Packet 10: Bonus 10
For a function f between manifolds M and N, one of these two operations maps differential forms on N to differential forms on M, while the other maps tangent vectors on M to tangent vectors on N. For 10 points each:
[10h] Give these two terms whose names are used in several mathematical subfields to denote operations that transfer data from the domain to the range of a function and vice-versa.
ANSWER: pullback AND pushforward [accept in either order]
[10m] The pullback of n-forms can be used to prove that, when changing variables, the integrand gets multiplied by this quantity’s absolute value. When converting from spherical coordinates, this quantity is “rho-squared sine phi.”
ANSWER: Jacobian determinant [or determinant of the Jacobian matrix; prompt on determinant]
[10e] The pullback “phi-star f” equals this operation of f and phi. The chain rule is used to compute a derivative after this operation has been performed on two functions.
ANSWER: function composition [or word forms like compose or composite; prompt on “f of g” with any letters sandwiching “of”]
<TM, Other Science (Math)> | NAFTA-Packet-10
| Heard | PPB | E % | M % | H % |
|---|---|---|---|---|
| 30 | 16.33 | 77% | 67% | 20% |
Conversion
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total | Parts |
|---|---|---|---|---|---|---|
| Lex | Maybe Ever | 0 | 0 | 10 | 10 | E |
| The Worst Trade Deal in the History of Trade Deals | WashU | 0 | 10 | 10 | 20 | ME |
| W&L+Friend | St. Marx | 0 | 10 | 10 | 20 | ME |
| four high school generalists shock natives with fluent stock in Tocharian B | [Redacted] | 10 | 10 | 10 | 30 | HME |
Summary
| Tournament | Edition | Match | Heard | PPB | E % | M % | H % |
|---|---|---|---|---|---|---|---|
| 2026 NAFTA at Stanford | 01/17/2026 | ✓ | 4 | 20.00 | 75% | 75% | 50% |
| 2025 NAFTA Online | 02/14/2026 | ✓ | 4 | 20.00 | 100% | 75% | 25% |
| 2026 NAFTA at Vanderbilt | 02/14/2026 | ✓ | 3 | 10.00 | 67% | 33% | 0% |
| 2025 NAFTA at Toronto | 09/13/2025 | ✓ | 1 | 20.00 | 100% | 100% | 0% |
| 2025 NAFTA at Maryland | 09/27/2025 | ✓ | 1 | 10.00 | 100% | 0% | 0% |
| 2025 NAFTA at Harvard | 10/04/2025 | ✓ | 3 | 20.00 | 100% | 100% | 0% |
| 2025 NAFTA at Oxford | 10/11/2025 | ✓ | 3 | 6.67 | 33% | 33% | 0% |
| 2025 NAFTA at Chicago | 11/08/2025 | ✓ | 6 | 16.67 | 67% | 83% | 17% |
| 2025 NAFTA at Columbia | 11/08/2025 | ✓ | 5 | 18.00 | 80% | 60% | 40% |