Vector Calculus Problem

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  • #1
Nicolaus
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Homework Statement


Let R be the unit square such that R= [0,1] x [0,1]
Find a sequence of partitions of R such that the limit as k ->inf of the area of the largest sub-rectangle of the partition (where k is number of partitions) goes to zero but the mesh size does not go to zero.
Depicting the first couple of partitions in the sequence graphically suffices.


Homework Equations




The Attempt at a Solution


I know how to show the converse, i.e. showing that if the mesh size goes to zero, then the area of the largest sub rectangle goes to zero. In this case, the mesh size is the Euclidean norm of the i'th sub-rectangle: if I divide the square into 4 unequal sub-rectangles, then: 0 < 4(maxA(R)) < 4mesh^2 (i.e. Euclidean Norm squared) so by taking the limit of the mesh, and if it tends to zero, then the max area will tend to zero by squeeze. I'm having trouble starting the aforementioned problem, though.
 

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  • #2
LCKurtz
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Homework Statement


Let R be the unit square such that R= [0,1] x [0,1]
Find a sequence of partitions of R such that the limit as k ->inf of the area of the largest sub-rectangle of the partition (where k is number of partitions) goes to zero but the mesh size does not go to zero.
Depicting the first couple of partitions in the sequence graphically suffices.


Homework Equations




The Attempt at a Solution


I know how to show the converse, i.e. showing that if the mesh size goes to zero, then the area of the largest sub rectangle goes to zero. In this case, the mesh size is the Euclidean norm of the i'th sub-rectangle: if I divide the square into 4 unequal sub-rectangles, then: 0 < 4(maxA(R)) < 4mesh^2 (i.e. Euclidean Norm squared) so by taking the limit of the mesh, and if it tends to zero, then the max area will tend to zero by squeeze. I'm having trouble starting the aforementioned problem, though.

Think about long skinny rectangles.
 
  • #3
Nicolaus
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Thanks
 

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