18:52, 2 Гуравдугаар сар 2021-ий байдлаарх засвар

ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠭᠲᠠᠷᠭᠤᠢ ‍ᠢᠢᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠨᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠶᠤᠮ᠃

ᠷᠢᠮᠠᠨ^Riemann  ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ^geometry ᠨᠢ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠳᠠᠷᠭᠤ ᠪᠤᠶᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ᠳᠡᠭᠡᠷ᠎ᠡ ᠬᠡᠷᠡᠭᠵᠢᠳᠡᠭ᠃^[1]

ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠦᠨ ᠡᠨᠡ ᠬᠡᠯᠪᠡᠷᠢ ‍ᠢᠢ ᠮᠥᠨ ᠧᠯᠯᠢᠫᠰ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ^{elliptic geometry} ᠭᠡᠳᠡᠭ᠃^[2] ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋^Euclid ‍ᠦᠨ ᠪᠤᠶᠤ ᠤᠯᠠᠮᠵᠢᠯᠠᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠲᠡᠭ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠡᠷ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ^{Лобачевский} ᠬᠢ ᠨᠢ ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢ ᠳᠦᠷᠰᠦᠯᠡᠳᠡᠭ ᠪᠢᠯᠡ᠃

ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠦᠳᠦᠭᠦᠷ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠵᠠᠭᠪᠤᠷ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃

ᠷᠢᠮᠠᠨ ᠥᠪᠡᠷ ‍ᠦᠨ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠢ ᠬᠣᠶᠠᠷ ᠬᠡᠮᠵᠢᠭᠳᠠᠬᠦᠨᠲᠦ ᠬᠠᠪᠲᠠᠭᠠᠢ᠂ ᠭᠤᠷᠪᠠᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠬᠢᠭᠡᠳ ᠲᠡᠭᠦᠨ ᠡᠴᠡ ᠳᠡᠭᠡᠭᠰᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠳ᠋ᠤ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠢᠳᠠᠭ᠃

ᠠᠻᠰᠢᠣᠮ

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠦᠭ ‍ᠢᠢᠡᠷ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠵᠤ ᠪᠤᠢ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠪᠠᠨ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠨᠢᠭᠡ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠰ ᠪᠦᠷᠢ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃ ^[3]

ᠪᠠᠰᠠ᠄

ᠬᠡᠷᠪᠡ $g$ ᠰᠢᠯᠠᠭᠤᠨ ᠱᠤᠭᠤᠮ ᠪᠥᠭᠡᠳ $P$ ᠴᠡᠭ ᠨᠢ ᠲᠡᠭᠦᠨ ‍ᠦ ᠭᠠᠳᠠᠨ᠎ ᠎ᠠ ᠣᠷᠤᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠪᠣᠯ $g$ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠ $P$ ᠴᠡᠭ ‍ᠢ ᠳᠠᠮᠨᠠᠭᠰᠠᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ $g$ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠦᠯᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠨᠢᠭᠡ ᠴᠦ ᠰᠢᠯᠠᠭᠤᠨ $h$ ᠪᠠᠢᠢᠬᠤ ᠦᠭᠡᠢ᠃^[4]

ᠪᠠᠰᠠ᠄

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠧᠷᠫᠧᠨ᠍ᠡᠳᠢᠻᠦ᠋ᠯᠠᠷ^{perpendicular} ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠪᠠᠨ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃

ᠶᠡᠷᠦ ᠨᠢ᠂ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠨᠢᠭᠡ ᠲᠠᠯ ‍ᠠ᠋ ᠳᠠᠬᠢ ᠪᠦᠬᠢ ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ ‍ᠤᠳ ᠲᠡᠷᠡ ᠬᠦ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ "ᠦᠨᠡᠮᠯᠡᠬᠦᠢ ᠲᠤᠢᠢᠯ" ᠬᠡᠮᠡᠬᠦ ᠭᠠᠭᠴᠠ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠨᠥᠭᠦᠭᠡ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ ‍ᠤᠳ ᠴᠤ ᠮᠥᠨ ᠨᠢᠭᠡ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃

ᠬᠣᠶᠠᠷ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠲᠤᠢᠢᠯ ‍ᠤᠳ ᠨᠢᠭᠡ ᠯᠡ ᠲᠤᠢᠢᠯ ᠪᠠᠢᠢᠨ᠌ ᠊ᠠ᠋᠃

ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠬᠣᠶᠠᠷ ᠵᠦᠭ ᠲᠦ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠪᠢᠲᠠᠭᠦᠦ ᠪᠠᠢᠢᠳᠠᠭ᠂ ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠠᠡ ᠬᠡᠯᠡᠪᠡᠯ ᠬᠣᠶᠠᠷ ᠦᠵᠦᠭᠦᠷ ᠨᠢ ᠠᠭᠤᠯᠵᠠᠨ ᠨᠡᠢᠢᠯᠡᠳᠡᠭ᠃^[5]

ᠰᠢᠯᠠᠭᠤᠨ ᠭᠡᠵᠦ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠪᠦᠰᠡᠯᠡᠭᠦᠷ ᠲᠣᠭᠤᠷᠢᠭ ‍ᠤᠳ ‍ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠴᠢᠩᠬ᠎ᠠ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍᠋ᠠ᠋ ᠳ᠋ᠤ ᠪᠠᠢᠢᠷᠢᠯᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠳ ‍ᠦᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯᠵᠤ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠰᠢᠯᠠᠭᠤᠨ ‍ᠢ ᠢᠩᠭᠢᠵᠦ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠭᠰᠠᠨ ‍ᠢᠢᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ ᠨᠢ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ‍ᠢᠢᠡᠷ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠳᠠᠨ᠎ ‍ᠠ᠋᠄

$L=\pi r$

ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄
L ᠨᠢ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ᠂
r ᠨᠢ ᠬᠡᠭᠡᠰᠦ (ᠷᠠᠳᠢᠦ᠋ᠰ^radius)᠃

ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠢᠮᠠᠭᠲᠠ 180° ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃^[6]

ᠮᠥᠨ ᠲᠥᠰᠦᠲᠠᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠨᠤᠭᠤᠳ ᠵᠠᠪᠠᠯ ᠻᠣᠨ᠍ᠭ᠌ᠷᠦ᠋ᠧᠨ᠍ᠲ^congruent ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ

1. ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠨᠢᠭᠡ ᠯᠡ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃ (ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)
2. ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᢩᠪᠥᠭᠡᠳ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠣᠷᠤᠰᠢᠬᠤ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃ (ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)
3. ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠦᠭᠡᠢ᠃ (ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)

ᠷᠢᠮᠠᠨ ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋᠂ ᠭᠠᠦ᠋ᠰᠰ^Gauss᠂ ᠪᠣᠯᠢᠶᠠᠢ^Boliay᠂ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠨᠠᠷ ‍ᠤᠨ ᠣᠨᠣᠯ  ᠢ ᠪᠦᠭᠦᠳᠡ ᠶᠢ ᠨᠢᠭᠡ ᠳᠡᠭᠡᠪᠦᠷ ᠳᠣᠣᠷ᠎ᠠ ᠣᠷᠣᠭᠤᠯᠵᠤ ᠥᠭᠭᠦᠭᠰᠡᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃
ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ᠂ ᠲᠡᠳᠡᠭᠡᠷ  ᠢ ᠦᠯᠡᠮᠵᠢ ᠥᠷᠭᠡᠳᠬᠡᠵᠦ᠂ ᠡᠮᠦᠨ᠎ᠡ ᠨᠢ ᠬᠡᠨ  ᠦ ᠴᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠵᠦ ᠪᠠᠶ᠋ᠢᠭᠠᠳᠦᠢ ᠨᠡᠩ ᠰᠢᠨ ᠎ᠡ ᠪᠣᠯᠤᠮᠵᠢ ᠶᠢ ᠬᠦᠮᠦᠨ ᠲᠥᠷᠥᠯᠬᠢᠲᠡᠨ  ᠦ ᠣᠶᠤᠨ ᠤᠬᠠᠭᠠᠨ  ᠳᠤ ᠨᠡᠭᠡᠭᠡᠵᠦ ᠥᠭᠴᠡᠢ᠃
ᠪᠣᠳᠠᠲᠤ ᠹᠢᠽᠢᠻ^physic ᠣᠷᠤᠨ ᠵᠠᠢ᠂ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠴᠢᠨᠠᠷ ‍ᠤᠳ ᠨᠢ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠢᠰᠢ ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠬᠠᠷᠭᠠᠯᠵᠠᠳᠠᠭ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃^[7]

