In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.
In this part we deal with vector fields. These are useful for describing fluid flow or force.
Imagine a fluid moving in a pipe in a steady flow. (In physical applications the vector field may also depend on time, but we do not consider such instances here.) Then at each point we can draw a velocity vector corresponding to the velocity of the fluid at this particular point to obtain the velocity field of the fluid. If we put a particle in the fluid, then, at each point, the direction of the velocity vector is tangent to the movement of the particle and the length of the vector corresponds to the speed of the particle. See
here for a picture on fluid flow or here for some animations on fluid flow. Similarly one can draw force fields, see for instance here and here. The force field of a magnetic dipole is shown here.
Mathematically, a vector field is a function from to . This means, at every point in we assign a vector
To visualize vector fields, one draws the vector with its foot at (rather than the origin ).
Examples (Vector fields)
Vector fields :
(These vector fields are generated using Matlab. For example, to generate , type:
>> u=inline(’-y’,’x’,’y’)
>> v=inline(’x’,’x’,’y’)
>> x=linspace(-2,2,11);
>> y=linspace(-2,2,11);
>> [X,Y]=meshgrid(x,y);
>> U=u(X,Y); V=v(X,Y);
>> quiver (X,Y,U,V)
>> axis image)
Some java applets in 2d can be found here and in 3d here.
As we have seen in the example on fluid flow, a particle in the fluid moves along a curve such that the velocity field is tangent to the curve at each point. Hence, given a starting point , we obtain a curve . Mathematically, this is expressed in the following definition.
Definition (Integral curve)
Let be a vector field. An integral curve is a curve such thatIn other words, the is the velocity field of .
Example
Let . Find the integral curves which satisfy .
There is also a numerical method for solving ordinary differential equations based in these ideas which is called Euler’s method.