Vector field help
One of my assignments is to create a vector field resembling a tornado, shape-wise, at least...
I've gotten the basic shape of the tornado, given by the function f(x,y) = 2*ln(x^2+y^2)... How would I go about converting this function to a vector field resembling this shape?
Thanks in advance!
I assume the function you've given is a surface where the magnitude of the velocity is constant. You need to also say how the speed changes when you move away from the surface.
So you'll need a function of the form:
To get this function, you'll probably want to simplify the equation for your surface, then write an expression that is constant on your surface and increases when decreases.
The direction at (x,y,z) will be (y,-x,0), which will make the wind go in a circle clockwise around the z-axis. It needs to be a unit vector, so:
and to get your vector field, just multiply speed by direction.
Post again if you're still having trouble.
Thank you for your reply!
Let's say if I were to model the speed with something like 1/e^(x^2+y^2) because it's a tornado and tornadoes have 0 outside and inside of the "wall"... (Well, 0 is impossible without piecewise functions, so really, really small magnitudes will suffice)... How would I go about doing the actual modeling and integrating with the direction vector?
Sorry, I've just gotten into vector fields and my teacher threw this assignment at me so I don't really have the hang of it yet... Please bear with me.
Ok. You can make the speed . Of course, that will mean that everything is independent of z. So we don't really need to keep z around - we can just use 2 dimensions. Vector fields still work in 2 dimensions, so that's not really a problem.
A vector field is just a vector for every point, and a vector is just magnitude and direction. If the direction is a unit vector , you just multiply the unit vector by the magnitude r to get the actual vector ui + vj.
If the components u and v of the vector are functions of the coordinates x and y, you have a vector field u(x,y)i + v(x,y)j. In three dimensions, it would look like u(x,y,z)i + v(x,y,z)j + w(x,y,z)k.
So you have the speed (which is the magnitude of the velocity vector) and the direction unit vector (which I gave you in my previous post). Multiply them together to get the equations of the vector field:
, and substituting gives us:
Hope that helps! Post again if there's something you don't understand.
Thank you! I think I understand the concept a lot more now...
But the problem still remains.
My goal is to create a vector field that resembles a tornado and the surface z = 2*ln(x^2+y^2), but I'd assume, as you said, it's a constant function... Is it possible to integrate the properties of 1/e^(x^2+y^2) with the surface z = 2*ln(x^2+y^2)?
Sorry if I sound a little lost!
Let's set the speed to be 1 on z = 2*ln(x^2+y^2) and behaving as 1/e^(x^2+y^2) when off the curve.
Let q0 be the value e^(z/2) of x^2+y^2 when speed=1. Let q=x^2+y^2. So we want speed=1 when q=q0 and speed goes like 1/e^q. So the equation is speed=1/e^(q-q0). Substituting in gives:
speed = 1/e^(x^2-y^2-e^(z/2))
Is that what you're looking for?