I’ve been working thru Leonard Susskind’s The Theoretical Minimum course, and one component I’ve discovered spirited is the ubiquity of tensors – they appear to pop up everywhere in physics. I’ve been making an try to construct some intution within the support of what makes them so extensively applicable, and I wished to part my notes on this within the hopes that others would possibly well well well also also fetch this basic. I’d also welcome any insights or corrections.

Most typically, a tensor is defined as being the rest that transforms like a tensor. Namely, a tensor $T$ is a bunch of numbers, one for every coordinate (in regardless of coordinate blueprint you’re the usage of), such that as soon as you happen to turn coordinates from $x$ to $x’$, the ingredients of $T$ (i.e., the numbers that fabricate up $T$) remodel per the tensor transformation law.

[T’^nu=frac{delta x’_nu}{delta x_mu}T^mu]

Where $nu$ and $mu$ imprint indices of $T’$ and $T$ (each of fallacious 1 – this isn’t the more classic transformation law), respectively.

I don’t deserve to dig into this too much for now, because even supposing this is the definition that tends to be presumably the most beneficial in practice, I haven’t discovered it too helpful for building more intuition for the place tensors come from. That said, this definition does show one thing wanted – transformations.

Let’s initiating by asking the query – why would a transformation property be the defining property of an object, i.e., why would we prefer some object $T$ (a tensor) to remodel if the underlying coordinates broken-all of the system down to stipulate that object switch?

The foundation right here is that we deserve to checklist an object that doesn’t depend on the coordinates that somebody chooses to manufacture measurements. If I fabricate measurements in polar coordinates and also you fabricate them in cartesian coordinates, we would possibly well well well also simply tranquil each be in a situation to compose calculations and come in up with the same consequence – the felony guidelines of physics would possibly well well well also simply tranquil not depend on programs of coordinates. So if I fabricate some measurements in my coordinate blueprint and then switch those coordinates pretty, then my measurements, the ingredients of $T$, would possibly well well well also simply tranquil switch in a capability such that they constantly checklist the same object.

Non-public of converting items. Suppose that you just weigh an object and fetch that it weights 100 kilograms, and also you’d snatch to convert this to pounds. If we have the items (kilograms/pounds) as a coordinate, then in a capability we’re altering the coordinate blueprint broken-all of the system down to manufacture our dimension. Nonetheless, converting items clearly doesn’t fabricate the object lighter, so as to ‘compensate’ the incontrovertible fact that pounds are a smaller unit than kilograms, we don’t factual depart the 100 there – we multiply it to be sure that the classic dimension doesn’t switch. This offers some notion of how transformations assist be sure we’re coordinate invariant.

So by having a seek on the tensor transformation law given above, it seems to be cheap to guess that there would possibly be about a notion of sameness fervent. But why that person transformation law – why need to tensors remodel in that implies in repeat to withhold regardless of object the tensor represents?

It seems to be that there would possibly be yet every other definition of tensors that is same to the one given above. Namely, a tensor is one thing that takes a bunch of vectors and spits out a staunch number. This tranquil sounds reasonably abstract, so let’s drill loyal into about a examples to stare why converting vectors into staunch numbers would possibly be a basic component to total.

Let’s initiating with measurements of distance. Suppose that you just stare an object pass from situation $x_0$ at $t_0$ to $x_1$ at $t_1$. From your point of view, $dt$, the switch in time is $t_1 – t_0$, and $dx$, the switch within the placement of the object, is $x_1 – x_0$. We can construct a vector out of these measurements: $initiating up{bmatrix} dx dt cease{bmatrix}$.

Now stammer that there’s yet every other observer also having a seek at that object, however that observer is consuming at practically the roam of light. When this observer tries to construct a vector recording their measurements, they’ll file a much smaller charge of $dt$, since clocks slack down as you capability the roam of light, and also a smaller charge of $dx$ since distances contract.

The problem for us is that we prefer a capability of formulating the felony guidelines of physics that would possibly well well well also simply also be broken-all of the system down to give the same results regardless of the body of reference measurements are made in. Right here’s tricky – since two observers can gape entirely a lot of values of $dx$ and $dt$ for the same object!

There’s an enticing consequence in special relativity that claims that for any measurements of $dx$ and $dt$, even supposing two observers would possibly well well well also find entirely a lot of values, they’ll constantly agree on the price of $dx^2 – dt^2$. We call this charge $ds$, or the biggest distance traversed by the object, and it is invariant in all frames of reference. So if we give you felony guidelines that biggest depend on $ds$, then they’ll also be authentic in all frames of reference!

Electromagnetism is yet every other spirited instance of a case the place observers can file entirely a lot of measurements reckoning on their body of reference. For occasion, have a pair of magnet consuming relative to a wire. From the point of stare of the wire, the consuming magnet creates an electrical discipline, which creates a force on the wire, producing a fresh.

On the a lot of hand, from the point of stare of the magnet, the magnetic discipline created by the magnet interacts with the consuming wire to manufacture a force on the wire, increasing a fresh. So reckoning on the body of reference, it’s both an electrical or magnetic discipline that’s doing the work. The wire biggest experiences an electrical discipline, whereas the magnet thinks that it’s factual a magnetic discipline.

All every other time, it seems to be there’s a capability to bring these two a lot of descriptions of the same fact collectively. It seems to be that $textbf{E}^2 – textbf{B}^2$, the place $textbf{E}$ is the electrical discipline and $textbf{B}$ is the magnetic discipline, is constantly the same in every coordinate body.

