<prompt> Acha varphix, y iwe kitendakazi chenye derivatives za sehemu za mpangilio wa pili kama vile1. varphi_x xvarphi_y yvarphi_x0 kwenye ndege iliyotobolewa mathbbR2 backslash0,2. r varphi_x mshale wa kulia fracx2 pi r na r varphi_y rightarrow fracy2 pi r kama r sqrtx2y2 mshale wa kulia 0.Acha C_R iwe mduara x2y2R2. Tathmini kiunga cha mstariint_C_R exkushoto-varphi_y d xvarphi_x d ykulia  <response>: Fikiria eneo la mwaka mathcalA kati ya miduara ya radius r na R, kisha kwa Theorem ya Greenanzailiyopangwaint_R exkushoto-varphi_y d xvarphi_x d ykulia-int_r eyleft-varphi_y d xvarphi_x d yright nint_partial mathcalA exleft-varphi_x d xvarphi_x d y kulia niint_mathcalA dleftexleft-varphi_y d xvarphi_x d ykuliakulia niint_mathcalA-ex varphi_y y d y kabari d xleftex varphi_xex varphi_x x kulia d x kabari d y niint_mathcalA exleftvarphi_x xex varphi_x xvarphi_x kulia d x kabari d y0mwishoiliyopangwakuonyesha kuwa kiunganishi hakitegemei radius r. Sasa, kuweka kigezo cha mduara wa radius ranzailiyopangwaint_C_r exkushoto-varphi_y d xvarphi_x d ykulia nint_02 pi er cos thetakushotovarphi_yr cos theta, r sin theta sin thetavarphi_xr cos theta, r dhambi theta cos thetakulia r d thetamwishoiliyopangwalakini wakati r rightarrow 0 integrand inapobadilika kuwa sawafracsin theta2 pi cdot sin thetafraccos theta2 pi cdot cos thetafrac12 pikwa hivyo muunganisho unakaribia 1 wakati r rightarrow 0 na hiyo ndio thamani ya muunganisho. 