Can someone please explain the intuition behind tensors? Like an example or something of the similar kind that I should keep in mind reading the theorems about it? I can't visualize it
Thanks in advance!
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asked Feb 20, 2015 at 3:29
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1$\begingroup$ Linear maps $V\to\Bbb R$, linear maps $V\to V$, and bilinear maps $V\times V\to\Bbb R$ are among the basic examples. If you mean tensor fields on a manifold, locally, just make coefficients functions instead of constants. $\endgroup$– Ted ShifrinFeb 20, 2015 at 3:40
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$\begingroup$ originally quantities in physics with multiple indices that changed correctly if a different coordinate system were used. The invariant mathematical definition came a good deal later. $\endgroup$– Will JagyFeb 20, 2015 at 3:40
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1$\begingroup$ Have you looked at this: math.stackexchange.com/questions/10282/…? $\endgroup$– Hans LundmarkFeb 20, 2015 at 7:44
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A possible approach to tensors, is to see them as multi-way (or multi-array). Here is a way to get an intuition of it:
- A set of numbers can be put together (respecting a given order) to obtain a vector (= tensor of order $1$). $$ x_1,\ldots, x_n\in\Bbb R \qquad \to\qquad \begin{pmatrix}x_1\\\vdots \\x_n\end{pmatrix}\in\Bbb R^{n}$$
- A set of vectors (of identical dimension) can be put together (respecting a given order) to obtain a matrix (= tensor of order $2$). $$ \begin{pmatrix}x_{1,1}\\\vdots \\x_{m,1}\end{pmatrix},\ldots,\begin{pmatrix}x_{1,n}\\\vdots \\x_{m,n}\end{pmatrix}\in\Bbb R^{n} \qquad \to\qquad\begin{pmatrix} x_{1,1}&\dots& x_{1,n}\\\vdots & & \vdots \\ x_{m,1}&\dots & x_{m,n}\end{pmatrix}\in \Bbb R^{m\times n}$$
- A set of matrices (of identical dimension) can be put together (respecting a given order) to get a tensor of order $3$. $$\begin{pmatrix} x_{1,1,1}&\dots& x_{1,n,1}\\\vdots & & \vdots \\ x_{m,1,1}&\dots & x_{m,n,1}\end{pmatrix},\ldots,\begin{pmatrix} x_{1,1,k}&\dots& x_{1,n,k}\\\vdots & & \vdots \\ x_{m,1,k}&\dots & x_{m,n,k}\end{pmatrix}\in \Bbb R^{m\times n}\\ \qquad \to\qquad \big(x_{i_1,i_2,i_3}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k}}\in\Bbb R^{m\times n \times k}$$
- A set of tensors of order $3$ (of identical dimension) can be put together (respecting a given order) to get a tensors of order $4$. $$\big(x_{i_1,i_2,i_3,1}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k}},\ldots,\big(x_{i_1,i_2,i_3,p}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k}}\in\Bbb R^{m\times n \times k} \qquad \to\qquad \big(x_{i_1,i_2,i_3,i_4}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k\\ 1 \leq i_4 \leq p}}\in\Bbb R^{m\times n \times k\times p}$$
- etc...
Note that I made here an example for real tensors, but, for example, you can replace $\Bbb R$ by $\Bbb C$ and the idea stays the same.
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$\begingroup$ But then tensors of a given order are a vector space. Does that mean that all tensors are vectors? And vectors are tensors? How are they different, then? $\endgroup$ Jul 6, 2019 at 7:47
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$\begingroup$ @user2316602 Yes tensors of a given order and dimension form a vector space. "Does that mean that all tensors are vectors? And vectors are tensors?" Every finite dimensional real vector space is isomorphic to $\Bbb R^n$. Nevertheless, for instance, matrices are interesting for other reasons than just being vectors (e.g. they represent a linear transformation and can be multiplied). The same goes for tensors, yes they are vectors, but they have much more structure than simply being a collection of numbers. $\endgroup$– SurbJul 6, 2019 at 10:52