So there’s the same pattern again. Observers can file a lot of measurements of time and distance, however agree on the price of $ds$. Observers can file a lot of measurements of the electrical and magnetic fields, however all agree on the price of $textbf{E}^2 – textbf{B}^2$. There are more examples of this – vitality and momentum, and the metric tensor – factual to call about a.

Right here’s the core advise that tensors enable us to resolve – they checklist a rule that maps vectors, or sets of ‘connected’ measurements (for our functions), to one thing known as a scalar – a bunch that every observers agree on. A tensor hyperlinks the ingredients of those vectors to one thing that is more classic. We can then construct physical felony guidelines that elevate this property into legend, and these felony guidelines will then be applicable in all frames of reference.

Introducing some notation, we checklist a tensor $T$ as a blueprint between a bunch of vectors $V$ to a scalar, which is a staunch number:

[T : V times V times dots to textbf{R}]

We’re tranquil lacking one key portion of the puzzle. It factual so occurs that most mappings between vectors and scalars that we fetch in physics are linear. This completes the definition of tensors – they aren’t factual any rule for taking vectors and spitting out scalars, however also they’re linear. This property has essentially spirited implications for how we work with tensors.

Let’s initiating with our definition of a tensor – a linear rule for mapping a rep 22 situation of vectors to a staunch number. For now, let’s factual fill in mind the case the place $T$ factual ‘takes’ two vectors, even supposing the following dialogue will apply to any selection of enter vectors. It helps to have $T$ as a machine that takes two vectors and spits out a staunch number.

Now let’s strive to construct a fresh machine given the one now we fill already acquired, $T$. Let’s call this fresh machine $T’$. The capability this machine will work is that as soon as we first construct it, we’ll ‘store’ or ‘veil’ some vector, let’s call it $V_1$, inner it. It also has its luxuriate in duplicate of $T$.

Then, at any time when this machine is given yet every other vector (of form $V$), let’s call this vector $V_2$, this would possibly well maybe well factual elevate $T$, feed it $V_1$ (which used to be hidden inner $T’$) and $V_2$. $T$ takes two vectors, so this would possibly well maybe well output a scalar, which $T’$ will then output as successfully.

So starting from $T$, a machine to blueprint two vectors loyal into a scalar, we’ve built a fresh machine $T’$ that maps a single vector loyal into a scalar. Internally, this fresh machine is factual the usage of $T$ and a hidden vector, however from the initiating air, it seems to be like one thing fresh – it takes a single vector and maps it to a scalar. Which is the definition of a tensor! So starting from a tensor that took two parameters, now we fill constructed a fresh tensor that takes a single parameter.

There’s a tremendous capability to categorize tensors. The unique tensor, $T$, is taken into legend a (2, 0) tensor since it takes two arguments. The fresh tensor we constructed is named a (1, 0) tensor. So factual to recap:

[T : V times V to textbf{R}]

[T’ : V to textbf{R}]

But what about the 2d number in that representation – the place’s that 0 coming from? It seems to be that the fresh tensor we built, the (1, 0) tensor, is an component of a vector space – so it’s a vector. A brand fresh more or much less vector, that is a lot of from the forms of vectors that $T$ takes – let’s call it $U$. So $U$ is two issues: it’s a tensor ($V to textbf{R}$), and it is mostly a vector.

[U : V to textbf{R} tag{1}]

Up to now what we’ve carried out is taken $T$ and broken-down it to construct a $U$ vector. But what does $U$ deserve to total with $V$ and $T$?

Now comes the wanted step – the vectors that $T$ consumes, $V$, will also be regarded as a rule that would possibly well well elevate a $U$ vector and blueprint it to a scalar by making affirm of itself to it! So a $V$ can elevate a $U$, affirm equation $(1)$ and factual feed itself as enter into the $U$, then output the scalar. So $V$, which used to be a vector, would possibly be a tensor (i.e. a blueprint between $U$ and $R$):

[V : U to textbf{R} tag{2}]

So $V$ takes a $U$ – which is a recipe to blueprint a $V$ loyal into a scalar – and then applies it to itself, giving a scalar. Right here’s also a tensor! This would possibly well maybe well well be regarded as a (0, 1) tensor – because it maps a single $U$ vector to a scalar.

There’s a symmetry between $(1)$ and $(2)$. Given a $U$, we can convert it to a $V$, and vice versa. $U$ is named a dual vector or a covector of $V$.

This symmetry leads us to a remaining modification within the definition of tensors. No longer biggest can a tensor elevate vectors $V$ as enter, however also covectors. So any tensor $T$ is a linear blueprint between a bunch of vectors and covectors to a scalar:

[T : V times V times dots times U times U times dots to textbf{R}]

Lastly, it seems to be that any tensor – a linear blueprint from vectors to scalars – also follows the transformation felony guidelines given within the origin of this post. So starting from two requirements:

• We prefer a capability to blueprint a lot of sets of physical measurements to a charge that all individuals can agree on.
• The above mapping would possibly well well well also simply tranquil be linear.

We’ve built up a capability of systematically categorizing these mappings – or tensors, with the tremendous property that even vectors would possibly well well well also also be regarded as tensors and tensors would possibly well well well also also be regarded as vectors. Right here’s what makes tensors so considerable – they elevate a pretty runt rep 22 situation of ‘requirements’ and provide a capability to manipulate objects that match those requirements (thru the tensor transformation felony guidelines), and also provide a capability to construct and decompose tensors from a lot of tensors – making the abstraction your entire more basic, as these same transformation felony guidelines would possibly well well well also also be broken-down repeatedly again.