Page11-228x300.jpg 228w, https://gravitacion.es/wp-content/uploads/2022/11/Page11-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2022/11/Page11-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2022/11/Page11-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2022/11/Page11-1560x2048.jpg 1560w, https://gravitacion.es/wp-content/uploads/2022/11/Page11-1080x1418.jpg 1080w" sizes="auto, (max-width: 1904px) 100vw, 1904px" /></a>

<hr>
<h2>Ejemplos</h2>
<h4>Vectores tangentes para una superficie regular</h4>
<figure class="wp-block-gallery columns-1 is-cropped">
<ul class="blocks-gallery-grid">
<li class="blocks-gallery-item">
<figure><a href="https://gravitacion.es/wp-content/uploads/2020/10/Page1-2.jpg"><img loading="lazy" decoding="async" width="1905" height="2500" src="https://gravitacion.es/wp-content/uploads/2020/10/Page1-2.jpg" alt="" data-id="1432" class="wp-image-1432" srcset="https://gravitacion.es/wp-content/uploads/2020/10/Page1-2.jpg 1905w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-229x300.jpg 229w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-780x1024.jpg 780w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-768x1008.jpg 768w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-1170x1536.jpg 1170w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-1561x2048.jpg 1561w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-1200x1575.jpg 1200w, https://gravitacion.es/wp-content/uploads/2020/10/Page1-2-600x787.jpg 600w" sizes="auto, (max-width: 1905px) 100vw, 1905px" /></a></figure>
</li>
<li class="blocks-gallery-item">
<figure><a href="https://gravitacion.es/wp-content/uploads/2020/10/Page15.jpg"><img loading="lazy" decoding="async" width="1891" height="895" src="https://gravitacion.es/wp-content/uploads/2020/10/Page15.jpg" alt="" data-id="1433" class="wp-image-1433" srcset="https://gravitacion.es/wp-content/uploads/2020/10/Page15.jpg 1891w, https://gravitacion.es/wp-content/uploads/2020/10/Page15-300x142.jpg 300w, https://gravitacion.es/wp-content/uploads/2020/10/Page15-1024x485.jpg 1024w, https://gravitacion.es/wp-content/uploads/2020/10/Page15-768x363.jpg 768w, https://gravitacion.es/wp-content/uploads/2020/10/Page15-1536x727.jpg 1536w, https://gravitacion.es/wp-content/uploads/2020/10/Page15-1200x568.jpg 1200w, https://gravitacion.es/wp-content/uploads/2020/10/Page15-600x284.jpg 600w" sizes="auto, (max-width: 1891px) 100vw, 1891px" /></a></figure>
</li>
</ul>
</figure>
<p><!-- 

<fieldset>

<legend>Observación. Sobre el abuso de notación</legend>




<p>En este ejemplo hemos trabajado con una parametrización de tercero <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-6b03c72ff79968d4e247f565894f4996_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#88;&#40;&#117;&#44;&#118;&#41;" title="Rendered by QuickLaTeX.com" height="31" width="93" style="vertical-align: -7px;"/>. En ella, el objeto <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-a93b6b8871731871e6bbb454dea26266_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;" title="Rendered by QuickLaTeX.com" height="16" width="22" style="vertical-align: 0px;"/> tiene dos roles bien diferentes:</p>




<p>1) de un lado, <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-a93b6b8871731871e6bbb454dea26266_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;" title="Rendered by QuickLaTeX.com" height="16" width="22" style="vertical-align: 0px;"/> representa <strong>la primera coordenada</strong> del plano <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-f516a7b3843b5cdf1cc90272250c911c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;&#94;&#50;" title="Rendered by QuickLaTeX.com" height="27" width="35" style="vertical-align: 0px;"/> (mientras que <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-34997c2ad43ede4ca82544c1bab65f6b_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#118;" title="Rendered by QuickLaTeX.com" height="15" width="16" style="vertical-align: 0px;"/> es la segunda). A este respecto, recordemos que dicho plano se llama en tercero el <strong>plano de coordenadas</strong> o <strong>plano coordenado</strong>, y</p>




<p>2) de otro, <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-a93b6b8871731871e6bbb454dea26266_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;" title="Rendered by QuickLaTeX.com" height="16" width="22" style="vertical-align: 0px;"/> representa la función coordenada <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-8cf782e2f3a420290d6515eca335f072_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;&#58;&#86;&#92;&#114;&#105;&#103;&#104;&#116;&#97;&#114;&#114;&#111;&#119;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#82;&#125;" title="Rendered by QuickLaTeX.com" height="22" width="147" style="vertical-align: 0px;"/>, es decir, la primera función coordenada de la carta <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-b2586ac4fe596878a32ec222d6c75ce6_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#92;&#118;&#97;&#114;&#112;&#104;&#105;" title="Rendered by QuickLaTeX.com" height="22" width="20" style="vertical-align: -6px;"/> que, a su vez, es la inversa de la parametrización <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-977525b20b733831b6519668681769a3_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#88;" title="Rendered by QuickLaTeX.com" height="22" width="21" style="vertical-align: 0px;"/>.</p>




<p>En el caso (1), la función <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-a93b6b8871731871e6bbb454dea26266_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;" title="Rendered by QuickLaTeX.com" height="16" width="22" style="vertical-align: 0px;"/> hace el papel de <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-7bfb4b90bf680f252deb3a1fb3dc5845_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;&#94;&#49;" title="Rendered by QuickLaTeX.com" height="27" width="32" style="vertical-align: 0px;"/> en las definiciones anteriores, mientras que en el caso (2) la función <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-a93b6b8871731871e6bbb454dea26266_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#117;" title="Rendered by QuickLaTeX.com" height="16" width="22" style="vertical-align: 0px;"/> es la función coordenada <img loading="lazy" decoding="async" src="https://gravitacion.es/wp-content/ql-cache/quicklatex.com-d6ee36302088c8c4c3062f01eff26e56_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="&#120;&#94;&#49;" title="Rendered by QuickLaTeX.com" height="27" width="28" style="vertical-align: 0px;"/>. Este abuso de notación es muy frecuente y lo usaremos asiduamente para abreviar y economizar. Pronto os acostumbraréis.</p>


</fieldset>

 --></p>
<h4>Coordenadas polares en el plano</h4>
<figure class="wp-block-gallery columns-1 is-cropped">
<ul class="blocks-gallery-grid">
<li class="blocks-gallery-item">
<figure><a href="https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230.jpg"><img loading="lazy" decoding="async" width="1906" height="1606" src="https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230.jpg" alt="" data-id="1434" class="wp-image-1434" srcset="https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230.jpg 1906w, https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230-300x253.jpg 300w, https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230-1024x863.jpg 1024w, https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230-768x647.jpg 768w, https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230-1536x1294.jpg 1536w, https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230-1200x1011.jpg 1200w, https://gravitacion.es/wp-content/uploads/2020/10/Page2-2-e1602579533230-600x506.jpg 600w" sizes="auto, (max-width: 1906px) 100vw, 1906px" /></a></figure>
</li>
</ul>
</figure>
<p>La entrada <a href="https://gravitacion.es/vectores-tangentes/">Vectores tangentes</a> aparece primero en <a href="https://gravitacion.es">Gravitación</a>.</p>
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