Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 2 } z ^ { 2 } \mathbf { j } + y ^ { 2 } z ^ { 2 } \mathbf { k }$ . Let C be the rectangular path from $( 1,1,2 )$ to $( 3,1,2 )$ to $( 3,5,2 )$ to $( 1,5,2 )$ to $( 1,1,2 )$ . Use Stokes' Theorem to evaluate the line integral $\int _ { C } \mathbf { F } \cdot \mathbf { T } d \mathrm {~s}$ , where T is the unit tangent vector to C.

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The Answer of Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 2 } z ^ { 2 } \mathbf { j } + y ^ { 2 } z ^ { 2 } \mathbf { k }$ . Let C be the rectangular path from $( 1,1,2 )$ to $( 3,1,2 )$ to $( 3,5,2 )$ to $( 1,5,2 )$ to $( 1,1,2 )$ . Use Stokes' Theorem to evaluate the line integral $\int _ { C } \mathbf { F } \cdot \mathbf { T } d \mathrm {~s}$ , where T is the unit tangent vector to C.

Let F(x,y,z)=x2y2i+x2z2j+y2z2k\mathbf { F } ( x , y , z ) = x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 2 } z ^ { 2 } \mathbf { j } + y ^ { 2 } z ^ { 2 } \mathbf { k }F(x,y,z)=x2y2i+x2z2j+y2z2k

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