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FloatHeadPhysics Physics Science

I never intuitively understood Tensors...until now!

FloatHeadPhysics rebuilds the tensor from scratch by staging a dialogue with Feynman, who never defines the word but asks one question: how many numbers do you need to fully describe conductivity? Starting from Ohm's law in its vector form and an anisotropic crystal that conducts one way and not the other, the count goes from one number to two to four, then to nine in 3D and n squared in n dimensions, which is exactly what makes conductivity a rank two tensor. The pattern generalizes into a rank ladder (scalars at n to the zero, vectors at n to the one) and uncovers more rank two tensors in moment of inertia and stress, plus rank three and rank four beasts in piezoelectricity and elasticity. It closes by defining a tensor as a coordinate independent multilinear machine and showing why Einstein's field equations must be written in rank two tensors.

Published May 16, 2025 23:24 video 20 min read Added Jun 16, 2026 Open on YouTube →

At a glance

A tensor is one of those words that physics throws at you with a definition that explains nothing: "it is kind of like a matrix," or "think in terms of indices." Mahesh Shenoy of FloatHeadPhysics read Feynman's lectures on the subject and noticed something strange: Feynman never once defines the word, yet by the end of the chapter it all clicks. This video reconstructs that experience. The whole thing is a staged dialogue between Mahesh and Feynman, pretending to know nothing, rediscovering the object from scratch.

The entry point is Ohm's law rewritten as current density proportional to electric field, J proportional to E, with conductivity as the proportionality constant. Feynman's one question drives everything: how many numbers do you need to fully describe conductivity? The naive answer is one. Then it becomes two, for an anisotropic crystal like graphite. Then, after one decisive move, rotating the coordinate axes off their lucky alignment, it becomes four. The pattern generalizes to nine in 3D and n squared in n dimensions, and that single counting fact is what a rank two tensor is.

The page below walks the same path, in order: Ohm's law in vector form, the crystal that conducts one way and not the other, the "two numbers" trap and why it was luck, the rotation that exposes the four numbers, why that is exactly matrix multiplication, the rank ladder from scalars to vectors to tensors, three more rank two tensors hiding in plain sight (moment of inertia, stress), the rank three and rank four beasts, the clean two rule definition, and why Einstein's field equations cannot be written without tensors.

Starting from Ohm's law, in vectors

Everyone meets Ohm's law first as I proportional to V: the current through a wire is proportional to the voltage across it. Feynman's first move is to upgrade it to its local, vector form. Instead of total current I, use current density J, the current per unit area. Instead of voltage V, use the electric field E. The law becomes:

J proportional to E.

It is the same law, just written point by point inside the material rather than end to end across a wire. And it makes physical sense. Push a stronger electric field through an area and the electrons there feel a stronger force, drift faster, and you get a stronger current. The proportionality constant is the conductivity, written sigma. A bigger sigma means a better conductor. So far this is freshman physics.

Then Feynman asks the question the entire video hangs on: how many numbers do you need to fully describe conductivity?

The instant answer is one. If sigma is three, then you get three units of current density per unit of electric field, done. One material, one number. That is the trap, and it springs slowly over the next ten minutes.

The crystal that conducts one way and not the other

One number is only enough if the material conducts the same in every direction. Feynman points at a counterexample: graphite. Graphite conducts beautifully along its sheets of carbon and poorly across them. Its conductivity depends on direction. Materials like this are anisotropic, the opposite of isotropic (same in all directions).

Take an extreme idealized anisotropic crystal. Along the horizontal it conducts perfectly; along the vertical it does not conduct at all. Put an electric field along the horizontal and you get a healthy current density. Put the same field along the vertical and you get nothing. So one number clearly cannot capture this. Mahesh's fix is the obvious one: use components. Two numbers, one per axis. Along x the conductivity is, say, three units; along y it is zero. In three dimensions you would use three numbers, one per axis. Problem solved, or so it seems.

Figure 1. The anisotropic crystal at the heart of the video. It conducts along the horizontal layers and not across them. Push the field straight along a layer and the current follows it. Push it across the layers and nothing flows. Push it diagonally and only the horizontal piece survives, so the current density comes out purely horizontal, pointing in a different direction from the field. That misalignment is the seed of the whole tensor idea.

