SUNY Geneseo Department of Mathematics
Wednesday, September 28
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
Based on “Curvature” in section 2.3 of our textbook.
The terminology and formulas seem harder than usual to wade through. This is true, there are a lot of both.
The fact that there are choices to make, e.g., 3 different formulas for curvature to choose between when given a curvature problem, is part of it. Trying to match the requirements of the formulas to specific problems can help (e.g., one of the formulas only works for 2-dimensional problems, so you wouldn’t consider it in 3 dimensions), but a lot of times the choice just comes down to which formula seems easiest to work with for you.
The fact that terminology cascades makes it hard too. For example, there are formulas that refer to a curve’s unit tangent vector (T), which has its own definition in turn, and so forth. For this, you just have to be willing to trace back through definitions or their formulas wherever you need to (don’t expect reading math — or other technical material — to be quick, unfortunately).
Curvature measures how sharply or tightly a curve bends.
There are formulas for calculating curvature, but they are definitely things to remember where to look up rather than things to be memorized in their own right.
Consider the curve r(t) = ⟨ √3 cos t, cos t, 2 sin t ⟩. It’s a circle of radius 2, perpendicular to the xy plane and forming a 30 degree angle with the x axis.
Find the curvature of this curve, using Mathematica to help with the tedious parts.
The main new Mathematica idea here was “variables” as a way to name things and store them for later use. We did the curvature calculation as a series of smaller calculations that eventually led to a number for curvature. There were also a few places where Mathematica failed to simplify things as much as we could. In those cases, we replaced Mathematica’s results with our own to keep later results simple. You can download the notebook with this work in it from Canvas.
Based on “The Normal and Binormal Vectors” in section 2.3.
The normal and binormal vectors work with the unit tangent vector to create a coordinate system local to someone moving along a curve: the tangent points in the direction that person is moving, the normal in the direction towards which the person is turning (i.e., the direction of greatest curvature), and the binormal is perpendicular to the other two.
The normal and binormal vectors are calculable from the unit tangent vector; like with curvature, knowing this and where to look up the formulas is more important than memorizing the exact formulas.
Now that we can work with functions that produce multi-dimensional outputs, let’s look at functions with multi-dimensional inputs (so that towards the end of the semester we can put the two ideas together in “vector fields”).
Please read “Functions of Two Variables,” “Level Curves,” and “Functions of More than Two Variables” in section 3.1 of the textbook. (You can skip the section on graphing functions of 2 variables for now, since we’ll look at that separately via Mathematica, on Monday.)