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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.

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• 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.
• 1. How long has the study of vector fields been around? I'm asking because most of the resulting images are (seem to be) computer generated. If the study of vector fields is very old, or somewhat old, computer generated results would not have been available, or depending on the era, most likely not yielding the images we see here.
• 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