Lecture24 - F igÙ Öe Øh eÖF Ó ÖÑ ×Ó fØh eC ha...

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Unformatted text preview: F igÙ Öe : Øh eÖF Ó ÖÑ ×Ó fØh eC ha iÒÙ Ðe ÒÓÙ Ö iÒ ØÖÓdÙ c Ø iÓÒ ØÓ Øh eÑ Ù ÐØ iÚa Ö ia Øecha iÒ ÖÙ ÐeÛ ecÓÒ × id eÖed Øh e× iØÙa Ø iÓÒ iÒÛ h ich z = f ( x, y ) Û iØh x aÒd y eachd eÔ eÒd eÒ ØÙÔÓÒa Øh iÖdÚa Ö iab Ðe t ÓÛ eÚ eÖÛ eÑ aÝeÒ cÓÙÒ ØeÖÔ ÖÓb ÐeÑ × iÒÛ h ich x aÒd y a ÖeØh eÑ ×e ÐÚ e×g iÚ eÒbÝ Ñ Ù ÐØ iÚa Ö ia Øe fÙÒ c Ø iÓÒ × C h a iÒÙ Ðe# ÙÔÔÓ ×eØha Ø z = f ( x, y ) Û h iÐe x = g 1 ( s, t ) aÒd y = g 2 ( s, t ) ×eeØh eØÖeed iag ÖaÑ Ðab e ÐÐeda ×F igÙ Öe Ò Øh i×× iØÙa Ø iÓÒÛ eÑ aÝ Ú ieÛ z a ×b e iÒgd eÔ eÒd eÒ ØÙÔÓÒ s aÒd t Û iØhÔa ÖØ ia Ðd eÖ iÚa Ø iÚ e×g iÚ eÒbÝ ∂z ∂s = ∂z ∂x ∂x ∂s + ∂z ∂y ∂y ∂s aÒd ∂z ∂t = ∂z ∂x ∂x ∂t + ∂z ∂y ∂y ∂t . hÝ?Óca ÐcÙ Ða Øe ∂z ∂s a ÐÐÛ ehaÚ eØÓdÓ i×Ú ieÛ t a ×cÓÒ ×ØaÒ ØaÒdÙ ×eÓÙ Ö Ô ÖeÚ iÓÙ ×Ú eÖ× iÓÒÓ fØh echa iÒ ÖÙ Ðe !h eÓÒ ÐÝd ieÖeÒ ce i×Øha ØÛ h eÖea ×Øh e ÖeÕÙ iÖedd eÖ iÚa Ø iÚ e×Ó f x aÒd y Ù ×ed ØÓb eÓ Öd iÒa ÖÝd eÖ iÚa Ø iÚ e×Øh eÝa ÖeÒÓÛ Ôa ÖØ ia Ðd eÖ iÚa Ø iÚ e× h i×Ôa ÖØ icÙ Ða ÖÚ eÖ× iÓÒÓ fØh echa iÒ ÖÙ ÐeÛ iÐÐb eaÒe××eÒ Ø ia ÐØÓÓ ÐiÒ Ða ØeÖ cÓÙ Ö×e×b ecaÙ ×e iØa ÐÐÓÛ ×Ù ×ØÓcÓÒÚ eÖØd eÖ iÚa Ø iÚ e× fÖÓÑ ÓÒ e ØÛ Ód iÑ eÒ × iÓÒa Ð cÓÓ Öd iÒa Øe×Ý ×ØeÑ ØÓaÒÓ Øh eÖ E ÜaÑ Ô Ðe : ÙÔÔÓ ×e z = f ( x, y ) fÛ e iÒ ØÖÓdÙ ceØh echaÒgeÓ fÚa Ö iab Ðe× x = r cos θ y = r sin θ Øh eÒÛ ecaÒ eÜÔ Öe×× z rr iÒ ØeÖÑ ×Ó fØh ed eÖ iÚa Ø iÚ e×Ó f z Û iØh Öe×Ô ec ØØÓ x aÒd y a × fÓ ÐÐÓÛ × : • Óge Ø z rr Û eÑ Ù ×Ø×Øa ÖØÛ iØh z r : z r = ∂z ∂r = ∂z ∂x ∂x ∂r + ∂z ∂y ∂y ∂r = z x cos θ + z y sin θ. • ÓÛÛ e jÙ ×ØhaÚ eØÓd ieÖeÒ Ø ia ØeØh i×Öe×Ù ÐØÛ iØh Öe×Ô ec ØØÓ r : z rr = ∂ 2 z ∂r 2 = ∂ ∂r parenleftbigg ∂z ∂r parenrightbigg = ∂ ∂r ( z x cos θ + z y sin θ ) = cos θ ∂ ∂r ( z x ) + sin θ ∂ ∂r ( z y ) = cos θ bracketleftbigg ∂z x ∂x ∂x ∂r + ∂z x ∂y ∂y ∂r bracketrightbigg + sin θ bracketleftbigg ∂z y ∂x ∂x ∂r + ∂z y ∂y ∂y ∂r bracketrightbigg = cos θ [ z xx cos θ + z xy sin θ ] + sin θ [ z yx cos θ + z yy sin θ ] = z xx cos 2 θ + 2 z xy sin θ cos θ + z yy sin 2 θ C ÓÑÑ eÒ Ø× : h eÖ igh ØhaÒd × id eÓ fÐiÒ e aÔÔ ea Ö×ØÓhaÚ eaÑ iÜ ØÙ ÖeÓ ffÙÒ c Ø iÓÒ × Ó f ( x, y ) aÒd ( r, θ ) bÙ ØÛ ea Öe fÖeeØÓ ÖeÔ ÐaceØh eÚa Ö iab Ðe× iÒ z x aÒd z y Ù × iÒgÓÙ ÖchaÒgeÓ fÚa Ö...
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This note was uploaded on 04/05/2010 for the course MATH 119 taught by Professor Harmsworth during the Spring '08 term at Waterloo.

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Lecture24 - F igÙ Öe Øh eÖF Ó ÖÑ ×Ó fØh eC ha...

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