Let $\mathbf { F } ( x , y , z ) = 3 x y \mathbf { i } + y ^ { 2 } \mathbf { j } - x ^ { 2 } y ^ { 4 } \mathbf { k }$ and let S be the surface of the tetrahedron with vertices $( 0,0,0 ) , ( 1,0,0 ) , ( 0,1,0 )$ , and $( 0,0,1 )$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

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The Answer of Let $\mathbf { F } ( x , y , z ) = 3 x y \mathbf { i } + y ^ { 2 } \mathbf { j } - x ^ { 2 } y ^ { 4 } \mathbf { k }$ and let S be the surface of the tetrahedron with vertices $( 0,0,0 ) , ( 1,0,0 ) , ( 0,1,0 )$ , and $( 0,0,1 )$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Let F(x,y,z)=3xyi+y2j−x2y4k\mathbf { F } ( x , y , z ) = 3 x y \mathbf { i } + y ^ { 2 } \mathbf { j } - x ^ { 2 } y ^ { 4 } \mathbf { k }F(x,y,z)=3xyi+y2j−x2y4k

Let $\mathbf { F } ( x , y , z ) = 3 x y \mathbf { i } + y ^ { 2 } \mathbf { j } - x ^ { 2 } y ^ { 4 } \mathbf { k }$