E-mail policy

Please be sure that the numeral 285 appears in the subject field of your emails.
Please (try to) send clear text only (no attachments, no .htm, no encryptions.)

Test 10sep99

This test covering Chapter 1 will be restricted to the following material:
Slope fields, solution curves, isoclines.
Problems 1-30 in the review section of the text.

Supplementary Classnotes

First, a comment on Chapter 1 of our textook. It is customary, if regrettable, that authors of introductory textbooks on differential equations feel obliged to deluge the student with a bewildering and voluminous collection of examples and details of how differential equations occur in the "real world". Since reviewers generally read only the first chapter of a proposed manuscript, and publishers rely on them to insure their profit margins, successful authors cast their initial nets in that direction. So the first chapter of every DE text I have ever taught from is really an introduction and should be studied carefully just before the final examination. But we have a syllabus (prepared by a committee of my colleagues) to comply with, so we began the semester with Chapter 1, just like the other sections of this course.

Slope Fields

Here is a quick review of the "review" part of the lecture on Wednesday. A direction or slope field consists of elements, each of which is a point and a direction or "tangency". We draw such a thing by placing a short line segment through a point. We can't draw them all, so we draw enough to give the general idea. One way of drawing them is to use isoclines.

Did you understand just how the isoclines of a DE act like venetian blinds. The flow of "parallel" solution curves (think of a fluid flowing) must cross the isoclines tangent to the tangent field elements (the shutters). Of course, if the venetian blind is closed then the isocline is also a solution curve (or flow line). Each isocline is labeled by the slope m of the slope field along that curve.

Question 1: For the m=0 isocline to be a flow line it must be ......?

To solve for the isoclines of a DE F(x,y,y')=0 the trick it to substitute a constant m in place of y' (because, by definition, an isocline is where the slope of the field is constant).

Question 2: What if F(x,y,m)=0 is never true? [Hint: recall the example xy'=1.]

Torricelli's Law

At the end of Wednesday (continued Friday) I discussed Torricelli's Law. He asked: Given a container with a hole, find the escape velocity of the liquid in the container. At first sight it would seem that there isn't enough information here. Surely the shape of the container matters, perhaps the mass-density of the liquid. How about the shape of the hole? How about the altitude of the container ... gravity varies from Mt. Everest to the Dead Sea! If, like certain elements of our society, we kept making such objections we would never even get started. The classical scientist makes simplifying assumptions and gets a first approximation to an answer. After all, one can always do an experiment to check the model.

Here is Torricelli's answer: The (v)elocity of the liquid exiting the hole is proportional to the sqare-root of the vertical distance (y) from the hole to the surface.

Consider a drop falling through a distance y from the surface to the hole. From Newton's Law, F=ma, and the F = mg, we get a = g for the constant acceleration. From dv/dt =a we get v =gt. From v = ds/dt, where s is the vertical di(s)placement from the top, we get s = (g/2)t*2. Solving y = s_final = (g/2)(t_final)^2 = (g/2)(v_final/g)^2 = (v_final)^2/2g , whence v_final = \(2gy).

Note: You should copy this to a piece of paper, using standard rather than "ascii" mathematical notation. In ascii mathematical notation ^ means superscript and _ means subscript. Other notation we make up as we go along. It is useful for text editors that can use only the standard keyboard (ascii). You should use it when you email.

If the question is how long it takes for the tank to empty, we need some functional relation between the (V)olume to be emptied and the he(y)ght. There are examples in the text of this kind of question.

An equally interesting question is how one might measure the velocity of the escaping liquid. Imagine a spigot attached to the barrel so that the water must exit horizontally. So now our unknown v is its horizontal velocity. Neglecting air resistance, after a (T)ime, a droplet has horizontal displacement X = vT. So if the vertical displacement above a flat floor of the spigot is Y, then (from our free-fall solution above, t_final = \(2y/g) ) we have T = \(2Y/g). Solving this for v we find, with Torricelli, that the exit velocity is proportional to the horizontal stream displacement X. So if the height y above the hole varies directly with the squared distance, X^2, we have verified Torricelli's law. Anyone have a water-cooler handy? And a mop?

Question 3: Verify the above to make sure I didn't make any more mistakes than I found already.