Hi. I'm Dan Fleisch. When people hear that
the subject of my new students guide is vectors and tensors, a reasonably high
percentage of them have the same question: What's a tensor? My goal for this video
is to take about 12 minutes to answer that question, not using a bunch of
mathematical equations, but instead some simple household objects including
children's blocks, small arrows, a couple of pieces of cardboard, and a pointed stick. I think the very best route to
understanding tensors is to begin by making sure that you're solid on your
understanding of vectors. If you've taken any college-level
physics or engineering, you probably think of a vector is
something like this: an arrow representing a quantity that has both
magnitude and direction, where the length of the arrow is
proportional to the magnitude of the quantity and the orientation of the
arrow tells you the direction of the quantity. This could represent the force of
gravity on an object, or the strength and direction of the Earth's magnetic field,
or the velocity of a particle in a flowing fluid. But vectors can represent
other things as well, such as an area. How does a vector represent area? It's
pretty straightforward: you simply make the length of the vector
proportional to the amount of the area (the number of square meters in the area)
and then you make the direction of the arrow perpendicular to the surface. So in that way this can represent an
area as well. So vectors can represent lots of things. But if you want to take
the step beyond thinking of vectors representing quantities with magnitude
and direction, to understanding that vectors are members of a wider class of
object called tensors, then you have to make sure you understand vector components and basis vectors. If you're even going to think about the
components of a vector, you better get yourself one of these. This represents a coordinate
system - in this case I picked the simplest one with the x-axis the y-axis and
z-axis all meeting at right angles. This represents the Cartesian coordinate
system, and the thing to remember about coordinate systems is they come along
with coordinate basis vectors. You probably ran into these as "unit
vectors" and the thing to remember about these little guys is they have a length
of one. One what? One of whatever the units are that you're going to express the length
of your vector in. The direction of the basis vectors or unit vectors is in the
direction of the coordinate axes, so this might represent the unit vector in the x
direction that's often called "x" with a little hat over it or sometimes "i-hat". That's the x-hat unit vector - it points
in the direction of increasing x coordinate. Likewise the y-hat (sometimes called the
"j-hat") unit vector points in the direction of increasing y, and the z-hat or "k-hat"
unit vector points in the direction of increasing z. Once you have the coordinate system and
the unit vectors in place, now you're in a position to find the components of
your vector. How exactly do you do that? I think it's
easiest to understand how to find vector components if you begin with a vector in
the (x,y) plane, so i'm going to lay this vector in the (x,y) plane at some angle to
the x-axis. In order to find the x- component, I'm going to project this vector onto
the x-axis. In order to find the y- component, I'm going to project this vector onto
the y-axis. And how am I going to do those projections? Here's one way: I've darkened the room
because I want to use this lamp to project the vector onto the x- and y- axes. First I'm gonna shine the light
perpendicular to the x-axis (that is parallel to the y-axis) and look for the
shadow of the vector on the x- axis. That will be the x-component of
this vector. As you can see the shadow of the vector on the x-axis ends right here.
This is the x-component of this vector. If I make the vector have a greater
angle to the x-axis, notice the shadow moves this way - the x-component is
getting smaller. And if I make the vector lie entirely along the x-axis, then the
shadow and the vector are the same length - the x-component is the length of the
vector in that case. Now I've got my lights shining
perpendicular to the y-axis (that is parallel to the x-axis) and the shadow
cast by the vector onto the y-axis gives me the y-component of the vector. Notice
that as I increase the angle to the x-axis and decrease the angle to the y-
axis, the y-component is getting bigger. Another way of visualizing vector
components is to ask yourself: "To get from the base of the vector to the tip
of the vector, how far do I have to go in the x-
direction and how far do I have to go in the y-direction?" In other words how many x-hat (or i-hat)
unit vectors and how many y-hat (or j-hat) unit vectors would it take to get from the base to
the tip of this vector? I can show you this if I get rid of
these axes and just line up some x-hat basis vectors (these are going to go in the x-direction
obviously), and some y-hat basis vectors. So in other words this vector is made up
of about four x-hat plus three y-hat. That means that instead of drawing an
arrow for this vector you could simply say four of these, plus three of these.
And if you want to be complete (since there's no z-component of this vector), zero
of these. That is the same as this. In other words, this is a perfectly valid
representation of that vector, and of course if you know the basis vectors, you wouldn't even have to put these on,
would you? You could simply use these components as your vector. You could
write him in a little array. You could even stack them up, and put a nice set of parentheses around
them. This looks just like the way you see
column vectors written. Of course these three components pertain
only to the vector we had lying on the table a minute ago. To generalize this to
vector capital A, for example, we can replace these components with A sub x, and A sub y, and A sub z. Of course, A sub x is the component that pertains to the x-hat
basis vector, A sub y pertains to the y- hat basis vector, and A sub z pertains to the
z-hat basis vector. Notice that we need one index for each
of these, because there's only one directional indicator (that is one basis
vector) per component. This is what makes vectors "tensors of
rank one" - one index, or one basis vector per component. By the same token, scalars can be
considered to be tensors of rank zero, because scalars have no directional
indicators, therefore need no indices. Those are
tensors of rank zero. I'll see in a minute why it's so
powerful to represent tensors as this combination of components and basis
vectors, but first I want to show you some examples of higher-rank tensors.
