### Vector Field and Stokes' TheoremLet \( \mathbf{F} = \langle xyz, xy, x^2yz \rangle \).Use Stokes' Theorem to evaluate\[ \iint_S \text{curl} \, \mathbf{F} \cdot d\mathbf{S}, \]where \( S \) consists of the top and the four sides (but not the bottom) of the cube with one corner at \((-1, -1, -1)\) and the diagonal corner at \((4, 4, 4)\).There is an empty space below this statement that is likely meant for student input or solution steps.#### HintUse the fact that if \( S_1 \) and \( S_2 \) share the same boundary curve \( C \), then\[ \iint_{S_1} \text{curl} \, \mathbf{F} \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_{S_2} \text{curl} \, \mathbf{F} \cdot d\mathbf{S}. \]This suggests using the equivalence of the surface integrals of the curl of \( \mathbf{F} \) over different surfaces sharing the same boundary to simplify calculations.### Explanation of Concepts**Stokes' Theorem:**Stokes' Theorem relates a surface integral over surface \( S \) to a line integral over the boundary curve \( C \) of that surface.\[ \iint_S \text{curl} \, \mathbf{F} \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}. \]This theorem is useful for converting a difficult surface integral into a more manageable line integral, or vice versa.**Application to the Problem:**Given the specific vector field \( \mathbf{F} = \langle xyz, xy, x^2yz \rangle \) and the surface \( S \) that is a cube excluding the bottom face, the problem directs you to use Stokes' Theorem. The hint suggests that the boundary curve \( C \) of the surface \( S \) can be used to simplify the evaluation of the integral.**Diagrams or Graphs:**

Name: ### Vector Field and Stokes' TheoremLet \( \mathbf{F} = \langle xyz,
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: ### Vector Field and Stokes' TheoremLet \( \mathbf{F} = \langle xyz, xy, x^2yz \rangle \).Use Stokes' Theorem to evaluate\[ \iint_S \text{curl} \, \mathbf{F} \cdot d\mathbf{S}, \]where \( S \) consists of the top and the four sides (but not the bott