In high school, I learned that physical
quantities are either scalers, 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 tener? 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 fon lectures and
although he has not even defined even
once what 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 the 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 Fman, where do we start?
Fman says, "Mah, let's start with a very
familiar problem, ohms 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 Fman
says, "Mahes, 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 Fman asks me a
question, the central question of this
entire video. You ready? He asks Mahes,
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 Fman, you just need
one number. And Fman smirks and says,
"Mahes, that would be true if the
conducting property was the same in all
the direction. 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 Mahes 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 Fman 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
numbers to fully represent conductivity
in general in such an isizotopic
crystals. But Fman 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 vectorally to
find the total current density. And I
can do this for any arbitrary direction
of electric field. So yes, Fineman, I am
sure I need only two numbers in 2D, of
course, three numbers in 3D to fully
represent conductivity in any material.
But Fineman says, Mahesh, nope, you need
more numbers here. You just got
lucky. What do you mean? Fman says,
Mahesh, for a second, forget about
coordinate axis and everything and just
look at what we got. we see that even
though the current 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
fineman. Thank you for noting that. But
it still does not explain why you said I
got lucky. So here's what Fineman 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
xaxis 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 yaxis just happened
to be along that direction. So Fman says
Mesh, 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
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 fman 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
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 a line 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 Fineman 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 comp 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 matrix. It makes perfect
sense now. But now Fman asks, Mahes,
this is in two dimensions. How many
numbers would you need for three
dimensions? Again, you 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 * 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
n² numbers. So there we have it.
Conductivity in its full glory is a
mathematical object that requires
n² numbers in n
dimensions. But fon, what do we call
this thing? And are there other physical
quantities like this? Well, Fman says,
"Mahes, 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 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 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 a same
bucket and call them all something. We
call them
tenszers. And so scalars are just rank
zero tensors. Vectors are rank one
tensor. And so we're going to call
conductivity as rank two tensors.
But of course our next question is are
there more quantities that are ranked
two tensors and how do we again
rediscover them? Fman says Mahesh if you
go back what made conductivity a rank to
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 are
is angular velocity, 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 both, you know, 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 moment of inertias 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 forces and area
always aligned? Well let's see. Well
area is always you know prop
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 what
and all are forcing it. So, clearly
force and area 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 tensor. 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. Finan, can we
intuitively think about even higher rank
tensor 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 piso electric
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
pisoelectric 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 pisoelectric constant
should be? Think about it. It needs to
take in n² values and for each of those
n square values spit out n values
meaning it needs n to the power three
numbers to fully represent it in n
dimensions. Piso electric 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 square inputs and
for each input, it needs to spit out n²
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 it 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 Fineman's approach.
Amazing Fineman, 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 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. It should 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 tenzor 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 spac-time 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 space-time
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
space-time curvature like rigid tensor,
the metric tensor, Einstein's tensor and
even the rank four beast, the reman
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.