Birinchi o'zgarish - First variation

Qo'llaniladi matematika va o'zgarishlarni hisoblash, birinchi o'zgarish a funktsional J(y) chiziqli funktsional sifatida aniqlanadi ${displaystyle deltasi J (y)}$ funktsiyani xaritalash h ga

${displaystyle delta J (y, h) = lim _ {varepsilon o 0} {frac {J (y + varepsilon h) -J (y)} {varepsilon}} = chap. {frac {d} {dvarepsilon}} J (y + varepsilon h) ight | _ {varepsilon = 0},}$

qayerda y va h funktsiyalar va ε skalar. Bu sifatida tanilgan Gateaux lotin funktsional.

Misol

Ning birinchi o'zgarishini hisoblang

${displaystyle J (y) = int _ {a} ^ {b} yy'dx.}$

Yuqoridagi ta'rifdan,

${displaystyle {egin {hizalangan} delta J (y, h) & = chap. {frac {d} {dvarepsilon}} J (y + varepsilon h) ight | _ {varepsilon = 0} & = chap. {frac { d} {dvarepsilon}} int _ {a} ^ {b} (y + varepsilon h) (y ^ {prime} + varepsilon h ^ {prime}) dxight | _ {varepsilon = 0} & = chap. {frac {d} {dvarepsilon}} int _ {a} ^ {b} (yy ^ {prime} + yvarepsilon h ^ {prime} + y ^ {prime} varepsilon h + varepsilon ^ {2} hh ^ {prime}) dxight | _ {varepsilon = 0} & = left.int _ {a} ^ {b} {frac {d} {dvarepsilon}} (yy ^ {prime} + yvarepsilon h ^ {prime} + y ^ {prime} varepsilon h + varepsilon ^ {2} hh ^ {prime}) dxight | _ {varepsilon = 0} & = left.int _ {a} ^ {b} (yh ^ {prime} + y ^ {prime} h + 2varepsilon hh ^ {prime}) dxight | _ {varepsilon = 0} & = int _ {a} ^ {b} (yh ^ {prime} + y ^ {prime} h) dxend {aligned}}}$

Shuningdek qarang

• O'zgarishlar hisobi
• Funktsional lotin

Bu matematik tahlil - tegishli maqola a naycha. Siz Vikipediyaga yordam berishingiz mumkin uni kengaytirish.