Tensor (intrinsic definition)

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the modern <a href="/facts/Component-free/cVuyeeAK">component-free</a> approach to the theory of a tensor views a tensor as an <a href="/facts/Abstract_object/uNshak6S">abstract object</a>, expressing some definite type of <a href="/facts/Multilinear_map/YosSfBKE">multilinear</a> concept. Their  properties can be derived from their definitions, as <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> or more generally; and the rules for manipulations of tensors arise as an extension of <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> to <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>.
</p><p>In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, an intrinsic geometric statement may be described by a <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a> on a <a href="/facts/Manifold/rXPERJtu">manifold</a>, and then doesn't need to make reference to coordinates at all. The same is true in <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, of tensor fields describing a <a href="/facts/Physical_property/ujcMUceW">physical property</a>. The component-free approach is also used extensively in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> and <a href="/facts/Homological_algebra/Jfm8w52l">homological algebra</a>, where tensors arise naturally.
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Tensor (intrinsic definition) open-in-new

Tensor (intrinsic definition)