Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y z \right) \mathbf { i } + x ^ { 2 } y \mathbf { j } + x z ^ { 2 } \mathbf { k }$ and let S be the surface of the solid bounded by the spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

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The Answer of Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y z \right) \mathbf { i } + x ^ { 2 } y \mathbf { j } + x z ^ { 2 } \mathbf { k }$ and let S be the surface of the solid bounded by the spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Let F(x,y,z)=(x3+yz)i+x2yj+xz2k\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y z \right) \mathbf { i } + x ^ { 2 } y \mathbf { j } + x z ^ { 2 } \mathbf { k }F(x,y,z)=(x3+yz)i+x2yj+xz2k

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