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## Multivariable calculus

### Course: Multivariable calculus>Unit 1

Lesson 4: Visualizing vector-valued functions

# Fluid flow and vector fields

A neat way to interpret a vector field is to imagine that it represents some kind of fluid flow.  Created by Grant Sanderson.

## Want to join the conversation?

• So am I right in saying that the size of a vector is representative of the speed of the particle as it moves through that point? • This is my speculation and reasoning:

This is what the fluid flow model is meant to represent - the output indicates velocity (and so the magnitude/size of the vector of course indicates speed), while the input vectors are the location. By having the output mean velocity, which is derived from location, you can nicely map the velocities on the location map, which happens to be the input map.

I think that for any vector field to make sense, you would have to be mapping the derivative (the "output" as described in the video) onto the "input" (from which the derivative would be derived) - since there would be no point in mapping the output arrows into the input, if the output vectors/the vector field does not directly affect or model the changes in the input (at least it would not be a very useful or intuitive representation). The easiest way to make sense of the vector field model is using velocity (first derivative, "output") and location, with the model of the fluid flow. The vector field can be used to represent other cases as well, that don't involve time.

This also would mean that a vector field can't be used to model all types of multivariable/multidimensional equations. (Of course, like he said in the video, there must be the same number of inputs as outputs, so these ideas about #ofinput=#ofoutput and derivative/input are probably related.)

• Can anyone provide an online source that does something akin to the animations that are shown in the video? • Which software has been used to show the fluid flow? • What software is he using to simulate this fluid flow? I have the Grapher app but couldn't find any option for this simulation. • How does a vector field differ from a slope field?
(1 vote) • Starting at about , the diagonal particles along the path y=x can be seen compressing inwards at the origin and not moving thence, I know this is mathematics, but if this kind of equation was used to describe actual movement of water, as in a simulation, what could be done to negate this effect?
(1 vote) • Even if the vectors aren't drawn to scale, are all their tails at their true starting positions?
(1 vote) • Based on how vector fields can be correlated to fluid flow:
*)In the stream line flow of fluids(Which is the only flow that can be analysed in contrast to transient and turbulent flow) each fluid particle passing through a particular point has the same velocity when it passes through that point (i.e. velocity vector) and hence the vectors in the vector field should in this case start from their true starting points,as the velocity varies with he point being analysed.
*)This may also be generalised for most cases.Thus,the tails of the vectors are at their true starting points.
(1 vote)