Let s start with vectors. Vectors are usually visualized as arrows.
A vector has a length . . . and a direction. We can also specify a vector using a list
of components, or coordinates. This vector s components are 2 and 3. It means its size is 2
along the x-axis, . . . and 3 along the y-axis. In this video series, we ll add the labels of
the axes inside the component list. This will help us later.
Here are a few more example vectors with their component lists.
Vectors reside in what s called a vector space. This one is two-dimensional,
so it has two axes, and each vector is specified using two components.
Here s a three-dimensional vector space. In this space,
there are three axes, and each vector is specified using three components.
Now let s take two 3-dimensional vector spaces. We gave them different colors to
distinguish between them, and also labeled the axes of the red one using capital letters.
And here is an example tensor. It is a 3 by 3 matrix, having a total of 9 components. We
just picked some random numbers for this example, but here too we added a label to each component,
same as we did with the vectors. So let s start explaining what these labels mean.
Take this one for example: x-y. It means this component is associated with the x-axis of the
blue space, . . . and the y-axis of the red space. It actually has some deeper meaning,
but for now let s just say it s associated with them. We ll gradually clarify this
relationship as we continue. Similarly, this one is associated with the z-axis of the blue space,
and the x-axis of the red space, . . . and we have here all nine possible combinations.
Now let s put it to use with an example.
Here s a tiny metal cube. Let s attach springs to it, . . . and pull . . .
This stretches our cube, . . . though the effect is too small to be shown here.
Now we re compressing it. We can independently stretch or
compress the other faces, . . . but we always make sure to keep the cube in equilibrium,
so it won t move. If apply a force on one face, the opposite face gets the opposite force.
We can also pull this way. This generates what s
called a shearing force. Such a force attempts to deform the cube like this.
Of course in reality the cube will simply rotate. But we want our cube to be in equilibrium:
it shouldn t move and it shouldn t rotate. So if we want to apply such a force,
we ll need to balance it with another shearing force acting on the top face.
The forces acting on the cube faces are called stress forces. There are many ways to combine them
while keeping the cube in equilibrium. Now let s model this using what
s called a stress tensor. We ll use two vector spaces again.
We can think of the blue one as responsible for selecting a face.
The red one will specify a force.
And here s an example stress tensor. The first row is entirely labeled with blue
x s. This means it is associated with the face pointing in the x direction.
And it contains the three components of force . . . this is the force acting on this face.
The second row is entirely labeled with blue y s. So it similarly defines the
force acting on the top face . . . And the last one on the front face.
We assume the forces are balanced, to prevent motion,
so the opposite faces get the opposite forces.
Stretch and compression forces are specified by the diagonal of the matrix. That s because the
direction of the face and the direction of the force are the same. A negative value indicates
a compression force, and the arrows face inward. Shearing forces are specified by matrix elements
that are off-diagonal. Here s a pure one. Recall that we need to balance them, to prevent rotation.
This makes the matrix symmetric around the diagonal.
Any symmetric matrix represents a valid stress tensor.
This cube is not really an ordinary object. It s actually a small region inside an object,
such as a beam of a bridge. This beam is under stress due to the load it carries.
Our tensor describes the stress forces felt at a particular small region inside it.
Different regions may experience different stress forces, so each one has its own stress tensor.
The regions are actually infinitely small, so each stress tensor describes one point in space.
Now let s focus on the original point we started with.
Of course it s hidden inside the beam and we can t directly observe it. But
imagine we could slice the beam open, . . . and peek at the stress forces inside.
Let s mark the point our tensor describes. Since we revealed the face pointing in the x direction,
we ll see the stress force specified by the first row of the matrix.
And now we see the force specified by the second row, . . . and now the third . . .
But what if we slice diagonally? This slice is not aligned with our axes,
but our choice of axes is arbitrary. So this slice is as good as any,
and we should get some stress force here too. Our tensor gives us this information as well,
according to the following rule. We exposed a face pointing in the direction of this vector.
Its components are 0.7, 0.6, and 0.4. The stress force acting on this slice
is the weighted sum of the three forces specified by our matrix.
So we need to take the first row . . . scale it by 0.7, which means multiplying each component by
this value. . . scale the second row by 0.6 . . . and the third by 0.4 . . . then add them together.
And this is the force acting on our slice.
This computation is the same as vector-matrix multiplication.
So now let s try a new visualization. Previously we ve drawn the three stress forces specified by
the rows of our matrix on the faces of a cube. Now let s draw the same forces on a sphere. We can
compute forces in other directions using the rule we just saw, as a weighted sum of these three.
Here is for example the direction we just saw. And this is the force we computed it to be. And here s
another example. Since it s closer to the y-axis, the y component is large, and the stress force
here will be more similar to the force at the top. So let s draw them all around the sphere.
This is a good visual metaphor for what a stress tensor really is. It shows the
stress force felt in every possible direction we can slice the beam.
We ll soon show what it looks like for various values of the matrix, but first
here are some general notes about tensors. Let s return to vectors for a second.
We said we can visualize them as arrows, or specify them using an array of components.
But these components depend on our arbitrary choice of axes. We can choose different ones.
Here s for example a different coordinate system. We ll call
the axes here x and y . Here the vector has different components.
But note our vector didn t change. We just changed the coordinate system and rotated the camera. So
we can write this equality . . . these are two different ways to represent the same vector.
The same thing is true for tensors.
We have a visual representation of a tensor. We can change our point of view,
but this doesn t change the object. And we can also represent a tensor using an array
of components. This representation does depend on a particular coordinate system.
The array describes only three vectors, those that happen to lie on our chosen axes. And they are
broken down into their components using the same coordinate system, ending up with 9 components.
The rest of the vectors can be deduced using the weighted sum rule,
so this array describes the tensor in full. We can choose a different coordinate system . . .
and we ll get different components. We can say this system describes the tensor
from this point of view. And we can write equality here too . . . it s the same tensor.
There are formulas for transforming an array given how our coordinate system has changed.
This is especially useful in relativity, where switching between coordinate systems is crucial.
In this video, we ve only shown simple rotations. Later we ll show scaling and skewing as well,
where things get more complicated. Also, we only constructed tensors over
two vector spaces. These are second-order tensors. Here is a third-order tensor.
It has 27 components, one for each combination of labels from the three spaces. We are yet
to explain the full meaning of these label combinations. A first-order tensor is simply
a vector. And of course everything works with spaces of any number of dimensions, not just 3-d.
Now let s visualize second-order tensors. Let s think of it as a stress tensor for
now. We ll start from an all-zero matrix, and add components of the diagonal one by one.
Each component of the diagonal controls the size of the bulge along one axis.
And we can also have negative bulges.
Now let s add shearing forces off-diagonal components - but we ll keep the matrix
symmetric as required for a stress tensor. The forces that lie on the axes are slanted now,
because they contain shearing components. But overall, we got the same kind of oval shape,
except the bulges are not aligned with the axes. It can be shown we can always find a
coordinate system that is aligned with the bulges. In this coordinate system, the matrix is diagonal.
The three stress forces here are called principal stresses .
Here are a few more examples of matrices with shearing forces.
Let s now drop the requirement for a symmetric matrix. These
are now general second-order tensors.
This is an antisymmetric matrix. The diagonal is zero,
and the part below the diagonal is the negative of the upper part. It might
remind you of electromagnetic fields, . . . and there is a connection. Electromagnetic
tensors have antisymmetric matrices, but the details here are a bit more complicated.