### 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:**

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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

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Let F = < xyz, xy, xʻyz > . Use Stokes' Theorem to evaluate curlF · d5, 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). Hint: Use the fact that if S1 and S2 share the same boundary curve C that curlF · d5 = | F - . dī curlF · dS S2

Let F = < xyz, xy, xʻyz > . Use Stokes' Theorem to evaluate curlF · d5, 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). Hint: Use the fact that if S1 and S2 share the same boundary curve C that curlF · d5 = | F - . dī curlF · dS S2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Vector Field and Stokes' Theorem

Let $ \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.

#### Hint

Use 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:**

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