How to Remember Type I, II, and III Regions in Multivariable Calculus

by Justin Skycak (x.com/justinskycak) on

Type I pairs with the variable that runs vertically in the usual representation of the coordinate system. The remaining types are paired with the rest of the variables in ascending order.

Want to get notified about new posts? Join the mailing list.

I’ve always had trouble remembering the definitions of type I, II, and III regions. In particular, it’s tricky to remember which type (I, II, III) is paired with which axis ($x,$ $y,$ $z$).

In $2$-dimensional space, type I and II regions are paired with the $x$ and $y$ axes, respectively:

  • A region is of type I if it can be decomposed into vertical segments parallel to the $y$-axis, with each $x$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
  • A region is of type II if it can be decomposed into horizontal segments parallel to the $x$-axis, with each $y$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.

But in $3$-dimensional space, type I, II, and III regions are paired with the $z,$ $x,$ and $y$ axes, respectively:

  • A region is of type I if it can be decomposed into vertical segments parallel to the $z$-axis, with each $xy$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
  • A region is of type II if it can be decomposed into horizontal segments parallel to the $x$-axis, with each $yz$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
  • A region is of type III if it can be decomposed into horizontal segments parallel to the $y$-axis, with each $xz$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.

It seems as though these definitions aren’t even consistent with each other! Type I pairs with $y$ in $2$-dimensional space, but with $z$ in $3$-dimensional space.

Further complicating the matter, they aren’t even consistent in being inconsistent! Type II pairs with $x$ in $2$-dimensional space, and again with $x$ in $3$-dimensional space.

However, I recently noticed a trend that seems to generalize well:

  • Type I pairs with the variable that runs vertically in the usual representation of the coordinate system.
    • In $2$-dimensional space, that's $y,$ and in $3$-dimensional space, that's $z.$
  • The remaining types are paired with the rest of the variables in ascending order.
    • In $2$-dimensional space, we've already assigned $y$ to type I, and the only remaining variable is $x,$ so we assign it to type II.
    • In $3$-dimensional space, we've already assigned $z$ to type I, and the remaining variables in alphbetical order are $x,y,$ so we assign $x$ to type II and $y$ to type III.

Generalizing to $N$-dimensional space with coordinates $(x_1, x_2, x_3, \ldots, x_N),$ we have the following pairings:

  • Type I is paired with $x_N$
  • Type II is paired with $x_1$
  • Type III is paired with $x_2$
  • Type IV is paired with $x_3$
  • (and so on)


Want to get notified about new posts? Join the mailing list.