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Let the Differential 1-Form = Dz

Question 42

Multiple Choice

Let the differential 1-form Let the differential 1-form = zdy dz + xdz dx + ydx dy be defined in a star-like domain .(a) Is closed?(b) Is exact on D? If so, find a differential 1-form such that = d . A) is neither closed nor exact B) is both closed and exact, = (xz - + ) dx + (yz + sin(z) ) dz C) is both closed and exact, = (xz + + ) dx + (yz + sin(x) ) dy D) is closed but not exact E) is both closed and exact, = zxdx+ xydy + yzdz = zdyLet the differential 1-form = zdy dz + xdz dx + ydx dy be defined in a star-like domain .(a) Is closed?(b) Is exact on D? If so, find a differential 1-form such that = d . A) is neither closed nor exact B) is both closed and exact, = (xz - + ) dx + (yz + sin(z) ) dz C) is both closed and exact, = (xz + + ) dx + (yz + sin(x) ) dy D) is closed but not exact E) is both closed and exact, = zxdx+ xydy + yzdz dz + xdz11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dx + ydx11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dy be defined in a star-like domain 11ee77e1_7790_5b15_a0f8_bd3aa301ff2f_TB9661_11 .(a) Is 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 closed?(b) Is 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 exact on D? If so, find a differential 1-form 11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11such that 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 = d11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 .


A) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is neither closed nor exact
B) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact, 11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 = (xz - 11ee77e1_7790_8226_a0f8_456aa09d7d2b_TB9661_11 11ee77e1_7790_8227_a0f8_abc2d7df0122_TB9661_11 + 11ee77e1_7790_8228_a0f8_2b77fd4d264e_TB9661_11 ) dx + (yz + sin(z) ) dz
C) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact,11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 = (xz + 11ee77e1_7790_8229_a0f8_ab3af347a04d_TB9661_11 11ee77e1_7790_822a_a0f8_f57ddbd860ec_TB9661_11 + 11ee77e1_7790_822b_a0f8_9d419232909e_TB9661_11 ) dx + (yz + sin(x) ) dy
D) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is closed but not exact
E) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact, 11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11= zxdx+ xydy + yzdz

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