ᠲᠡᠦᠬᠡᠨ ᠳᠠᠷᠠᠭᠠᠯᠠᠯ  ᠤᠨ ᠬᠤᠪᠢ  ᠳᠤ ᠷᠢᠮᠠᠨ  ᠤ ᠭᠧᠣᠮᠡᠲ᠋ᠷ ᠨᠢ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠧᠡᠲ᠋ᠧᠷ ᠡᠴᠡ ᠬᠣᠵᠢᠭᠤ᠂ 1854 ᠣᠨ  ᠳᠤ ᠭᠠᠷᠴᠤ ᠢᠷᠡᠭᠰᠡᠨ᠃

ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠦᠨ ᠰᠠᠭᠤᠷᠢ  ᠶᠢ ᠷᠢᠮᠠᠨ ᠪᠥᠷᠢᠨ ᠬᠢᠨᠠᠵᠤ ᠦᠵᠠᠭᠡᠳ᠂ ᠲᠡᠭᠦᠨ ᠦ ᠣᠷᠣᠨ ᠳᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ ᠥᠪᠡᠷ  ᠦᠨ ᠵᠠᠷᠴᠢᠮ  ᠤᠳ  ᠢ ᠰᠠᠨᠠᠯ ᠪᠣᠯᠭᠠᠵᠠᠢ᠃

ᠡᠭᠦᠪᠡᠷ  ‍ᠢᠶ᠋ᠡᠨ ᠧᠦᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠡᠲ᠋ᠧᠷ  ᠦᠨ ᠣᠯᠠᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠪᠠᠶ᠋ᠢᠵᠤ᠂ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠶᠢᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠥᠭᠡᠷ ‍ᠠ᠋ ᠪᠦᠲᠦᠴᠡ ᠶᠢ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠵᠤ ᠪᠣᠯᠬᠤ ᠶᠢ ᠲᠤᠩᠬᠠᠭᠯᠠᠭᠰᠠᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃

ᠷᠢᠮᠠᠨ ᠡᠳᠡᠭᠡᠷ ᠪᠣᠯᠤᠮᠵᠢᠲᠤ ᠬᠤᠪᠢᠯᠪᠤᠷᠢ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠡᠩ  ᠦᠨ ᠨᠢᠭᠡᠨ ᠲᠣᠬᠢᠶᠠᠯᠳᠤᠯ ᠪᠣᠯᠬᠤ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠣᠨ ᠵᠠᠢ ᠶᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠢ ᠲᠠᠨᠢᠯᠴᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠪᠢᠯᠡ᠃᠎^[5]

ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ‍ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ ‍ᠢ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠧᠢᠢᠨ᠍ᠱᠲ᠋ᠠᠢᠢᠨ^Einstein ᠥᠪᠡᠷ ᠲᠦ ᠪᠡᠨ ᠬᠡᠷᠡᠭ ᠲᠡᠢ ᠦᠨᠡᠨ ᠵᠥᠪ ᠮᠠᠲᠾᠧᠮᠠᠲᠢᠻ^mathematic ᠬᠡᠷᠡᠭᠰᠡᠯ ‍ᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠡᠴᠡ ᠣᠯᠵᠤ^[8]᠂ ᠲᠡᠭᠦᠨ ‍ᠳᠤ ᠲᠦᠰᠢᠭᠯᠡᠭᠰᠡᠨ ᠪᠢᠯᠡ᠃

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ

ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - Riemannian geometry - геометрия Римана

ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - elliptic geometry - эллиптическая геометрия

ᠪᠥᠮᠪᠦᠷᠴᠦᠭ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - spherical geometry - сферическая геометрия

ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - positive curvature - положительная кривизна

ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - negative curvature - отрицательная кривизна

ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ - perpendicular - перпендикуляр

ᠫᠠᠷᠠᠯᠯᠧᠯ - parallel - параллель

ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ - intersect - пересекаться

ᠬᠠᠪᠲᠠᠭᠠᠢ - plane - плоскость

ᠣᠭᠲᠠᠷᠭᠤᠢ᠂ ᠣᠷᠤᠨ ᠵᠠᠢ - space - пространство
ᠦᠨᠡᠮᠯᠡᠬᠦᠢ ᠲᠤᠢᠢᠯ - absolute pole - абсолютный полюс

ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍- sphere - сфера

ᠲᠣᠭᠤᠷᠢᠭ - circle - окружность

ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨ - dimension - измерение

ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ

↑ Наука. Геометрия Римана. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28
↑ Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02
↑ Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23
↑ Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0
↑ ^5.0 ^5.1 О.А.Старова. Геометрия Римана. 2012/12. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]
↑ WolframMathWorld. Elliptic Geometry. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02
↑ В.А.Фок. Теория пространства, времени и тяготения. ГИТТЛ. Москва 1955. с. 253
↑ Max Born. Physics in My Generation (Springer, 1969), ISBN 0-387-90008-X. p. 329

[1] Наука. Геометрия Римана. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28

[2] Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02

[3] Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23

[4] Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0

[:0-5] 5.0 ^5.1 О.А.Старова. Геометрия Римана. 2012/12. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]

[6] WolframMathWorld. Elliptic Geometry. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02

[7] В.А.Фок. Теория пространства, времени и тяготения. ГИТТЛ. Москва 1955. с. 253

[8] Max Born. Physics in My Generation (Springer, 1969), ISBN 0-387-90008-X. p. 329

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@@ 4-р мөр: / 4-р мөр: @@
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 <br>
-ᠷᠢᠮᠠᠨ<sup>Riemann</sup>  ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ<sup>geometry</sup> ᠨᠢ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠡᠭ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠳᠠᠷᠭᠤ ᠪᠤᠶᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ᠳᠡᠭᠡᠷ᠎ᠡ ᠬᠡᠷᠡᠭᠵᠢᠳᠡᠭ᠃<ref>[https://science.wikia.org/ru/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0 Наука. Геометрия Римана]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28</ref>
+ᠷᠢᠮᠠᠨ<sup>Riemann</sup>  ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ<sup>geometry</sup> ᠨᠢ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠳᠠᠷᠭᠤ ᠪᠤᠶᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ᠳᠡᠭᠡᠷ᠎ᠡ ᠬᠡᠷᠡᠭᠵᠢᠳᠡᠭ᠃<ref>[https://science.wikia.org/ru/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0 Наука. Геометрия Римана]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28</ref>
 ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠦᠨ ᠡᠨᠡ ᠬᠡᠯᠪᠡᠷᠢ ‍ᠢᠢ ᠮᠥᠨ ᠧᠯᠯᠢᠫᠰ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ<sup>elliptic geometry</sup> ᠭᠡᠳᠡᠭ᠃<ref>[https://www.booksite.ru/fulltext/1/001/008/080/804.htm Н. В. Ефимов. Неевклидовы геометрии]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02 </ref> ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋<sup>Euclid</sup> ‍ᠦᠨ ᠪᠤᠶᠤ ᠤᠯᠠᠮᠵᠢᠯᠠᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠲᠡᠭ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠡᠷ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ [[ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠊ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ|ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ]] ‍ᠢᠢᠨ<sup>Лобачевский</sup> ᠬᠢ ᠨᠢ ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢ ᠳᠦᠷᠰᠦᠯᠡᠳᠡᠭ ᠪᠢᠯᠡ᠃
-ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠡᠶᠡᠷᠡᠭ ᠮᠤᠷᠤᠢᠢᠯᠲᠠᠲᠠᠢ ᠭᠦᠳᠦᠭᠦᠷ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠵᠠᠭᠪᠤᠷ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃
+ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠦᠳᠦᠭᠦᠷ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠵᠠᠭᠪᠤᠷ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃
-ᠷᠢᠮᠠᠨ ᠥᠪᠡᠷ ‍ᠦᠨ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠢ ᠬᠣᠶᠠᠷ ᠬᠡᠮᠢᠵᠳᠠᠬᠦᠨᠲᠦ ᠬᠠᠪᠲᠠᠭᠠᠢ᠂ ᠭᠤᠷᠪᠠᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨ ᠲᠦ ᠬᠢᠭᠡᠳ ᠲᠡᠭᠦᠨ ᠡᠴᠡ ᠳᠡᠭᠡᠰᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠳ᠋ᠤ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠢᠳᠠᠭ᠃  <br><br>
+ᠷᠢᠮᠠᠨ ᠥᠪᠡᠷ ‍ᠦᠨ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠢ ᠬᠣᠶᠠᠷ ᠬᠡᠮᠵᠢᠭᠳᠠᠬᠦᠨᠲᠦ ᠬᠠᠪᠲᠠᠭᠠᠢ᠂ ᠭᠤᠷᠪᠠᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠬᠢᠭᠡᠳ ᠲᠡᠭᠦᠨ ᠡᠴᠡ ᠳᠡᠭᠡᠭᠰᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠳ᠋ᠤ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠢᠳᠠᠭ᠃  <br><br>
 == ᠠᠻᠰᠢᠣᠮ ==
@@ 41-р мөр: / 41-р мөр: @@
 ᠬᠣᠶᠠᠷ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠲᠤᠢᠢᠯ ‍ᠤᠳ ᠨᠢᠭᠡ ᠯᠡ ᠲᠤᠢᠢᠯ ᠪᠠᠢᠢᠨ᠌ ᠊ᠠ᠋᠃
-ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠬᠣᠶᠠᠷ ᠵᠦᠭ ᠲᠦ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠪᠢᠲᠠᠭᠦ ᠪᠠᠢᠢᠳᠠᠭ᠂ ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠠᠡ ᠬᠡᠯᠡᠪᠡᠯ ᠬᠣᠶᠠᠷ ᠦᠵᠦᠭᠦᠷ ᠨᠢ ᠠᠭᠤᠯᠵᠠᠨ ᠨᠡᠢᠢᠯᠡᠳᠡᠭ᠃<ref name=":0">[http://www.e-osnova.ru/PDF/osnova_3_24_3845.pdf О.А.Старова. Геометрия Римана. 2012/12]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]</ref>
+ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠬᠣᠶᠠᠷ ᠵᠦᠭ ᠲᠦ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠪᠢᠲᠠᠭᠦᠦ ᠪᠠᠢᠢᠳᠠᠭ᠂ ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠠᠡ ᠬᠡᠯᠡᠪᠡᠯ ᠬᠣᠶᠠᠷ ᠦᠵᠦᠭᠦᠷ ᠨᠢ ᠠᠭᠤᠯᠵᠠᠨ ᠨᠡᠢᠢᠯᠡᠳᠡᠭ᠃<ref name=":0">[http://www.e-osnova.ru/PDF/osnova_3_24_3845.pdf О.А.Старова. Геометрия Римана. 2012/12]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]</ref>
-ᠰᠢᠯᠠᠭᠤᠨ ᠭᠡᠵᠦ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠪᠦᠰᠡᠯᠡᠭᠦᠷ ᠲᠣᠭᠤᠷᠢᠭ ‍ᠤᠳ ‍ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠴᠢᠩᠬ᠎ᠠ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍᠋ᠠ᠋ ᠳ᠋ᠤ ᠪᠠᠢᠢᠷᠢᠯᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠳ ‍ᠦᠨ ᠣᠯᠤᠨᠯᠢᠭ ᠪᠣᠯᠵᠤ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃
+ᠰᠢᠯᠠᠭᠤᠨ ᠭᠡᠵᠦ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠪᠦᠰᠡᠯᠡᠭᠦᠷ ᠲᠣᠭᠤᠷᠢᠭ ‍ᠤᠳ ‍ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠴᠢᠩᠬ᠎ᠠ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍᠋ᠠ᠋ ᠳ᠋ᠤ ᠪᠠᠢᠢᠷᠢᠯᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠳ ‍ᠦᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯᠵᠤ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃
 ᠰᠢᠯᠠᠭᠤᠨ ‍ᠢ ᠢᠩᠭᠢᠵᠦ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠭᠰᠠᠨ ‍ᠢᠢᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ ᠨᠢ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ‍ᠢᠢᠡᠷ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠳᠠᠨ᠎ ‍ᠠ᠋᠄
@@ 55-р мөр: / 55-р мөр: @@
 </span>
 </span><br><br>
-ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠢᠮᠠᠭᠲᠠ <span style="writing-mode: horizontal-tb;">180°</span> ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃<ref>[https://mathworld.wolfram.com/EllipticGeometry.html WolframMathWorld. Elliptic Geometry]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02</ref>
+ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠢᠮᠠᠭᠲᠠ <span style="writing-mode: horizontal-tb;">180°</span> ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃<ref>[https://mathworld.wolfram.com/EllipticGeometry.html WolframMathWorld. Elliptic Geometry]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02</ref>
 ᠮᠥᠨ ᠲᠥᠰᠦᠲᠠᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠨᠤᠭᠤᠳ ᠵᠠᠪᠠᠯ ᠻᠣᠨ᠍ᠭ᠌ᠷᠦ᠋ᠧᠨ᠍ᠲ<sup>congruent</sup> ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃
@@ 77-р мөр: / 77-р мөр: @@
 ᠡᠭᠦᠪᠡᠷ  ‍ᠢᠶ᠋ᠡᠨ ᠧᠦᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠡᠲ᠋ᠧᠷ  ᠦᠨ ᠣᠯᠠᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠪᠠᠶ᠋ᠢᠵᠤ᠂ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠶᠢᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠥᠭᠡᠷ ‍ᠠ᠋ ᠪᠦᠲᠦᠴᠡ ᠶᠢ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠵᠤ ᠪᠣᠯᠬᠤ ᠶᠢ ᠲᠤᠩᠬᠠᠭᠯᠠᠭᠰᠠᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃
-ᠷᠢᠮᠠᠨ ᠡᠳᠡᠭᠡᠷ ᠪᠣᠯᠤᠮᠵᠢᠲᠤ ᠬᠤᠪᠢᠯᠪᠤᠷᠢ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠡᠩ  ᠦᠨ ᠨᠢᠭᠡᠨ ᠲᠣᠬᠢᠶᠠᠯᠳᠤᠯ ᠪᠣᠯᠬᠤ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠡᠭ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠣᠨ ᠵᠠᠢ ᠶᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠢ ᠲᠠᠨᠢᠯᠴᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠪᠢᠯᠡ᠃᠎<ref name=":0" />
+ᠷᠢᠮᠠᠨ ᠡᠳᠡᠭᠡᠷ ᠪᠣᠯᠤᠮᠵᠢᠲᠤ ᠬᠤᠪᠢᠯᠪᠤᠷᠢ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠡᠩ  ᠦᠨ ᠨᠢᠭᠡᠨ ᠲᠣᠬᠢᠶᠠᠯᠳᠤᠯ ᠪᠣᠯᠬᠤ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠣᠨ ᠵᠠᠢ ᠶᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠢ ᠲᠠᠨᠢᠯᠴᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠪᠢᠯᠡ᠃᠎<ref name=":0" />
 [[ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ᠊ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ|ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ‍ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ]] ‍ᠢ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠧᠢᠢᠨ᠍ᠱᠲ᠋ᠠᠢᠢᠨ<sup>Einstein</sup> ᠥᠪᠡᠷ ᠲᠦ ᠪᠡᠨ ᠬᠡᠷᠡᠭ ᠲᠡᠢ ᠦᠨᠡᠨ ᠵᠥᠪ ᠮᠠᠲᠾᠧᠮᠠᠲᠢᠻ<sup>mathematic</sup> ᠬᠡᠷᠡᠭᠰᠡᠯ ‍ᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠡᠴᠡ ᠣᠯᠵᠤ<ref>Max Born. Physics in My Generation (Springer, 1969), <nowiki>ISBN 0-387-90008-X</nowiki>. p. 329</ref>᠂ ᠲᠡᠭᠦᠨ ‍ᠳᠤ ᠲᠦᠰᠢᠭᠯᠡᠭᠰᠡᠨ ᠪᠢᠯᠡ᠃<br><br>
@@ 87-р мөр: / 87-р мөр: @@
 ᠪᠥᠮᠪᠦᠷᠴᠦᠭ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - spherical geometry - сферическая геометрия
-ᠡᠶᠡᠷᠡᠭ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - positive curvature - положительная кривизна
+ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - positive curvature - положительная кривизна
 ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - negative curvature - отрицательная кривизна
@@ 109-р мөр: / 109-р мөр: @@
 <br>
 == ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ ==
+<references />
+[[Ангилал:ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ]]
+[[Ангилал:ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]