Let $\vec { F } = 4 z ^ { 3 } y \vec { i } + \left( 4 x + 4 z ^ { 3 } x \right) \vec { j } + \left( x ^ { 2 } + 12 z ^ { 2 } x y \right) \vec { k }$

question 81

Essay

Let $\vec { F } = 4 z ^ { 3 } y \vec { i } + \left( 4 x + 4 z ^ { 3 } x \right) \vec { j } + \left( x ^ { 2 } + 12 z ^ { 2 } x y \right) \vec { k }$ (a)Evaluate the line integral $\hat { \mathrm { Q } } { \vec { F } \times d \vec { r } }$ , where C is the circle $x ^ { 2 } + y ^ { 2 } = 1$ on the xy-plane, oriented in a counter-clockwise direction when viewed from above.
(b)Without any computation, explain why the answer in part (a)is also equal to the flux integral $\dot {\mathrm { Q } _ { 1 } }\operatorname { curl } \vec { F } \times \vec { d { A } }$ where S₁ is lower hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 , z £ 0$ oriented inward.

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