"Are you sure?" The two number trap

Feynman pushes. Are two numbers really enough? Test it with an arbitrary field direction. Decompose the electric field into its x and y parts. Use sigma_x on the x part to get the current along x. Use sigma_y on the y part to get the current along y, which here is zero. Add the two vectorially and you have the total current density for any field direction. It works for every angle. Mahesh declares victory: two numbers in 2D, three in 3D, fully general.

Feynman: nope. You just got lucky.

The tell is hiding in plain sight. Forget the axes for a second and just look at the result. The current density is proportional to the electric field, yet the two vectors are not pointing in the same direction. For a diagonal field, the field tilts but the current comes out horizontal. That is genuinely new. Every vector equation up to this point in a physics education has the two sides aligned. In F = ma, force and acceleration point the same way. In p = mv, momentum and velocity point the same way. That is precisely why mass is a scalar, a single number, in those laws. Here, for the first time, is a proportionality where the output vector is twisted away from the input vector.

Why was that luck? Because of how the axes were chosen. The horizontal happens to be the one special direction where field and current line up. The vertical happens to be the other special direction, where the current vanishes. The coordinate axes were quietly parked exactly on top of those two special cases, which is what collapsed everything to two clean numbers. It was not generality. It was a rigged stage.

The climax: rotate the axes and watch the numbers multiply

So strip away the luck. Rotate the coordinate axes to an arbitrary angle, say aligned with the electric field itself, and redo the bookkeeping honestly. Now E points along the new x axis, but the current density J does not. J is tilted. So the single number sigma_x is no longer enough to say how much current you get, because the field along x is producing current in two directions at once.

This is the moment the video stops and tells you to pause. The honest count goes like this. The x component of the field, E_x, produces a current component along x, call the constant sigma_xx. But the same E_x also produces a current component along y, because J is tilted off the field. That needs its own constant, sigma_yx. So one component of the field already demands two numbers. Earlier, on the lucky axes, J had no y component, so sigma_yx was zero and invisible. That hidden zero is exactly what Feynman meant by "you got lucky." It was always there. It just happened to vanish on the special axes.

Now do the same for the y component of the field. E_y produces a current along x (sigma_xy) and a current along y (sigma_yy). Two more numbers. Total: four numbers to fully describe conductivity in two dimensions, not two.

Figure 2. The decisive count. On honest, rotated axes the field along x spills current into both x and y, so it takes two numbers (sigma_xx and sigma_yx) just for E_x. The field along y does the same, adding sigma_xy and sigma_yy. Four numbers total in two dimensions. The off diagonal terms are the ones that were secretly zero when the axes were parked on the crystal's special directions, which is why the naive count missed them.

Why this is exactly a matrix

Stack the four numbers into a grid and the bookkeeping turns into something instantly recognizable. J_x and J_y on the left, the four sigmas in a two by two block, E_x and E_y on the right. That is matrix multiplication, exactly. This is the payoff for the textbook line that conductivity "is kind of like a matrix." It is not a definition handed down from nowhere. The matrix is forced on you the moment you accept that each component of the field feeds every component of the current. The grid is just the natural ledger for "input component, output component."

Now scale the dimension. In three dimensions there are three components of current density to compute, and each one collects a contribution from all three field components. Three outputs times three contributions each is three times three, nine numbers. And the abstraction mathematicians love: in n dimensions there are n output components, each needing n numbers, for a total of n squared. Conductivity, in full generality, is an object that takes n squared numbers to pin down in n dimensions.

The rank ladder: scalar, vector, tensor

That n squared is the key that unlocks the word. Look back at the simpler objects through the same lens, counting numbers as a function of dimension.

A scalar like mass or energy needs exactly one number, no matter the dimension. One in 2D, one in 3D, one in any n. In other words, a scalar needs n to the power zero numbers.

A vector like velocity needs two numbers in 2D, three in 3D, n in n dimensions. So a vector needs n to the power one numbers.

Conductivity needs n squared, which is n to the power two.

The exponent is the only thing changing. That exponent is the rank. Scalars are rank zero tensors. Vectors are rank one tensors. Conductivity is a rank two tensor. They all belong in one family because they are the same kind of object at different ranks, and the rank is literally the power of n that counts their components.

Figure 3. The rank ladder. A tensor's rank is just the power of n that counts its components in n dimensions. Scalars (n to the zero, one number) and vectors (n to the one) are the bottom rungs of the same staircase that conductivity (n squared) and the higher beasts climb. Naming the family "tensors" is what lets a scalar, a vector, and a conductivity matrix sit in one bucket.