This is a representation of a rank-two tensor in three-dimensional space. Notice that instead of having three
components and three basis vectors, we now have nine components and nine sets
of two basis vectors. Notice also that the components no longer have a single index, they have two indices. Why might you need such a representation? Consider for example the forces inside a
solid object. Inside that object you can imagine surfaces whose area vectors point in
the x- or in the y- or in the z-direction. And on each of those types of surface,
there might be a force that has a component in the x- or in the y- or in the
z-direction. So to fully characterize all the possible forces on all the possible
surfaces, you need nine components, each with two indices referring to basis vectors. So for example A sub xx might refer to
the x-directed force on a surface whose area vector is in the x-direction, A sub
yx might refer to the x-directed force on a surface whose area vector is in the y-
direction, and so forth. This combination of nine components and
nine sets of two basis vectors makes this a rank-two tensor. This is a representation of a
rank-three tensor in three-dimensional space: 27 components each pertaining to one of
27 sets of three basis vectors. I'll zoom in a little bit here so you can see the
components better. Notice that now each component has three
indices: A sub xxx pertains to three x basis vectors, A sub xyx pertains to two x and one y basis vector, and so forth. This entire front slab has x as the third index and
pertains to these nine sets the basis vectors. The middle slab all has y as
the third index and pertains to these nine, and the back slab all has z as the
third index and pertains to those nine. So in three-dimensional space 27 components, 27 sets of three basis
vectors, and three indices on each component. You may be wondering what is it about
the combination of components and basis vectors that makes tensors so powerful. The answer is this all observers, in all
reference frames, agree. Not on the basis vectors, not on the compliments, but on
the combination of components and basis vectors. The reason for that is that the basis
vectors transform one way between reference frames, and the components
transform in just such a way so as to keep the combination of components and
basis vectors the same for all observers. It was this characteristic of tensors
that caused Lillian Lieber to call tensors "the facts of the universe". Thanks very much for your time. (Subtitles bei Majestik Moose)
I get why he took some time to give a primer about vectors, their components, and so on. It was something a lot of people already know super-well, but it was something many people don't really know as well as they think, and the fundamentals are good.
I get what he meant by "tensor of rank 0" for a scalar, and "tensor of rank 1" for a vector which can be stored in a 1-dimensional array. He definitely started to pick up the pace here, but since I'm from a software background it made sense here. To store a tensor of rank 3 I would need a 3-dimensioned array for all the coefficients.
But I think he really whooshed through the last parts. The kinds of math that can be answered by this. I just starting thinking of maybe FEA cells and fluid dynamics, still just rolling it around in my head a little, and foop, he was wrapping up his "independent of any observer" truth explanation.
Need to understand more of that.
This gives the unfortunate component-based understanding of vectors and tensors.
I think this is kind of like his preface for the book "A Student's Guide to Vector's and Tensors" since he links to the video on the book's site. Easily the most reader friendly book on the subject, but not the most thorough.
Conceptualizing tensors this way gets the job done, but it misses a lot of the beauty and utility. The best explanation I've seen explains tensors as linear operators. If you try to extrapolate this conceptualization to spinors, you're going to have a hell of a bad day.
Didn't watch the whole video but it pretty much just adds a dimension, compared to a matrix, am I right?
While this was an interesting video, I have a couple complaints. First, while the material itself was fine (if a bit simplistic), it was hard to get over his tone of voice. I felt like he was talking down to me the entire time, which isn't a good feeling and actively dissuades me from continuing to listen.
Second, I vaguely understand what a tensor is now, but I don't understand how they're really used or what they really represent. The brief example of "forces in an object" was intriguing, but he left it at that without going into why we would represent the forces in that way.
I was hoping that by the end of 12 minutes I'd have a high level understanding of the purpose of tensors, but instead I had to wait through about 9 minutes of "what are vector components" just to get to an incomplete description of what a tensor is (scaled combinations of basis vectors) and a handwavy statement about the "truths of the universe" which doesn't make sense to someone who knows nothing about tensors.
Don't get me wrong, I appreciate the folk who take the time to explain important concepts in math and science to the uninitiated, but this particular video felt lacking to me.
Isn’t the idea just breaking up the internal forces into components so we can add / subtract / multiple easily? It’s the same idea when you break up external forces into component vectors.
It’s the same as we’ve always learned with i,j,k vectors, dot and cross product, its the tensor notation that confuses, since now you are using linear algebra to do the math.
Maybe I’m wrong but when i took advanced mechanics we learned tensor notation but i already learned the concepts using the older vector notation. I struggled with the notation more than the concepts of the class.
The professor decided to teach it with tensors because that was the “new” method and any further courses would be based on it, so he felt inclined to introduce it to us. Be he said it was the same idea of breaking up into components that we all knew.
I have no idea what I just learned. But hey, at least I know it now.