Graphing Vector Fields: How to Determine Grid Size and Plot Vectors?

[LinearAlgebra]

What are the general rules that one should use in graphing vector fields. I'm having a lot of trouble doing this and don't really know where to start.

If you take F(x,y) = -yi + xj

What should be the next step in terms of graphing? They have it drawn in our book as a bunch of vectors that form a bunch of cirles within each other...

 

[HallsofIvy]

Choose some point (x,y), Caculate the vector F(x,y)= -yi+ xi, draw that vector starting at (x,y) (with its "tail" at (x,y)).

Choose another point (x,y), Repeat.

That's all there is to it.

 

[LinearAlgebra]

I know, its a stupid question. I just don't understand how to choose the points. The book has the points (1,0) (0,1) (-1,0) (0,-1) etc...how did they get this?

 

[D H]

You can't plot the field for every point because all you would see is black (assuming the vectors are in black). Therefore, you want to sample the space, usually on a grid. Coarsen the grid if you see so much black that you can't see the vectors, and refine the grid if the vectors are so widely spaced that you can't visualize the field.

 

[LinearAlgebra]

Okay, I'm sorry, i still just don't get it. Can someone just explain this step by step in terms of what i should be thinking or plotting?

 

[D H]

You want to view a vector field over a finite-sized region of $\mathbb R^2$. The first thing to do is to determine this region of interest. I will assume (just for illustration) that you want to look at the vector field from 0 to 10 in x and y. The next thing to do is to set up a grid on this interval. For example, a 1x1 grid. You will draw a vector at each grid intersection point. (In this example, this means 121 vectors.) For each grid intersection point (x,y), determine the vector field value F(x,y), and plot that vector with tail at (x,y).

You can do a bit better than just guessing how finely to make the grid. You will have a hard time seeing the field if the vectors cross multiple grid lines or if the vectors are a lot smaller than the space between grid lines. As a first guess, make the grid spacing about equal to the magnitude of the largest vector. Then fine-tune so it looks good.