What made it a tensor, and where else that happens

Strip conductivity down to the one feature that forced it up to rank two. It was not that current depends on field. It was that J is proportional to E but not necessarily aligned with it. If the two always pointed the same way, conductivity could collapse back to a single number, a scalar. The misalignment is the whole reason it must be rank two. So the hunt for other rank two tensors becomes a hunt for other proportional but misaligned vector pairs.

Moment of inertia. A satellite has momentum p = mv, and momentum and velocity always point the same way, which is exactly why mass is a scalar. But the rotational version is different. Angular momentum equals moment of inertia times angular velocity. Spin a satellite about one axis and about another, and its moment of inertia is generally different about those axes, because inertia depends on the axis of rotation. So even with the same angular velocity, the angular momentum differs from one axis to the next, and in general the angular momentum vector ends up tilted away from the angular velocity vector. Misaligned proportional vectors. Moment of inertia must be a rank two tensor.

Stress. Look at all the forces loading a dam. Slice a small enough cross section and the force on it is proportional to the area. Bigger area, more force. The constant is the stress. Are the force and the area vector aligned? The area vector is always perpendicular to the surface you cut. The net force across that surface can point any which way, depending on what is pushing and pulling. So force and area are not generally aligned. Stress must be a rank two tensor.

Three different corners of physics, conductivity, rotation, and the mechanics of solids, and the same structure surfaces in each: two vectors, proportional, not aligned, glued together by a rank two tensor.

Tensor	Relation	The two vectors	Aligned?
Conductivity	J = σ E	current density, electric field	no, in a crystal
Moment of inertia	L = I ω	angular momentum, angular velocity	no, off axis
Stress	F = σ A	force, area	no, in general
Mass (counterexample)	p = m v	momentum, velocity	yes, always

Figure 4. The signature of a rank two tensor. Every rank two example maps one vector to another that is proportional but tilted away from it; the misalignment is exactly what a single scalar cannot capture. Mass is the control case: momentum and velocity never tilt apart, so mass stays a humble scalar.

Going higher: rank three and the rank four beast

The same move keeps climbing. Rank three shows up in the piezoelectric effect: squeeze certain crystals and they separate charge, literally squeezing electricity out of a rock. The electric field produced is roughly proportional to the stress applied, E proportional to stress. But now the right side is a rank two tensor (stress), and the left side is a rank one tensor (the field). The piezoelectric constant has to take in n squared numbers of stress and, for each, hand back n numbers of field, which is n cubed numbers in all. It is a rank three tensor.

Rank four appears in elasticity. The deformation of a material, its strain, is proportional to the stress applied, strain proportional to stress. Strain, like stress, is a rank two tensor. So this equation maps one rank two tensor to another. The elastic constant (the stiffness tensor) must take in n squared values of stress and, for each, produce n squared values of strain: n to the fourth numbers. A genuine beast.

And the punchline of this whole stretch: none of it required a formal definition. No index gymnastics, no abstract axioms. Pure intuition, counting numbers, rediscovered conductivity, moment of inertia, stress, the piezoelectric constant, and the stiffness tensor in order of rank. That is the thing Mahesh finds beautiful about Feynman's path: the structure shows up before the definition does.

So what the heck is a tensor

Only now does the definition land cleanly, because there is something concrete for it to describe. A tensor is a machine that takes in a vector or a tensor and outputs another vector or a tensor, and its rank is set by the ranks of what goes in and what comes out. But not every machine qualifies. Two rules make it a tensor.

Rule one: the output must not depend on how you draw the axes. Change the coordinate system and the individual tensor components change, the four sigmas take new values, but they change in a coordinated way so that the physical output for a given input stays exactly the same. Nature does not care which way you drew your lines, so the answer cannot either. This is the deep meaning behind "a tensor transforms in a certain way," the line every textbook leads with and never motivates.

Rule two: the relationship is linear, in many components at once. Every example was a proportionality, J proportional to E, L proportional to omega, strain proportional to stress, but the proportionality runs across multiple components rather than through a single number. A tensor is a multilinear machine: linear, but in several slots.

Put together: a tensor is a multilinear machine that maps tensors to tensors, whose output is independent of the coordinate system you choose. Every earlier object obeys this. A scalar is the rank zero case, a vector the rank one case, and the conductivity matrix the rank two case of one single idea.

