SUNY Geneseo Department of Mathematics
Wednesday, October 3
Math 223 01
Fall 2018
Prof. Doug Baldwin
Today, 3:00 - 4:00, South 328
Learn what an actuary is and meet the actuary club
Friday, October 5, in class.
Covers material since the beginning of the semester (e.g., 3D space; lines, planes, and other surfaces; rectangular, polar, and cylindrical coordinates; vectors and their operations; vector valued functions and their calculus). Arc length and curvature may appear, but will probably be to some extent deferred to a future exam.
3 to 5 short-answer questions.
You’ll have the full 50-minute class period.
Open book, notes, computer as a reference or calculator; closed person.
Sample exam and solution are now available.
Find the principal unit normal to r(t) = 〈 -sin t, cos t, √3 t 〉.
The principal unit normal is the derivative of the unit tangent vector, scaled to unit length. So the basic calculation involves computing and scaling 2 derivatives.
Calculate the unit tangent vector by differentiating and scaling r:
Then calculate the principal unit normal by differentiating and scaling T:
What’s the geometric relationship between a constant-magnitude vector and its derivative? Figure it out by starting with an expression for magnitude (or at least its square), then take derivatives:
They’re perpendicular!
So the principal unit normal is always perpendicular to the tangent vector. Furthermore, because the principal unit normal is a derivative, it points in the direction the tangent is changing towards, i.e., the direction the curve is turning.
This particularly has implications for velocity and acceleration vectors: You can decompose acceleration into a component parallel to the tangent, which is the portion of acceleration responsible solely for speeding up or slowing down the object (since the tangent vector is parallel to velocity), and a component parallel to the principal unit normal, which is solely responsible for changing the object’s direction.
Principle unit normal as a vector that gives direction towards which a curve turns.
Formula for the principal unit normal.
Multivariable functions.
Read section 4.1.