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So now, let's start discussion of tensor calculus.
So we'll start by setting the rules, the rules of the game!
That's what chapter 2 is called, "The Rules of the Game".
Where everything's taking place, you have to have a starting point.
You have to start somewhere, with some concepts that are accepted as obvious, or natural.
So for us, that concept will be the concept of space.
Everything's happening in our... the space of our common physical experience.
We'll call it the Euclidean space, without much definition.
So, we all have an intuitive understanding of what our space is, and what we can do with space.
What we can do with space is basically what Euclid did with geometry:
You can draw straight lines, you can draw planes, you can draw circles.
You can talk about distances, we'll do that in a moment.
And we'll also talk about the quote-unquote "2-dimensional space", the plane.
We also understand what the plane is, it's something where you can draw straight lines.
All of these are called Euclidean spaces, that's what we're calling Euclidean spaces.
Some... what we agree on is the space of our everyday physical experience.
Where it is possible to draw straight lines and measure their lengths.
So I'll just give you an example of something that's not a Euclidean space:
A surface of a sphere.
Because we're not able to draw straight lines, I would not call a Euclidean space.
That's a space without straight lines.
You can still measure curved distances, but it's different.
So, some features are very much the same because curve- you have curves and they have lengths,
so you're in business for calculating lots of things.
But it's just not the same, you cannot draw straight lines.
In a moment you'll see that to being able to draw straight lines --
is equivalent to being able to draw and use a Cartesian coordinate system.
We never will use it, except when it's really called for,
but the possibility of having a Cartesian coordinate system --
is kind of an important thing to keep in the back of your mind.
So we're in this space, we also start with a concept of distance.
And it's very important to understand that distance exists on its own terms.
So we will say, I'm holding these pieces of chalk to define a unit of distance.
So we'll call this our unit of distance.
It will now allow us to calculate the area of squares and rectangles, and volumes of cubes,
and rectangular shapes, but also other shapes, like tetrahedra. And even curved shapes.
So we'll say that the volume of a sphere is a non-controversial concept.
We all understand what the volume of a sphere is without having to define it,
it's how much water you can pour into it.
And if you were to pour water out of it into a cubic shape, it's a rectangular shape --
we can put marks on it and see how much water we poured out.
We call that the volume of a sphere.
We also understand what a surface area of a curved shape is.
So in this course we won't struggle with defining those areas.
Actually, defining them precisely in the modern formal sense has proven somewhat difficult,
but without having to define those concepts, we can just use them as intuitively clear --
it would be our fundamental concepts.
What we will focus on is how to work with those concepts and how to measure them.
So, that's the space we're in.
What can we do with this space?
Well, we can define functions, it's very important that we are defining functions of space.
So, in this room, we can talk about temperature as a function of a point in this room.
So I can take a thermometer and measure the temperature at this point,
and then assign a particular temperature to this point.
That's the temperature at this point.
I don't need to introduce a coordinate system for that.
I'm bringing up a coordinate system by saying that we don't need it.
But I wouldn't have to really bring it up, I could just say at this point we have a certain temperature, there it is.
At another point you might have another temperature,
so our functions, initially, are just functions of space.
Like, we distinguish among points, and we say that at a certain point the function has a certain value.
So, that's nice.
That was a scalar function. Because we live in a Euclidean space, and we can have straight lines,
another fundamental concept will be a vector.
One of the most fundamental vectors is the position vector.
Let's limit ourselves to the plane,
let's limit ourselves to the plane.
Take an arbitrary point that we for the lack of a better word- a better term, we'll call the origin.
And then, to every point in the Euclidean space, draw a vector from that point.
So, here's another point, here's another vector.
We're gonna call this the vector R.
In the book it's denoted by bold letters,
on the blackboard i'll just put little vector signs over them.
These are vector quantities. What's a vector?
It's a segment... with a direction.
So every point, there could be a vector, and that would be a vector function.
So a vector function could look like this.
So, at this point it's this vector,
at this point it's this vector,
and so forth.
At each point, there is a vector.
So that would be called a vector valued function.
Let me go back and repeat what a vector is.
A vector, and I'll emphasize this over and over again, is a directed segment.
It is NOT a pair of numbers.
It would be to your great detriment to think of vectors as pairs of numbers.
Or, if you're coming from linear algebra, to think there is some sort of universal basis --
with respect to each of these vectors are decomposed
None of that. You have to accept these objects on their own terms.
These are directed segments. They're beautiful objects.
You can add them, you can multiply them by numbers,
in other words, you can form linear combinations,
more importantly, it's not new from what I just said, but you can subtract one from the other --
They're beautiful objects.
Even though they're basically pictures, they are just pictures.
