ѧѹͧͧҡͧ

ѧѹͧͧҡͧ

͹ؾѹ f ͧѧѹ f ͧ繿ѧѹ ԧ f 'ǹҡ繿ѧѹ͹ؾѹ ѧ f '͹ؾѹ ¹᷹ f'' , f ''͹ؾѹ¹᷹ f'''

f''(x) = Dx(f'(x))

f'''(x) = Dx(f''(x))

f (4)(x) = Dx(f'''(x))

..

¡ f'͹ؾѹѹѺ˹ (the first derivative) ͧ f

f''͹ؾѹѹѺͧ (the second derivative) ͧ f'

f'''͹ؾѹѹѺ(the third derivative) ͧ f''

ѹѺ n ٧ 3 ¹ f (n) ᷹ ͹ؾѹѹѺ n (nth

derivative) ͧ f

ҧ

͹ؾѹ繡͹ؾѹͧѧѹͧ
ͧǢ ѧѹµ(function of several variables) ͧҡɮվ鹰ҹͧѧѹµդᵡҧҡ
ɮվ鹰ҹͧѧѹ˹觵ҧҡդ¤֧Ѻ
ɮվ鹰ҹͧѧѹͧè֧Ƕ֧ѧѹͧǹ˭

z ¡ 繿ѧѹ(function)ͧͧ x y ФѹѺ (x,y)˹öҤ z §˹觤ҹѭѡɳ 繵 ᷹Ңͧѧѹ(x,y) ¹ 㹺ҧѭѡɳ ͹ѹ ¡ zҵõ(dependent variable)

¡ x y ҵ(independent variable)૵ͧش(x,y) Ҥ繨ӹǹԧ¡ (domain) ͧ ѧѹ

ҧ

ͧѧѹ

Ңͧ z 繨ԧ

ѧ ͧѧѹ

͹ؾѹ

͹ؾѹҢͧѧѹµºѺ˹µ ¡繤ҤǨ¡͹ؾѹҹ ͹ؾѹ¢ͧѧѹºѺù

f 繿ѧѹͧ x y ͹ؾѹ¢ͧ f(x,y) ºѺ x y ¹᷹

ѭѡɳѧ鹨͵ͧʴ÷繤Ҥ¹

Եҹ͹ؾѹҤش ѡ¹᷹

͹ؾѹѹѺ٧

Ѻѧѹ ͹ؾѹºѺ n ӹǹ ͹ؾѹҹѧ繿ѧѹͧ ¡͹ؾѹѹѺ˹觫е Ҩ͹ؾѹºѺ㴵˹ͷءǡ͹ؾѹ¢ͧ͹ؾѹѹѺ˹ ¡͹ؾѹѹѺͧѺ͹ؾѹѹѺ٧ͧǡѹ ͹ؾѹ¢ͧ͹ؾѹѹѺͧ ¡͹ؾѹѹѺ繵㹷СǶ֧͹ؾѹѹѺͧѹѺҹ ѡǡöзѹѺ٧աµͧ

͹ؾѹѹѺͧ

繿ѧѹö͹ؾѹ·˹ 觵ҧ繿ѧѹͧ x y Ƕö͹ؾѹա¶֧Ӽŷ͹ѹºѺ x y ա˹ ¡͹ؾѹѹѺͧ觨շ 4 ѧ

Ѻѭѡɳ ¤͹ؾѹ¢ͧѧѹ ºѺ x ֧͹ؾѹ¢ͧѧѹ ºѺ y ǹ ¤͹ؾѹ¢ͧѧѹ ºѺ y ֧͹ؾѹ¢ͧѧѹ ºѺ x

ѧ ᵡҧѹӴѺͧ͹ؾѹ«觼ŷ·Ҩҡѹ㹡óշѧѹ 繿ѧѹͧ㹺dzҧ§ͧش(x,y) ö ( ) ( )ѧ繿ѧѹͧ㹺dzѧ ö ( ) =

͹ؾѹѹѺ

͹ؾѹѹѺͧ 觵ҧ繿ѧѹͧ x y ö͹ؾѹա¶֧ӼŷҡѹѺͧ͹ؾѹºѺ x y ա˹ ¡͹ؾѹѹѺմѧ

Ѻ͹ؾѹѹѺҡ仡㹷ӹͧǡѹ

ҧԧ

http://www.science.cmu.ac.th/department/math/Project499/JAMRAS/partial_differention10.html

ٻҾǢͧ

Ԫ


ͧṹ ?

ҧ :


Pt_Ben

ʶҹ :
෤ԤüԵ