Calculus III - Vector Fields

Section 16.1 : Vector Fields

2. Sketch the vector field for $\vec F\left( {x,y} \right) = \left( {y - 1} \right)\,\vec i + \left( {x + y} \right)\vec j$.

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Recall that the graph of a vector field is simply sketching the vectors at specific points for a whole bunch of points. This makes sketching vector fields both simple and difficult. It is simple to compute the vectors and sketch them, but it is difficult to know just which points to pick and how many points to pick so we get a good sketch.

So, let’s start off with just computing some vectors at specific points.

\[\begin{align*}\vec F\left( {0, - 1} \right) & = - 2\,\vec i - \vec j & \hspace{0.5in}\vec F\left( {1,1} \right) & = 2\vec j & \hspace{0.5in}\vec F\left( { - 1,0} \right) & = - \,\vec i - \vec j\\ \vec F\left( { - 2, - 1} \right) & = - 2\,\vec i - 3\vec j& \hspace{0.5in}\vec F\left( {1, - 1} \right) & = - 2\,\vec i& \hspace{0.25in}\vec F\left( {2,2} \right) & = \,\vec i + 4\vec j\\ \vec F\left( { - 2,1} \right) & = - \vec j\end{align*}\]

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Now we need to “sketch” each of these vectors at the point that generated them. For example at the point $\left( {0, - 1} \right)$ we’ll sketch the vector $ - 2\vec i - \vec j$.

Here is the sketch of these vectors.

$A sketch of the seven vectors computed in the previous step, each drawn with its tail at the point that generated it, on a set of axes running from $-2$ to 2 in both directions. The vector at $\left( {2,2} \right)$ is long and points up and slightly right, the one at $\left( {1,1} \right)$ points straight up, the one at $\left( { - 2,1} \right)$ is very short and points straight down, the ones at $\left( { - 1,0} \right)$ and $\left( { - 2, - 1} \right)$ point down and to the left, the one at $\left( {0, - 1} \right)$ points left and slightly down, and the one at $\left( {1, - 1} \right)$ points straight left. The vectors are sketched in the correct direction but not to scale.$

In the sketch above we didn’t sketch each of these vectors to scale. In other words we just sketched vectors in the same direction as the indicated vector rather than sketching the vector with “correct” magnitude. The reason for this is to keep the sketch a little easier to see. If we sketched all the vectors to scale we’d just see a mess of overlapping arrows that would be hard to really see what was going on.

Note as well that with the few vectors that we sketched it’s difficult to get a real feel for what is going on at any random point let along any trends in the vector field.

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Below is a better sketch of the vector field with many more vectors sketched in. We got this sketch by letting a computer just plot quite a few points by itself without actually picking any of them as we did in the previous step.

In general, this is how vector fields are sketched. Computing this number of vectors by hand would so time consuming that it just wouldn’t be worth it. Computers however can do all those computations very quickly and so we generally just let them do the sketch.

$A computer generated sketch of the vector field $\vec F\left( {x,y} \right) = \left( {y - 1} \right)\,\vec i + \left( {x + y} \right)\vec j$ with a great many vectors drawn on a grid running from $-2$ to 2 in both directions. The overall pattern swirls counterclockwise about the point $\left( { - 1,1} \right)$, where both components are zero and the vectors shrink away to almost nothing. Vectors in the upper right point up and to the right and are the longest in the sketch, those along the bottom point to the left, those in the lower left point down and to the left, and those in the upper left are short and point roughly to the left.$