What does the partition norm $\|P\| \to 0$ mean in the definition of the double integral?

Conditions

• Applies to the limit step in the definition of volume by double Riemann sums.
• $P$ represents the partition of the rectangular region.

Reasoning, step by step

1. Identify $\|P\|$ as the norm or mesh size of the partition.
2. Interpret $\|P\| \to 0$ as the largest subrectangle's area tending to zero.
3. Deduce that if the largest piece shrinks to zero, all smaller pieces must also shrink to zero.
4. Conclude that the number of subrectangles must increase to infinity to cover the same fixed region.

Example

The speaker explains verbally: 'when the length of P goes to zero, the largest rectangle goes to zero area, all the other rectangles go to zero as well, and the number of rectangles goes to infinity.' A green annotation on the board points to $\|P\| \to 0$ and reads 'Largest Rectangle going to zero area'.

Common misconceptions

• Thinking $\|P\| \to 0$ refers to a single specific rectangle shrinking while others stay the same.
• Believing the number of rectangles decreases as the partition gets finer.
• Confusing the partition norm with the index $k$ of a specific subrectangle.

Watch the explanation

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