SUNY Geneseo Department of Mathematics
Tuesday, October 4
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
Beware of confusing “Equation 2.2.1” with “Theorem 2.2.1.” Several questions refer to “Theorem 2.2.1” as something that allows you to take the derivative of a vector-valued function by taking derivatives of each component; it’s identified in the text by being labeled as a “Theorem.” Equation 2.2.1, on the other hand, is part of the limit definition of the derivative (exactly what I don’t want you to use when I mention Theorem 2.2.1), and has no particular label other than the number in the right margin.
These typographical conventions, by the way, are fairly common in math books, i.e., equations might be numbered by just putting a number on the margin of the page next to the equation, whereas other numbered things (theorems, definitions, etc.) generally have a word (e.g., “Theorem,” “Definition,” etc.) next to the number to indicate what the numbered thing is.
Use the BoxRatio option to control the aspect ratio (i.e., ratio of lengths of axes as drawn) of plots in Mathematica.
BoxRatio->Automatic asks Mathematica to use aspect ratios proportional to the actual coordinate ranges.
BoxRatio->{x,y,z} sets the ratio of x to y to z widths of the plot region to x : y : z.
You can download a notebook from Canvas that demonstrates some examples of this option.
From “Limit of a Function of Two Variables” in section 3.2 of the textbook.
The familiar limit laws from 1-variable functions apply to multivariable functions, or have close analogs.
The tricks for finding limits that you may be used to from single-variable functions mostly work for multivariable functions, except for L’Hôpital’s Rule.
Find lim(x,y)→(2,-1) (x2 + xy - y2) / (x + y)
Find this limit by just substituting the limit values for x and y (the ability to do this is technically a consequence of the limit laws for multivariable functions).
How about lim(x,y,z)→(-2,2,1) (x2 - y2) / z(x + y)
Simple substitution on this leads to an indeterminate form 0/0, and L’Hôpital’s Rule isn’t available. But factoring the numerator gives a factor of x + y that cancels with the same term in the denominator, leaving something you can do substitution in:
From “Continuity of Functions of Two Variables” in section 3.2.
The definition of continuity is the same as for single-variable functions, namely that a function is continuous at a point if the function’s value equals the limit at that point (which implies that the function’s value and limit are both defined too).
Some examples of continuity and discontinuity, using the functions from earlier:
We can’t really do much with discontinuity due to non-existent limits until we have a better sense of how to tell if a limit of a multivariable function does or doesn’t exist in the first place.
That’s what we’ll talk about tomorrow.
Please read (if you haven’t already) the material on limits that don’t exist at the end of the “Limit of a Function of Two Variables” subsection in section 3.2 of the textbook.