Exercise 12. Let F, G: U C R2 → R' be a pair of mappings such that (F,G) = 0. Prove that (D,F, G) = -(F, D,G).
Exercise 12. Let F, G: U C R2 → R' be a pair of mappings such that (F,G) = 0. Prove that (D,F, G) = -(F, D,G).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 12
![Exercise 12. Let F, G: U c R² → R³ be a pair of mappings such that
(F,G) = 0. Prove that (D,F,G) = -(F, D,G).
Now recall that (D,X, N) = 0. Hence the previous exercise yields:
(D,X, D,N) = -(DajX, N) = -lij.
Combining the previous two lines of formulas, we get: (D,N, DµN) = E Sizlyki
which in matrix notation is equivalent to
[(D,N, D,N)| = [S,],].
Finally, recall that det[(D,N, D¿N)] = ||D,N × D¿N||?, det[Sj] = K, and
det [4] = Kg. Hence taking the determinant of both sides in the above
equation, and then taking the square root yields the desired result.
5
Next, we discuss the second method for proving that ||DỊN × D2N|| =
METHOD 2. Here we work with a special patch which makes the computa-
tions easier:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2F2e3d5113-4962-44fc-9f5c-37272b42cda6%2Frvt0fio_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 12. Let F, G: U c R² → R³ be a pair of mappings such that
(F,G) = 0. Prove that (D,F,G) = -(F, D,G).
Now recall that (D,X, N) = 0. Hence the previous exercise yields:
(D,X, D,N) = -(DajX, N) = -lij.
Combining the previous two lines of formulas, we get: (D,N, DµN) = E Sizlyki
which in matrix notation is equivalent to
[(D,N, D,N)| = [S,],].
Finally, recall that det[(D,N, D¿N)] = ||D,N × D¿N||?, det[Sj] = K, and
det [4] = Kg. Hence taking the determinant of both sides in the above
equation, and then taking the square root yields the desired result.
5
Next, we discuss the second method for proving that ||DỊN × D2N|| =
METHOD 2. Here we work with a special patch which makes the computa-
tions easier:
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