Math2111: Chapter 2: Vector fields and the operator ∇. Section 1: Vector fields

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 \boldsymbol{F} is a function from \mathbb{R}^n to \mathbb{R}^n. This means, at every point in (x_1,\ldots, x_n) \in \mathbb{R}^n we assign a vector

\displaystyle \boldsymbol{F}(x_1,\ldots, x_n) = (F_1(x_1,\ldots,x_n),\ldots,F_n(x_1,\ldots,x_n)).

To visualize vector fields, one draws the vector \boldsymbol{F}(\boldsymbol{x}) with its foot at \boldsymbol{x} (rather than the origin \boldsymbol{0}\in\mathbb{R}^n).
Examples (Vector fields)
Vector fields \boldsymbol{F}:\mathbb{R}^2\to\mathbb{R}^2:

  1. \boldsymbol{F}(x,y)=(ax,ay), where {}a is a constant with a>0;
  2. \boldsymbol{F}(x,y)=(ax,ay), where {}a is a constant with a<0;
  3. \boldsymbol{F}(x,y)=(-ay,ax), where {}a is a constant with a>0;
  4. \boldsymbol{F}(x,y)=(y/\sqrt{x^2+y^2},-x/\sqrt{x^2+y^2})
  5. \boldsymbol{F}(x,y)=(x-y,x+y)
  6. \boldsymbol{F}(x,y)=(\sin y, \sin x)

(These vector fields are generated using Matlab. For example, to generate \boldsymbol{F}(x,y)=(-y,x), 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.

Integral curve

As we have seen in the example on fluid flow, a particle in the fluid moves along a curve \mathcal{C} such that the velocity field is tangent to the curve \mathcal{C} at each point. Hence, given a starting point p_0, we obtain a curve \mathcal{C}. Mathematically, this is expressed in the following definition.

Definition (Integral curve)
Let \boldsymbol{F}:\mathbb{R}^n\to\mathbb{R}^n be a vector field. An integral curve is a curve \boldsymbol{c}:D\subseteq\mathbb{R}\to \mathbb{R}^n such that

\displaystyle \boldsymbol{c}^\prime(t)= \boldsymbol{F}(\boldsymbol{c}(t)).

In other words, the \boldsymbol{F} is the velocity field of \boldsymbol{c}.

Example
Let \boldsymbol{F}(x,y) = (-y,x). Find the integral curves \boldsymbol{c} which satisfy \boldsymbol{F}(\boldsymbol{c}(t)) = \boldsymbol{c}^\prime(t).

\rhd There is also a numerical method for solving ordinary differential equations based in these ideas which is called Euler’s method.

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