SUNY Geneseo Department of Mathematics
Wednesday, March 8
Math 223
Spring 2023
Prof. Doug Baldwin
Why does Mathematica divide by “Log[e]” in integrals involving e^x?
This turns out to be Mathematica’s entirely logical response to one of its harder-to-remember features. Specifically, the built-in constant for the number 2.71… is E, not e (following the rule that built-ins start with capital letters). So when you ask Mathematica to integrate e^x, it thinks that lowercase “e” is some variable of yours that it doesn’t know about, and uses the rule that the integral of nx is nx / ln n (Log is the built-in function for natural logarithms). If you use E^x in the integral, or Exp[x], Mathematica will show the result in a form more like what you’d get by hand.
You can download a notebook that demonstrates these things.
Based on part of section 3.2 in the textbook.
To show that a limit doesn’t exist at point P, show that it has different values along different approaches to P. This is exactly analogous to showing that a single-variable limit doesn’t exist by showing that the limits from the left and right are different (except that there are more ways to approach a point in 2 or more dimensions than there are in 1):
Show that lim(x,y)→(0,0) x/y doesn’t exist.
Consider approaching (0,0) along the lines x = 0 and x = y. These lead to different limits, thus the limit as a whole doesn’t exist:
More generally, any line of the form y = kx will produce a limit of 1/k.
Does lim(x,y)→(1,-1) (x + y) / (2x - 2) exist?
We weren’t really sure, but people tried approaching it along several lines and eventually saw different limits. Beware, in finding lines to approach along, that you use a line that really does pass through the point x and y are approaching. For example, approaching this limit along the line x = y wouldn’t work, because the point (1,-1) isn’t on that line.
Also from section 3.2.
An interior point of a set: Some circle (or sphere, hypersphere, etc. depending on how many dimensions) around the point contains only points inside the set.
A boundary point of a set: Every circle (sphere, hypersphere, etc.) around the point contains both points inside and points outside the set.
An open set: One whose boundary is not in the set.
A closed set: A set that contains its boundary.
A connected set: One that can’t be split into disjoint open subsets.
Is the set of points (x,y) defined by 0 < x < 1 and 2 < y < 3 open or closed? Connected or not connected? What is its boundary?
The definitions of a point P being on the “boundary” or in the “interior” of a set are clever, in that they ask whether all points near P are inside the set or not. Boundary means that no matter how small you make your definition of “near,” some nearby points will always be outside the region. Interior, on the other hand, means that there’s some sufficiently small notion of “near” for which every nearby point is inside the region.
A connected set is one that can’t be split into disjoint open subsets. For example, the set above can’t be split into open regions along the line x = 1/2, because if both subsets were open, neither would contain the line x = 1/2.
So the set is open and connected.
Derivatives of multivariable functions: partial derivatives.
Please read “Derivatives of a Function of Two Variables” and “Functions of More than Two Variables” in section 3.3 of the textbook.