![]() ![]() ![]() ![]() Now, a mathematician might say that the definition of a tensor is something more like “a multilinear map from vectors and dual vectors to real numbers”. So, we can simply define a tensor as any mathematical object whose components transform by the transformation law given above. In fact, this often works as the definition of a tensor. A different tensor generally follows the same pattern (there is one of these partial derivatives of the coordinates -terms for each index). Mathematically, the transformation law of the components of a tensor is as follows: Note that this is the transformation law for a tensor that has two downstairs indices. The key property of tensors is that a tensor is always the same in every coordinate system (in a technical sense, we say that a tensor transforms covariantly).įirst of all, what is a tensor anyway? A tensor is simply a “collection of objects” (these objects are its tensor components) whose components transform in a nice, predictable way between coordinate changes, while the tensor itself remains unchanged.Ī nice intuitive way to understand this is by looking at how a vector behaves under coordinate changes (a vector is, in fact, a tensor of “rank 1”): Essentially, tensors are mathematical objects that are used in many areas of physics (in general relativity, for example, because they have some very useful transformation properties) and also in many areas of mathematics (for example, differential geometry). ![]()
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