Evaluate $\oint _ { C } \left( 12 x ^ { 2 } y ^ { 2 } - 6 y z ^ { 2 } \right) d x + \left( 8 x ^ { 3 } y - 6 x z ^ { 2 } \right) d y - 12 x y z d z$

question 66

Short Answer

Evaluate $\oint _ { C } \left( 12 x ^ { 2 } y ^ { 2 } - 6 y z ^ { 2 } \right) d x + \left( 8 x ^ { 3 } y - 6 x z ^ { 2 } \right) d y - 12 x y z d z$ , where the path C is the curve of intersection of the paraboloid $3 z = x ^ { 2 } + y ^ { 2 }$ with the plane $3 x = 4 y + z = 12$ .

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