Why Einstein's field equations need tensors

The last beat cashes everything in on general relativity. How would you even encode the curvature of spacetime? Move a little in the x direction and you have to track how space stretches or squeezes in the y direction, and, independently, in the z direction, and, independently again, in the time direction. For each direction you move, you need several numbers to describe how the geometry responds. One number per direction is not enough; you need a number for every pairing of "direction moved" and "direction stretched." That is precisely a rank two object. So the Einstein field equations, which are statements about spacetime curvature, cannot be written in vectors. They are forced into the language of rank two tensors, the same n squared structure that conductivity revealed at the very start.

Mahesh signs off honestly: this is only one doorway. The full machinery of curved spacetime brings the Ricci tensor, the metric tensor, the Einstein tensor, and the rank four beast itself, the Riemann curvature tensor. All of it, he claims, is open to the same intuition first attack, and he offers to make those videos if people want them.

Key takeaways

A tensor is best understood not by definition but by counting: how many numbers does it take to fully describe a physical quantity in n dimensions. The exponent of n is the rank.
Scalars need n to the zero (one number, rank 0). Vectors need n to the one (rank 1). Conductivity needs n squared (rank 2). They are the same family at different ranks.
The trigger for rank two is misalignment: two vectors that are proportional but do not point the same way. A single scalar can only scale a vector, never tilt it.
The "two numbers are enough" intuition for an anisotropic crystal is an artifact of parking the axes on the crystal's special directions. Rotate to honest axes and the hidden off diagonal numbers appear, giving four in 2D, nine in 3D, n squared in general.
That n squared ledger is literally matrix multiplication, which is why conductivity is written as a matrix. The matrix is forced by "each input component feeds every output component," not handed down arbitrarily.
Moment of inertia and stress are also rank two tensors for the same reason: their input and output vectors are proportional but misaligned. Mass stays a scalar because momentum and velocity never tilt apart.
The piezoelectric constant is rank three (maps a rank two stress to a rank one field, n cubed numbers); the elastic stiffness tensor is rank four (maps rank two stress to rank two strain, n to the fourth numbers).
Formally, a tensor is a multilinear machine mapping tensors to tensors whose output is independent of the coordinate system. Coordinate independence is the rule that makes it physics, not bookkeeping.
Einstein's field equations need rank two tensors because encoding spacetime curvature requires a number for every pair of directions (the direction you move and the direction space responds), which is exactly the n squared structure.

Chapters

Timestamps are clickable. Click one and the player jumps there and keeps playing while you read. These are the creator's own chapter marks.

0:00 What exactly are tensors?
1:23 Analysing conductivity in anisotropic crystals
3:31 Is conductivity a vector? (hint: nope)
5:00 The key idea to understand tensors
7:07 Rotating the coordinate axes (climax)
10:48 Why are tensors written in matrix form
11:50 Conductivity is a rank 2 tensor
14:14 Rank 2 tensors in engineering and astronomy
17:48 Rank 3 and rank 4 tensors in material science
20:29 The most intuitive definition of tensors

Notable quotes

Most definitions I read were pretty vague and then I learned that it is kind of like a matrix but it's better to think in terms of indices. Yeah, none of that made any intuitive sense to me until now. Mahesh Shenoy, 0:20

How many numbers do you need to fully describe conductivity? Feynman, voiced by Mahesh, 1:18

Mahesh, nope, you need more numbers here. You just got lucky. Feynman, voiced by Mahesh, 3:00

For the very first time in my life I am seeing a vector equation in which the left hand side vector and the right hand side vector are not aligned. Mahesh Shenoy, 3:40

This is it folks. This is the moment. Please, please pause. Really pause and try to answer that question. Mahesh Shenoy, 7:30

This is why we write conductivity as a matrix. It makes perfect sense now. Mahesh Shenoy, 10:48

The fact that they're not aligned means the proportionality constant has to be a rank two tensor. Mahesh Shenoy, 13:20

Tensors are multilinear machines that relate one tensor or a vector to another tensor or a vector. And the output is independent of how you choose to draw your coordinate axis. Mahesh Shenoy, 21:50

