Is this the same thing as a slope field from calculus? Does this differ in that this extends to higher dimensions?
Staffjamesfong1
@rsha256 This is very similar. Slope fields are a special case of vector fields. Vector fields allow for the vector to point in any direction and have any magnitude. In contrast, slope fields only represent a direction. Vector fields are also trivial to extend to higher dimensions.
Vector fields are just a way to visualize a function that has a multi-dimensional input and output.
LeslieTrue
Just for fun and better understanding. 3B1B has a bunch of videos visualizing fields or fluid flows like this one:https://www.youtube.com/watch?v=rB83DpBJQsE.
andrewhuang56
If we have, say, a current, or wave coming in from outside and to the left of our field, do we simply add a constant gradient over the whole region? Or do we add it over the lefthand side of the region and let future iterations take care of dispersing the current/wave?
rsha256
@jamesfong1 How is magnitude shown in these figures? Is it through the color or the length of the vectors?
Staffjamesfong1
@rsha256 The magnitude is shown as the length of the arrow. Longer arrows mean that the gradient is stronger in that direction.
Each plot here is colored according to a special scalar value related to each vector field. The left-most plot is showing a function whose gradient is the vector field plotted here. The other two plots are divergence and curl, which are explained on the following slides.
modatberkeley
I understand the length represents magnitude, but could you clarify what the colors mean?
StaffDanCubed
@modatberkeley The color corresponds to the scalar value associated with the vector field. So for each coordinate (x, y), we have a number for it, and the colors are just a way to visualize these values (lighter means higher/lower, and darker means the opposite etc). Functions like the gradiant, divergence and curl (whose outputs are the vectors all use these pre-defined scalar values in their calculations.
Is this the same thing as a slope field from calculus? Does this differ in that this extends to higher dimensions?
@rsha256 This is very similar. Slope fields are a special case of vector fields. Vector fields allow for the vector to point in any direction and have any magnitude. In contrast, slope fields only represent a direction. Vector fields are also trivial to extend to higher dimensions.
Vector fields are just a way to visualize a function that has a multi-dimensional input and output.
Just for fun and better understanding. 3B1B has a bunch of videos visualizing fields or fluid flows like this one:https://www.youtube.com/watch?v=rB83DpBJQsE.
If we have, say, a current, or wave coming in from outside and to the left of our field, do we simply add a constant gradient over the whole region? Or do we add it over the lefthand side of the region and let future iterations take care of dispersing the current/wave?
@jamesfong1 How is magnitude shown in these figures? Is it through the color or the length of the vectors?
@rsha256 The magnitude is shown as the length of the arrow. Longer arrows mean that the gradient is stronger in that direction.
Each plot here is colored according to a special scalar value related to each vector field. The left-most plot is showing a function whose gradient is the vector field plotted here. The other two plots are divergence and curl, which are explained on the following slides.
I understand the length represents magnitude, but could you clarify what the colors mean?
@modatberkeley The color corresponds to the scalar value associated with the vector field. So for each coordinate (x, y), we have a number for it, and the colors are just a way to visualize these values (lighter means higher/lower, and darker means the opposite etc). Functions like the gradiant, divergence and curl (whose outputs are the vectors all use these pre-defined scalar values in their calculations.