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## Multivariable calculus

### Course: Multivariable calculus > Unit 1

Lesson 4: Visualizing vector-valued functions# Vector fields, introduction

Vector fields let you visualize a function with a two-dimensional input and a two-dimensional output. You end up with, well, a field of vectors sitting at various points in two-dimensional space. Created by Grant Sanderson.

## Want to join the conversation?

- The vector fields kind of look like slope fields. Are the two interchangeable, or are they separate in the way that one is the function while the other is derivative?(23 votes)
- You are right that they are similar, but the difference between a vector field and a slope field is the same as the difference between a single vector and a single line. That is, a vector has magnitude and direction, but the line only really gives a direction. In this way, a vector field packs more information than a slope field.(50 votes)

- can someone explain exactly the fundamental process of knowing that the 2-input and output vectors result to a 4-D model? I'm not sure if i worded that right but its in the beginning of the video.(9 votes)
- Maybe it would be helpful to start with simpler functions and work up.

1 input -> 1 output: to show this as a graph is simple -- you get a 2-D graph

e.g. a Cartesian x-y plane where y = f(x)

2 inputs -> 1 output: these were shown in earlier videos as 3-D graphs where z = f(x,y)

1 input -> 2 outputs: this will also be 3-D, but now you are generating y and z values for

each value x -- this will (typically) be a parametric curve

i.e. the vector

[ f(x) ]

[ g(x) ]

where y = f(x) and z = g(x)

More generally, if you want to graph a function with**m**inputs and**n**outputs, then each variable needs its own dimension so the total number of dimensions needed will be**m**+**n**.(33 votes)

- In the previous video (Parametric surfaces) you have a function with 2 inputs (t, s) and 3 outputs in a vector kind of way, but each row has both t and s. t and s could have easily been x and y, so i was wondering: how do you distinguish a vector field from a parametric surface? The only difference I see is that in the vector field each input parameter is the only one used in its own output row. Is that it?(4 votes)
- Yeah, I'm still a little bit confused too. The way I'm thinking of it is this: when dealing with parametric surfaces, you're using external information (like time or cost) to determine the exact location in space, whereas when dealing with a vector field, you're using your exact location in space to determine the external information (cost, speed, etc).

Would this be a correct way to think about this? I'm wondering if we'll have to be told this information beforehand in order to understand what a question is really asking.(3 votes)

- How do you distinguish between a vector field expression and a position vector expression? R (x,y) = x(t) i + y(t) j , a "tradional" position vector expression could be a vector field if we assign a vector R (x,y) = x(t) i + y(t) j for each value of "t" at the point x(t) i + y(t) j.........it seems you have to be told ahead of time that you have one or the other. Thank you for your great work!(4 votes)
- is this just a like a linear transformation T from R2 to R2 i.e T: R2 ----> R2(2 votes)
- Well if f(x,y) was a linear transformation, the angles of the vectors in the vector field would all have to be the same (ignoring orientation here now). (necessary, but I don't think sufficient condition for it to be a linear transformation; this would be akin to showing paralellity of the transformed grid-lines in a linear transformation; I don't think one could intuitively see or not see the 'evenly-spaced' part from this way of presenting the data, because only "intersection" points are transformed and not whole gridlines.(2 votes)

- Why can't functions having same number of inputs and output variables be represented in a different way?

Like a parametric curve or a parametric surface for example(2 votes) - How would I find the function for a vector field when working in, say, fluid dynamics?(2 votes)
- Thanks a lot for your lesson. Could you explain the difference between vector field and vector-valued function. they make me crazy... :'((2 votes)
- I have a question, which color represents the longest and which color represents the shortest(1 vote)
- Typically, colors closer to the red/orange end of the spectrum represent longer vectors. Colors closer to the purple/blue end of the spectrum represent shorter vectors.(2 votes)