Resources mentioned

Feynman's lectures on physics, specifically the chapter on tensors that never once defines the word, the inspiration for the whole video. By Richard Feynman.
Ohm's law and its local vector form using current density and electric field, the starting point of the derivation.
Electrical conductivity, the rank two tensor the video reconstructs from scratch, and anisotropy in crystals like graphite.
Matrix multiplication, the operation the four numbers turn out to encode.
Scalars (rank 0), vectors (rank 1), and the general notion of a tensor and its rank.
Other rank two tensors: moment of inertia with angular momentum and angular velocity, and stress in solids.
Higher rank tensors: the piezoelectric effect (rank 3) and linear elasticity with strain (rank 4).
Multilinear maps, the formal heart of the tensor definition.
Einstein's field equations of general relativity and the curvature of spacetime, plus the deeper objects teased for future videos: the metric tensor, Ricci tensor, Einstein tensor, and Riemann curvature tensor.
Brilliant, the video's sponsor, an interactive platform for math, science, programming, data, and AI.
FloatHeadPhysics, Mahesh Shenoy's channel.

Where it stands

This is a teaching video and on the physics it is sound. The counting argument for rank (how many numbers a quantity needs in n dimensions) is a legitimate and well known way to introduce tensors, and conductivity, moment of inertia, stress, the piezoelectric constant, and the stiffness tensor are textbook examples at ranks two, two, two, three, and four respectively. The "machine that takes vectors to vectors, independent of coordinates" picture matches the working definition physicists use.

Two honest footnotes for a careful reader. First, the clean component counts (n squared, n cubed, n to the fourth) are the counts before symmetry is taken into account. Real stress and strain tensors are symmetric, and the stiffness tensor has further symmetries, so the number of truly independent components in elasticity is far below the naive n to the fourth (21 for a general material in 3D, not 81). The video is counting raw slots, which is the right pedagogical move, not the final independent count. Second, "tensor" here means a tensor on a vector space at a point; the full general relativity story also needs tensor fields and how they connect across a curved manifold, which is exactly the deeper material Mahesh flags as future videos. Neither point dents the core lesson. It is a genuinely clarifying rebuild of an idea most courses present as an opaque definition.