They are these things that are drawn, nothing more, nothing less.
That's what they are.
And if you're used to, you know, not in my linear algebra class,
but in some linear algebra classes, probably most,
to immediately think of them as a pair of numbers,
or in the 3-dimensional space as a triplet of numbers, and so forth, don't do that here.
They're just vectors, and you can add them, not like they were numbers --
you can add them in their own right.
Vectors can be added by the parallelogram rule.
they can also be subtracted
that's not different from addition, you just multiply by a negative number and subtract
so if you can subtract things
and you can take derivatives of them
we'll discuss it in just a second
there's all sorts of things you can do with vectors
you can do just about everything you want to do, you can do with vectors
most interestingly you can take derivatives of vectors
if you have a vector as a function of a parameter
so, V.... it's not of an important at what point in space it is
but, if you have a vector V as a function of some parameter alpha
you can take the derivative of V with respect to alpha
that's a easy? we'll do that in a moment
the vector to which the operation of the derivative is most commonly applied is the position vector
you can see why its called the position v.. sometimes it's called the what???
all of these kind of make sense, it basically tells you the position of the point
and at this point, so this point, this is the position vector
but this is a more, this is a better way of visualizing it
vectors have length
why do vectors have length?
because the Euclidean Space gives you length
I could calculate the length of any vector if I had the unit vector.
for example this one is .9, no units, it chalks
right? no units
This one is .75
This one, hey that's 1.1
I hope I have a unit vector somewhere, I don't
I'm gonna draw one
there we go
there's a vector of unit length
right? so, Euclidean spaces give us length so we can measure the lengths of vectors
before I talk about differentiation, I will define another very important operation on vectors
it's the dot product
and i want the difference between this approach,
really the approach of vector calculus and physics, from the Linear Algebra approach
so,
u dotted with v
u
u dotted with v
is by definition
the length of u
times
the length of v
times
the cosine of the angle between them
did you learn this from physics?
it's very important to have a starting point, and then go from starting point
our starting point was the concept of length
because we're in Eucidean Space, which means that all of the Euclidean concepts like angle,
so we have lengths, we have angles those 2 they're our primary concepts
if we have those 2 concepts we can multiply one length by another and the cosine of the angle between them
and that's for better or worse is our definition of the dot product
that's some linear algebra that you probably remember, why that's such a useful combination
It just happens to be a super useful combination
the most super useful aspect of this operation
is to, is that if you dot a vector with itself, you get it's length squared
that's the number one thing to remember, dotting a vector with itself is its length squared.
and its linear in it's operator, so u1 plus u2 dotted with v, is u1 dotted with v plus u2 dotted with v
so its linear in its arguement
and again, and I think, behind the scenes,
the most important thing is that we are using our primary concepts of length and angle
geometric objects, we're only, so we've started with Euclidean Spaces with lengths and angles
that's all, that's all the elements
in addition to numbers
that we have, there will be no new concepts.
there's volume and area, but we take those for granted
ok,
one of very important thing that I wanted to you mention, and I forget what it is
what'd I wanna mention about this, ok, we'll come back to it
ok, so that's, that's the dot product
now let's talk about differentiating vectors
so let's look at d of alpha
let's , so one good example is
suppose let's say at one moment
lets this be v of one
v of 1
and let me draw v of 2
I'll just say what it is
let this be v of 2
let this be v of 1 and a half
let this
be v of 1 and a quarter
and so forth
so, I can write
I can, what'd I write , I can say
that the limit of v of alpha
as alpha approaches 1
is this vector
does that make sense?
this was 2, 1 and a half, 1 and a quarter, 1 and 1 eighth
so, do you see how I chose them?
that's just one example that I chose
so,
look
when you heard the word limit your used to see numbers
limit of a function, limit of a sequence
but vectors, because you can talk about the distance between 2 vectors
the distance between 2 vectors
and how small it is, right? because you have the length of the distance
so that's all you need to talk about limits
so vectors even though they're just pictures,
but the concepts of limit extends to vectors as pictures perfectly
think of these vectors as directed segments
don't ever imagine the background coordinate system
or a basis to help decompose these vectors and turn them into numbers
don't turn them into numbers
don't think of them, well, I can make sense of the limit if I think of the vector as a pair of numbers
and if I think of the vector as a pair of numbers, then I really understand
you don't need to go through that step
you can just see that this arrow approaches this arrow
and then you can also talk about the derivative
and the derivative, so I will write v prime
at time t
at time 1
this is real a bad example
it's the limit
it's the limit as some small number approaches zero
of v at time one plus h
subtracted v at time 1
divided by h
do you remember this as the definition of the derivative?
and do you realize that all the elements of this expression
and you can do all of these things with v