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A little background of my mathematical understanding/knowledge base may be beneficial to know when answering my question. I am currently in the beginning of my BC Calculus Course (Right now we are learning basic differential calculus concepts). I have spent a hefty sum of time, however, researching different math and physics concepts online that are not offered at my school, such as multivariate and vector calculus, Wave mechanics, Number theory, and quantum mechanics. I realize that this education is spotty at best compared to the rest of you, but I read all of your questions and answers on here and have a burning desire to eventually be able to read and understand about all of these concepts. I have wanted to be a theoretical physicist since 9th grade (I am now starting 12th), but reading about all of this math and physics I have no understanding of makes me worry about whether I have what it takes to have a future in this field. Physics is my passion, so these thoughts often worry me on a deep level.

Enough of that and back to math though, when reading many different answers on here and different articles online, I read about Tensors. I looked into them, and feel like I have a semi-understanding of how they work, but it is hard for me to understand anything above vectors intuitively. If any of you have a different or intuitive way of looking at them, it would be great to hear because Tensors are found all throughout physics.

I apologize for making you wade through my worrisome writing, but I honestly am afraid if I don't learn these concepts soon I won't be able to have the future I've hoped for.

Thank you, I hope you're having a nice day

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This is coming from an applied engineering background and how I first got a hold of tensors. Lets first look at physical vectors to see how they behave and maybe we can start to get some intuition on higher dimensional tensors (though it will be lacking in an abstract sense).

First, a physical vector (arrow in space) has a property that it is the same object in different coordinate systems. For example, if we look at the standard 3D basis (coordinate) we get $\hat{i} = \langle 1, 0, 0\rangle , \hat{j} = \langle0,1,0\rangle, \hat{k}=\langle0,0,1\rangle$. If we consider a new basis, just by negating the old, we get a new coordinate system: $\hat{i}^* = \langle -1, 0, 0\rangle , \hat{j}^* = \langle0,-1,0\rangle, \hat{k}^*=\langle0,0,-1\rangle$.

A vector described in the standard basis, like $\vec{v} = \langle 3, 2, 3\rangle$ would be described in the new basis as $\vec{v}^* = \langle -3, -2, -3 \rangle$. These numbers are simply nominal. The object, the arrow in space, remains the same. The arrow is still pointing in the same direction, its "matter-of-factness" remains the same, it's just described differently.

We can state this property by defining a coordinate transform $A$ and saying: $$ \vec{v}^* = A\vec{v} $$

That is, we have a means to move from one coordinate system to another, while still maintaining the properties of the system. Vectors are first rank tensors.

This property, invariance with respect to coordinate systems, holds for many other physical things. If you consider an infinitesimal cube of material in a mass and consider the stresses on the faces of that cube, then there are nine total stress. The three pressure terms, on each parallel face, and the shear terms on each face (Two types of shear in the other two directions). Imagine on each face, there are three basic ways to push on the face (a basis), $x, y,$ and $z$.

We can represent the state of stress by a $3\times 3$ matrix:

$$\left [ \begin{array}{ccc} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{y} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{z} \end{array} \right ] \overrightarrow{\text{Ex}} \left [ \begin{array}{ccc} 1& -2 & 0 \\ 0.1 & 8 & 0\\ 2&2 & -3 \end{array} \right ]$$

Those numbers are based on the the basis (coordinate system). If I rotate it, the way I name those stress (numerically) is different, but the "matter-of-factness" about the state of stress didn't change. This matrix has the same coordinate invariant property. If we define a special way to change the coordinate system, we can say all second rank tensors follow this rule.

Square Matrices, second rank tensors (given the property explained), have rows and columns. We can point to their elements with two indices $A_{ij}$.

Third rank tensors follow similarly. They contain three unique dimensions to put elements $A_{ijk}$. Think of them like 3D matrices (when you numerically name them). They still maintain coordinate invariance. Their "matter-of-factness" remains unchanged even if their numerical description changes. The physical properties they describe depends on their use.

This can be continued for higher rank tensors continually. They can encode properties of higher dimensional structures, or link other tensors. Tensors can be abstracted to a pure mathematical structure (one which I'm not versed in enough to do it justice). Tensor analysis can lead to understanding physical systems and their properties or describing laws in a much nicer structure than usual. This can be seen from fluid mechanics to general relativity and lots more.

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    $\begingroup$ Thank you. This is exactly what I was looking for, I really appreciate you took the time to do that. $\endgroup$ – Mike H Sep 20 '17 at 13:47

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