Full transcript

In high school, I learned that physical quantities are either scalars, like temperature, which is just a number, or vectors, like velocity, which is a number and a direction. But there's a far more powerful mathematical object that shows up everywhere from engineering to material science and even astronomy. It's called a tensor. To be specific, rank two and higher tensors. But here's a question. What the heck is a tensor? Most definitions I read were pretty vague and then I learned that it is kind of like a matrix but it's better to think in terms of indices. Yeah, none of that made any intuitive sense to me until now. I went through the entire lesson on tensors by Feynman's lectures and although he has not even defined even once what a tensor is in the entire chapter, by the end of it it all clicked and oh my god my mind was blown. So the goal of this video is we're going to pretend we don't know anything about tensors and start from scratch and step by step intuitively rediscover all these ideas ourselves. And if I do this right, we might even be able to intuitively understand why some of the most complex equations of physics like Einstein's field equations of general relativity has to be written in the language of higher order tensors, not just vectors. All right, so if you're ready for this, let's begin. So Feynman, where do we start? Feynman says, "Mahesh, let's start with a very familiar problem, Ohm's law." And I'm like, "That's awesome. I love Ohm's law. It states that current through a wire will always be proportional to the voltage across it." Which means current I is proportional to voltage V. But Feynman says, "Mahesh, let's look at the vector version in that instead of current, we use current density J, which is basically current per unit area, and instead of voltage, we have electric field. So we get J proportional to E. We obviously won't show that it's the same form as I proportional to V. But it makes intuitive sense, right? I mean through an area if you put a stronger electric field, the electrons over there will experience a stronger force. They will drift faster giving you a stronger current. So makes sense, right? And the proportionality constant over here is called conductivity. It's a measure of how good a conductor a material is. If the number is higher, it's a better conductor. But now Feynman asks me a question, the central question of this entire video. You ready? He asks Mahesh, how many numbers do you need to fully describe conductivity? I'm like one, you just need one number, right? I mean, if the conductivity is three units, for example, it means there will be three units of current density per unit electric field. So Feynman, you just need one number. And Feynman smirks and says, "Mahesh, that would be true if the conducting property was the same in all the directions. But what if you have a crystal like graphite which conducts very nicely along its layers, but it conducts very poorly across the layers. Conductivity in graphite depends on the direction. We call them anisotropic crystals. So what if you have an anisotropic crystal over here and for simplicity, let's say along the horizontal the material conducts very well. But let's say along the vertical it doesn't conduct at all. So putting electric field along the horizontal you get nice current density but you put an electric field along the vertical you get nothing. Now Mahesh do you think one number is enough to fully describe conductivity in such a material. I'm like ooh interesting this is getting complicated but nothing we cannot handle. I mean we can think in terms of components right? So here's what I say. All right let's use two numbers one for X and one for Y. So in this case for example along the X the conductivity might be three units but along the Y the conductivity would be zero. So I stand corrected Feynman you're right. I need two numbers to fully represent conductivity and that is in two dimensions. Of course if we're in three dimensions we have three axes. So we would need three in general in such anisotropic crystals. But Feynman asks Mahesh are you sure? Okay, let's double check. What if we put an electric field in some arbitrary direction? Are these two numbers enough to fully calculate the current density in this material? Let's see. If I were to decompose the electric field along the X and Y direction, I can now use this number to calculate the current density along the X direction. And then I can use this number to calculate the current density along the Y direction. In this case, it's zero. And then I can add those two together vectorially to find the total current density. And I can do this for any arbitrary direction of electric field. So yes, Feynman, I am sure I need only two numbers in 2D, of course, three numbers in 3D to fully represent conductivity in any material. But Feynman says, Mahesh, nope, you need more numbers here. You just got lucky. What do you mean? Feynman says, Mahesh, for a second, forget about coordinate axes and everything and just look at what we got. We see that even though the current density is proportional to electric field they are not in the same direction. And I'm like that is interesting because till now every single vector equation I have seen the left hand side vector was exactly in the same direction as the right hand side vector. For example consider F = ma, F and a will always be in the same direction. p = mv where p is the momentum v is the velocity, p and v always in the same direction. And so now for the very first time in my life I am seeing an equation, a vector equation in which the left hand side vector and the right hand side vector are not aligned. That is truly interesting Feynman. Thank you for noting that. But it still does not explain why you said I got lucky. So here's what Feynman says. All right this is the interesting part. Okay so in general in this particular crystal electric field and current density are not aligned isn't it? But along the horizontal they are aligned. That is a special case. And we chose our x axis to be in that special direction. Similarly along the y direction when you put an electric field you get no current density. That's another special case over here. And our y axis just happened to be along that direction. So Feynman says Mahesh, do you see what has happened? We accidentally chose our coordinate axes to align with these very special cases. That's why things got super simplified and it feels like only two numbers are enough. But if you want to analyze this in general, we have to take an arbitrary coordinate axis. So what say we rotate the coordinate axis to some arbitrary angle and then reanalyze the situation. I'm like that is interesting very interesting let's do this. So let's rotate this coordinate axis and align it with some arbitrary direction. Let's align it with this electric field direction itself. So this is our x and this is our y and Feynman says look Mahesh look now even though the electric field is along the