a5_soln - a Øh7 A ×× igÒÑ eÒ Ø5Ó ÐÙ Ø iÓÒ ×...

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Unformatted text preview: a Øh7 A ×× igÒÑ eÒ Ø5Ó ÐÙ Ø iÓÒ × eØ f : R 2 → R aÒdd eÒ e g ( x, y ) = f (sin y, cos x ) F iÒd g xx aÒd g yy Øa ØeaÒÝa ××ÙÑ Ô Ø iÓÒ ×ÝÓÙÒ eed ed ØÓÑ ak e Ó ÐÙ Ø iÓÒ :eØ u = sin y aÒd v = cos x h eÒa ××ÙÑ iÒg Øha Ø f i×d ieÖeÒ Ø iab ÐeÛ ehaÚ e g x = ∂f ∂v dv dx = (- sin x ) f v g y = ∂f ∂u du dy = (cos y ) f u Ó Øak e×ecÓÒdÔa ÖØ ia Ð×Û eÒ eed ØÓa ××ÙÑ eØha Ø f v aÒd f u a Öed ieÖeÒ Ø iab Ðeh eÒÛ ehaÚ e g xx = ∂g x ( x, u, v ) ∂v dv dx + ∂g x ( x, u, v ) ∂x = (- sin x ) f vv · (- sin x ) + (- cos x ) f v = sin 2 xf vv- cos xf v g yy = ∂g y ( y, u, v ) ∂u du dy + ∂g y ( y, u, v ) ∂y = (cos y ) f uu · (cos y ) + (- sin y ) f u = cos 2 yf uu- sin yf u eØ f, g : R → R Û h eÖe f aÒd g a ÖeØÛ iced ieÖeÒ Ø iab ÐehÓÛ Øha Ø u ( x, t ) = f ( x- at ) + g ( x + at ) i×a ×Ó ÐÙ Ø iÓÒÓ fØh eÛ aÚ eeÕÙa Ø iÓÒ : u tt = a 2 u xx Ó ÐÙ Ø iÓÒ :iÒ ce f aÒd g a ÖeØÛ iced ieÖeÒ Ø iab ÐeÛ ecaÒaÔÔ ÐÝ Øh echa iÒ ÖÙ ÐeØÓg eØ u x = f prime ( x- at )(1) + g prime ( x + at )(1) u xx = f primeprime ( x- at )(1) + g primeprime ( x + at )(1) u t = f prime ( x- at )(- a ) + g prime ( x + at )( a ) u tt = f primeprime ( x- at )(- a ) 2 + g primeprime ( x + at )( a ) 2 eÒ ce u tt = a 2 f primeprime ( x- at ) + a 2 g primeprime ( x + at ) = a 2 ( f primeprime ( x- at ) + g primeprime ( x- at )) = a 2 u xx eØ g : R → R aÒd ÐeØ f ( x, y ) = g ( u 2 v ) Û h eÖe u = e x aÒd v = x 2 + y 3 ×e Øh echa iÒ ÖÙ ÐeØÓÒd ∂ 2 f ∂x∂y Øa ØeaÒÝa ××ÙÑ Ô Ø iÓÒ ×ÝÓÙÒ eed ed ØÓÑ ak e Ó ÐÙ Ø iÓÒ :ÓÒd f y Û eÒ eed ØÓa ××ÙÑ eØha Ø g i×d ieÖeÒ Ø iab ÐeeØ t = u 2 v Øh eÒÛ ehaÚ e f y = dg dt ∂t ∂v ∂v ∂y = g prime ( t )...
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This note was uploaded on 11/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.

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a5_soln - a Øh7 A ×× igÒÑ eÒ Ø5Ó ÐÙ Ø iÓÒ ×...

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