29. Vector Fields

We've learned Calculus with various types of functions/structures:

$\large y=f(x)$, curves defined with a single intput and single output (derivatives and integrals)
$\large \vec{r}(t)=\left<x(t),y(t),z(t)\right>$, curves defined with a single input and multiple outputs (derivatives and ingetrals of vector valued functions, line integrals)
$\large z = f(x,y)$, surfaces defined with multiple inputs and a single output (partial derivatives, gradients, directional derivatives, double integrals, triple integrals, surface integrals)

We are ready for the full combination! A vector field is a function with multiple inputs and multiple outputs:

$$\large \vec{F}(x,y,z)=\langle F_1(x,y,z),F_2(x,y,z), F_3(x,y,z)\rangle$$

For a vector field, at each input point $\large (x,y,z)$, there is a vector output. Examples include blood flow:

heartFIeld

Wind velocity:

windMap

In physics, every force is a vector field (called a force field). Gravity, for example is a vector field modeled with Newton's Law:

$$\large \vec{F}(x,y,z) = -\frac{mMG}{|\vec{x}|^3}\langle x,y,z \rangle$$

A more common way to write the vector field above is:

$$\large \vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x}$$

Here's a drawing representing a gravity vector field:

gravityField

We visualize vector fields by plotting arrows, which is a task for a computer! CalcPlot3D has an option for vector fields. The following fields were made in Wolframalpha.com. Just type "plot vector field", and go from there.

Here's $\large \vec{F}(x,y) = \left<-y,x\right>$ and $\large \vec{F}(x,y) = \left<y,\sin(x)\right>$:

vectorField1 vectorField2

Terminology

The following is an example of a constant vector field: $\large \vec{F}(\vec{x}) = \left< 1,2,3\right>$ for all $\large \vec{x}$.
A unit vector field is a vector field where $\large \left| \vec{F}(\vec{x})\right|=1$ for all $\vec{x}$.
A radial vector field is a vector field where $\large \left| \vec{F}(\vec{x})\right|$ depends only on distance from the origin $\large \sqrt{x^2+y^2+z^2}$. For a radial vector field, all the arrows will point toward or away from the origin, and at a given distance from the origin, all vectors will be the same length.

The rest of this class will deal with the calculus (derivatives and integrals) of vector fields. This is at the heart of physics, electromagnetism, and fluid dynamics.