x axis the current density is not aligned. So do you think one number sigma x is enough to calculate how much current density you get. What do you think? This is it folks. This is the moment. Please, please pause. Really pause and try to answer that question. How many numbers do you need to calculate current density due to the electric field in just the x direction? Try to figure this out yourself because this is the moment to rediscover it. I promise you it's the best feeling that you will get. It's the best way to learn and it's one of the reasons why I love Brilliant, who's sponsoring this video. Why do I recommend Brilliant? Because just like in this particular video, it helps you intuitively rediscover ideas using a hands-on approach, but not just in math or physics, also in programming, data, and even AI. And you can get started for free. Take their course on vectors for example. It starts off like a puzzle and then using concrete interactive examples, you start visualizing things and pretty soon you'll start mastering all the basic operations, transformations, and eventually you even start applying it. And because they have bite-sized lessons, you can learn on the go and level up even with 15 to 20 minutes a day. I have been urging my students to use Brilliant from way before they started sponsoring me. So, you can try everything that Brilliant has to offer for free for a full 30 days by going to www.brilliant.org/floatheadphysics. The link is also in the description or you can just scan this QR code somewhere. And of course, you also get a 20% off on their annual premium subscription if you choose to buy it. Okay, back to the video. So, how do we represent conductivity? Now, J is not aligned along E. We can decompose J along the X and Y direction giving us JX and JY. So this means look the electric field component along the X direction gives us both JX and JY. It gives us two components of current density. This is the key. Therefore I would need two numbers. One number to represent how much I get JX due to EX. So we can write it this way. But I also need another number to calculate how much I get JY from EX. So look, I need two numbers over here. And earlier because J was aligned along EX, there was only JX. There was no JY. That's why we missed it. That's why that number was zero and it was hiding. This is what Feynman meant when he said I got lucky. We didn't see it. So we need two numbers just to calculate J from EX. And the same thing of course applies to the Y component of the electric field here as well. We need two numbers to calculate J. One number to tell us how much you get JX from EY and another number to tell us how much we get JY from EY. So look in total we need four numbers to fully describe conductivity not two, four. And by the way this itself took me some time to wrap my head around it because a lot is going on over here so at first if you don't get it please don't be disheartened this is not easy stuff mull over it it'll make sense. But the whole idea of why we need four numbers is if I want to calculate JX I need to add contributions from both EX and EY. That's why I need two numbers for JX. And similarly, I need two numbers for JY. And that's why I need total of four numbers, not two. But wait a second. Wait a second. Do you see? Do you see what I'm seeing? This looks like matrix multiplication. If we write it nicely in terms of matrix, it's a perfect matrix multiplication. Oh, this is why we write conductivity as a matrix. It makes perfect sense now. But now Feynman asks, Mahesh, this is in two dimensions. How many numbers would you need for three dimensions? Again, a great idea to pause and try to do this in your head. We have three components of current density to calculate. But for each component of current density, we need three numbers because there are three contributions that we need to add. And therefore we need three times three, nine numbers. And that's exactly what we see in three dimensions. We need nine numbers to fully represent conductivity. And now we can do what mathematicians love to do. Abstract the hell out of it. What about in n dimensions? Never mind what those n dimensions are. But how many numbers would we need if we had n dimensions? Now I have n components to calculate. For each of them, I need n numbers to add up their contributions, which mean I need a total of n squared numbers. So there we have it. Conductivity in its full glory is a mathematical object that requires n squared numbers in n dimensions. But Feynman, what do we call this thing? And are there other physical quantities like this? Well, Feynman says, "Mahesh, think of it this way." Remember scalars like energy and mass, you think of them as something that only has magnitude, one number, right? But think of it this way. Regardless of the number of dimensions, you always need one number to represent a scalar. Which means we could say that scalars require n to the power zero numbers where n is the number of dimensions, right? Okay. Similarly, think about vectors. You might think of vectors as something that has magnitude and direction. But another way to think about it is in terms of the fact that a vector requires two numbers in 2D, one for each component, three numbers in 3D. So in n dimensions it requires n numbers. In other words, in n dimensions it requires n to the power one numbers. And so from that we can see conductivity is just an extension of that. Oh my god. This is this is it. This is why it actually makes sense to club them all together and under the same bucket and call them all something. We call them tensors. And so scalars are just rank zero tensors. Vectors are rank one tensors. And so we're going to call conductivity a rank two tensor. But of course our next question is are there more quantities that are rank two tensors and how do we again rediscover them? Feynman says Mahesh if you go back what made conductivity a rank two tensor in the first place? Well it all boiled down to one and only one thing that J was proportional to E but not necessarily aligned. If it was always in the same direction, conductivity would have been a scalar. But the fact that it's not necessarily always aligned forced conductivity to be a rank two tensor. So the next natural question for us is are there other places where you have vectors proportional to each other but not necessarily aligned. Well, if you look at say a satellite going around the earth, you probably know that it has some momentum and momentum is mass times velocity. But when it comes to momentum, momentum and velocity are always aligned. They're always in the same direction, which is the reason why mass is a scalar. Okay. But what about its rotational versions? We can write rotational momentum or angular momentum equals rotational inertia times angular velocity. Now is angular momentum and angular velocity also always aligned? Let's see. You can spin them in multiple directions. For example, you can spin a satellite this way or you can spin a satellite