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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? •   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.
• 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. •  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.
• 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? • 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.
• 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! • is this just a like a linear transformation T from R2 to R2 i.e T: R2 ----> R2 • 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.
• 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 • How would I find the function for a vector field when working in, say, fluid dynamics? • Thanks a lot for your lesson. Could you explain the difference between vector field and vector-valued function. they make me crazy... :'(   