SUNY Geneseo Department of Mathematics
Friday, February 28
Math 223 01
Spring 2020
Prof. Doug Baldwin
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On arc length and curvature.
See handout for details.
Section 13.1.
Functions with multiple inputs (as opposed to a vector of outputs).
Level curves.
Plotting (with Mathematica or similar technology).
Lots of things you’ve already seen in math can be described as multivariable functions.
For example, the area of a rectangle, A = hw, can be described as a function A(h,w) = hw.
Other examples?
Physics of waves: the amplitude of a wave is a function of both position and time, A(x,t). For instance, a person standing beside a river can stand still and notice waves moving on the river over time, or walk along the river and notice waves rising and falling as position changes:
Force is a function of mass and acceleration: F =ma
Average of x, y, z: A(x,y,z) = (x+y+z)/3. Multivariable functions can have any number of inputs.
Think of standing on the ground outside South Hall as standing on the surface of a multivariable function (the function that gives elevation of the ground relative to some reference point).
Level curves are curves in the xy plane along which a multivariable function has a constant value. Compare to contour lines, which are lines at non-zero z along which the function has a constant value. So level curves are contour lines projected into the z = 0 plane. Confusingly, “contour lines” on a topographical (aka contour) map are more like level curves, in that they are curves in a plane that represent lines of constant elevation in a 3D surface.
Come up with a function for the “bowl” we were standing in: the bowl is basically a hemisphere, of radius, say, 10 feet. So start with the equation for a sphere, and solve for z, keeping in mind that we want the bottom half of the sphere:
Now remember that the bowl is half a sphere whose center isn’t really at the origin. So we have to offset the hemisphere equation by an amount that puts the center at (0,0,10):
Use the Plot3D function.
This notebook demonstrates, on the bowl function we found and another plausible alternative.
Limits of multivariable functions.
Read section 13.2