that way. And the key thing is that moment of inertia is not the same along these directions, moment of inertia changes with the axis of rotation. Okay. Therefore, even if the satellite was spinning with the exact same angular velocity in both these directions, because the moments of inertia are different, the angular momentum would be different. In one case, it will be higher, in the other case it'll be lower. Which means if you add them up together, look, the angular momentum will not be in the same direction as angular velocity in general. This means boom, moment of inertia has to be a rank two tensor. I mean, right now I don't really have the mental capacity to actually go through it and work out all the components, but it's the same as J and E. The fact that they're not aligned means the proportionality constant has to be a rank two tensor. Isn't that amazing? Next up, look at this dam. There are a ton of forces that are acting on it. But if you take a cross-sectional area then in general if this area is small enough the force acting on a cross-sectional area can be written to be proportional to the area. That makes sense right? Bigger the area more the forces. And the proportionality constant is called the stress. Now comes the big question. Are force and area always aligned? Well let's see. Well area is always you know perpendicular to the surface that we choose. And what about the net force? Well, there are multiple forces acting over here. So, the net force can be in any direction, right? It depends on the context and you know what and all are forcing it. So, clearly net force and area need not be in the same direction. And therefore, boom, stress has to be a rank two tensor. The same idea. So, we have so many examples of rank two tensors. We have conductivity, moment of inertia, stress and so on. They all follow the same pattern. They map two vectors. They're proportional to each other, but they're not necessarily aligned in general. They have to be a rank two tensor. But let's not stop here. Let's go one step further. Feynman, can we intuitively think about even higher rank tensors like rank three and rank four? Where would they appear? Well, when you squeeze certain crystals, charges get separated producing an electric field. We are literally squeezing electricity out of crystals. We call this effect the piezoelectric effect. And to a good degree, the electric field generated is proportional to how much you squeeze it or proportional to how much stress you're putting on this. So, we can write electric field is proportional to the stress applied. And of course the proportionality constant is called the piezoelectric constant. But now think about this. Look at this equation. On the left hand side you have a vector which is a rank one tensor. But on the right hand side you have a rank two tensor. We just saw stress is a rank two tensor. So for the very first time I'm now looking at an equation that maps a rank two tensor to a rank one tensor or a vector. And so now what kind of an object should the piezoelectric constant be? Think about it. It needs to take in n squared values and for each of those n squared values spit out n values, meaning it needs n to the power three numbers to fully represent it in n dimensions. Piezoelectric constant is a rank three tensor. And we can go even further if you think about elasticity. You probably know for any material the deformation produced depends on how much stress you put on it. In other words, the strain is proportional to the stress you put. And again, the proportionality constant this time is the elastic constant. And guess what? Just like stress, strain is also a rank two tensor. Meaning in this equation, we are mapping one rank two tensor to another rank two tensor. So the proportionality constant over here, it needs to take in n squared inputs and for each input, it needs to spit out n squared values of strain. So this would be a rank four tensor beast. Look at this point I have absolutely no mental capacity to dig into any of this in detail. Okay. But what's interesting is that even without defining tensors, we haven't even defined what tensors are. We haven't talked about index notations or anything like that. Just from raw intuition, we are able to rediscover all these amazing ideas. And that's what I find fascinating about Feynman's approach. Amazing Feynman, take a bow. But now I think it's time to finally answer our central question. What the heck is a tensor? Can we define it? Yes, we can. You can think of a tensor as a machine. A machine that takes in one vector or a tensor and outputs another vector or a tensor. And the rank of this machine depends on the ranks of the input and the output. But it's not just any machine. It's not just any function. It follows two powerful rules. The first one is that the output should not depend on how you draw your coordinate axis. I mean if you change the coordinate axis the tensor components will change but it should change in such a way that the output should remain the same. You should get the same output for the given input right? Because nature does not care how you draw your lines. And second, we noticed something across all our examples. We had proportionalities. So it was a linear relationship, but not with just one number, but with multiple components. That's what a tensor is. It gives you linear relationships between multiple components. A multi-linear machine. So putting it all together, tensors are multilinear machines that relate one tensor or a vector to another tensor or a vector. And the output is independent of how you choose to draw your coordinate axis. Finally, can we now use everything that we've learned to answer why Einstein's field equations must use tensors? Well, think about how could you encode spacetime curvature? Well, here's a way to think about it. If I move a little bit along the x direction for example, I need to think about how much the space stretches or squeezes in the y direction. But I should also think about how much the space stretches or squeezes in the z direction independently. It can also stretch and squeeze in the time direction which of course I haven't shown over here. So look for each component, each direction that I'm moving, I need multiple numbers to take care of curvature. In other words, I definitely need a tensor. That's why equations of general relativity involving spacetime curvature require tensors. In fact, they use rank two tensors. But the last thing I want to say over here is that we have barely scratched the surface. This is just one way to think about tensors. There's just so much more to talk about and there's so much more to explore when it comes to spacetime curvature like the Ricci tensor, the metric tensor, Einstein's tensor and even the rank four beast, the Riemann curvature tensor. And I think we can intuitively figure all of this out. So let me know if you want to get into all of that interesting stuff. Um, I'll make future videos on that. So that's it. Bye.