To compute the flux of the vector field $\mathbf{F}(x,y,z) = 6\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$ through the given rectangular region, we follow these steps:

Step-by-Step Solution

1. Identify the surface and orientation: The rectangular region has corners at $(1,1,0)$, $(0,1,0)$, $(0,0,2)$, and $(1,0,2)$. This region lies on a plane parallel to the x-y plane, but it's skewed in the z-direction.

2. Parameterize the surface: The surface can be parameterized using two parameters $u$ and $v$. Let: $\mathbf{r}(u,v) = (u, v, 2u)$ where $0 \leq u \leq 1$ and $0 \leq v \leq 1$.

3. Compute the tangent vectors: $\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u} = (1, 0, 2)$ $\mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v} = (0, 1, 0)$

4. Compute the normal vector: The normal vector $\mathbf{n}$ is given by the cross product of $\mathbf{r}_u$ and $\mathbf{r}_v$:

   \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{vmatrix} = (0 \cdot 0 - 2 \cdot 1)\mathbf{i} - (1 \cdot 0 - 0 \cdot 0)\mathbf{j} + (1 \cdot 1 - 0 \cdot 0)\mathbf{k} = -2\mathbf{i} + 0\mathbf{j} + 1\mathbf{k}$$ Thus, $$\mathbf{n} = -2\mathbf{i} + \mathbf{k}$$.

5. Normalize the normal vector: The normal vector $\mathbf{n}$ is already perpendicular to the surface. To ensure it points in the positive z-direction, we use $\mathbf{n} = -2\mathbf{i} + \mathbf{k}$.

6. Compute the flux: The flux $\Phi$ through the surface is given by: $\Phi = \iint_S \mathbf{F} \cdot d\mathbf{S}$ where $d\mathbf{S} = \mathbf{n} \, du \, dv$.

   Substituting $\mathbf{F}$ and $\mathbf{n}$: $\mathbf{F} \cdot \mathbf{n} = (6\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}) \cdot (-2\mathbf{i} + \mathbf{k}) = 6(-2) + 4(0) + 5(1) = -12 + 5 = -7$

7. Set up the integral: $\Phi = \iint_S \mathbf{F} \cdot d\mathbf{S} = \int_0^1 \int_0^1 -7 \, du \, dv$

8. Evaluate the integral: $\Phi = \int_0^1 \int_0^1 -7 \, du \, dv = -7 \int_0^1 \int_0^1 du \, dv = -7 \left( \int_0^1 du \right) \left( \int_0^1 dv \right) = -7 (1)(1) = -7$

Final Answer

The flux of the vector field $\mathbf{F}(x,y,z) = 6\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$ through the given rectangular region is $-7$.

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Do you want more details or have any questions about this problem?

Here are some related questions you might find interesting:

1. How to compute the flux of a vector field through a more complex surface?
2. What are the applications of flux in physics and engineering?
3. How do you parameterize a surface in different coordinate systems?
4. What is the divergence theorem and how is it applied?
5. How does the orientation of a surface affect the flux calculation?
6. What are some common mistakes to avoid when computing surface integrals?
7. How do you find the normal vector to a surface given its parameterization?
8. Can you explain the physical meaning of a negative flux value?

Tip: Always check the orientation of your surface normal when computing flux; it ensures the correct sign and direction for your integral result.