- Couldn't the output be thought of as points rather than vectors? It seems strange that the outputs correspond to the magnitude of a vector rather than a point.(1 vote)
- I think using a vector with a magnitude like this is just a representation. I think the problem is that if you want to display it as a point, you would need another two axes to show it's position. Showing it as the magnitude of the vector, you can draw that vector in the position of different inputs.(2 votes)

## Video transcript

- [Voiceover] Hello
everyone, so in this video I'm gonna introduce vector fields. Now these are a concept
that come up all the time in multi variable calculus,
and that's probably because they come up all the time in physics. It comes up with fluid
flow, with electrodynamics, you see them all over the place. And what a vector field is,
is its pretty much a way of visualizing functions
that have the same number of dimensions in their
input as in their output. So here I'm gonna write a function that's got a two dimensional input X and Y, and then its output is going to be a two dimensional vector and each of the components
will somehow depend on X and Y. I'll make the first one
Y cubed minus nine Y and then the second component,
the Y component of the output will be X cubed minus nine X. I made them symmetric here,
looking kind of similar they don't have to be, I'm
just a sucker for symmetry. So if you imagine trying to
visualize a function like this with a graph it would be really hard because you have two
dimensions in the input two dimensions in the output so you'd have to somehow visualize this thing in four dimensions. So instead what we do, we
look only in the input space. So that means we look
only in the X,Y plane. So I'll draw these coordinate axes and just mark it up,
this here's our X axis this here's our Y axis and for each individual input point like lets say one,two so lets say we go to one,two I'm gonna consider the
vector that it outputs and attach that vector to the point. So lets walk through an
example of what I mean by that so if we actually evaluate F at one,two X is equal to one Y is equal to two so we plug in two cubed whoops, two cubed minus nine times two up here in the X component and then one cubed minus nine times Y nine times one, excuse me down in the Y component. Two cubed is eight nine times two is 18 so eight minus 18 is negative 10 negative 10 and then one cubed is one,
nine times one is nine so one minus nine is negative eight. Now first imagine that this was if we just drew this vector where we count starting from the origin,
negative one, two, three, four, five, six,
seven, eight, nine, 10, so its going to have
this as its X component and then negative eight,
one, two, three, four, five, six, seven, we're gonna
actually go off the screen its a very very large vector so its gonna be something here and it ends up having
to go off the screen. But the nice thing about vectors it doesn't matter where they start so instead we can start it
here and we still want it to have that negative ten X component and the negative eight, negative one, two, three, four, five, six, seven, eight, negative eight as its Y component there and a plan with the vector field is to do this at not just one,two but at a whole bunch of different points and see what vectors attach to them and if we drew them all
according to their size this would be a real mess. There'd be markings all over the place and this one might have some
huge vector attached to it and this one would have some
huge vector attached to it and it would get really really messy. But instead what we do, just
gonna clear up the board here we scale them down, this is common you'll scale them down and
so that you're kind of lying about what the vectors themselves are but you get a much better feel for what each thing corresponds to. And another thing about this drawing that's not entirely faithful to the original function that we have is that all of these
vectors are the same length. I made this one just kind of the same unit this one the same unit, and over here they all just have the same length even though in reality
the length of the vectors' output by this function
can be wildly different. This is kind of common practice
when vector fields are drawn or when some kind of software
is drawing them for you so there are ways of getting around this one way is to just use
colors with your vectors so I'll switch over to a
different vector field here and here color is used to
kind of give a hint of length so it still looks organized
because all of them have the same length but the difference is that red and warmer
colors are supposed to indicate this is a very
long vector somehow and then blue would indicate
that its very short. Another thing you can
do is scale them to be roughly proportional
to what they should be so notice all the blue vectors scaled way down to basically be zero red vectors kind of stay the same size even though in reality this
might be representing a function where the true vector
here should be really long or the true vector should
be kind of medium length its still common for people
to just shrink them down so its a reasonable thing to view. So in the next video I'm
gonna talk about fluid flow a context in which vector
fields come up all the time and its also a pretty
good way to get a feel for a random vector field that you look at to understand what its all about.