SUNY Geneseo Department of Mathematics
Friday, September 30
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
Problem set 5, mainly on arc length and curvature of vector-valued functions, is ready.
Work on it next week, and grade it the week after.
See the handout for more information.
Based on “Functions of Two Variables,” “Level Curves,” and “Functions of More than Two Variables” in section 3.1 of the textbook.
Functions with more than variable (i.e., more than one input) exist.
Level curves are curves in the xy-plane (or generally, the space of inputs to a multivariable function) along which the function’s value remains constant.
Preview: multivariable functions must have derivatives, which seem to depend on the direction of change in the inputs.
There are lots of “real-world” examples of calculations that can be described as multivariable functions.
For example…
(In the last example, Cobb-Douglas functions, there is really a family of such functions, each member of the family differing in constants c and a. So technically those two symbols aren’t “variables” that are considered inputs to the function. You can also have 3-variable Cobb-Douglas functions, 4-variable, etc. We may see these functions again when we talk about optimization.)
We visited the hillside in the quad outside South Hall as a tangible example of a surface in 3 dimensions, where elevation of the hill depends on east-west (x) and north-south (y) position.
We used the hillside to explore curves you could walk along without changing your elevation (level curves) and the idea that if you’re not walking along a level curve, elevation changes at different rates as you walk in different directions (derivatives).
Level curves are curves in the xy-plane (or higher-dimensional space if the function has more than 2 variables) along which the function’s value doesn’t change.
To find level curves, you can set the function value to a constant and then figure out what shape curve the resulting equation in x and y describes. So notice that technically a level curve is just a curve in the xy-plane, not a curve with constant z in 3 dimensions. You generally get slightly different level curves according to what constant you set the function’s value to. For example…
Graphing multivariable functions with Mathematica.
But please read “Graphing Functions of Two Variables” in section 3.1 of the textbook for some underlying concepts.