Compute the flux of the vector field ⃗v = 5˜i − 5˜j − 2 ˜k through the rectangular region shown below, assuming it is oriented as shown and that a = 2 and b = 2. flux =

To compute the flux of the vector field \(\vec{v} = 5\hat{i} - 5\hat{j} - 2\hat{k}\) through the rectangular region, we can use the following formula for flux:

\[
\text{Flux} = \iint_S \vec{v} \cdot \hat{n} \, dS
\]

Where:
- \(\vec{v}\) is the vector field.
- \(S\) is the surface through which we are computing the flux.
- \(\hat{n}\) is the unit normal vector to the surface.
- \(dS\) is the differential area element.

### Given Information:
- The vector field is \(\vec{v} = 5\hat{i} - 5\hat{j} - 2\hat{k}\).
- The rectangular region has dimensions \(a = 2\) and \(b = 2\).
- The region lies in the \(xy\)-plane, so the normal vector \(\hat{n}\) points in the \(\hat{k}\) direction. Since the region is in the \(xy\)-plane, \(\hat{n} = \hat{k}\) or \(\hat{n} = -\hat{k}\) depending on the orientation.

Let's assume that the surface is oriented upward, so \(\hat{n} = \hat{k}\).

### Step 1: Compute the dot product \(\vec{v} \cdot \hat{n}\)
\[
\vec{v} \cdot \hat{n} = (5\hat{i} - 5\hat{j} - 2\hat{k}) \cdot \hat{k} = -2
\]

### Step 2: Compute the differential area element \(dS\)
Since the surface is a rectangle in the \(xy\)-plane, the area of the rectangle is:
\[
dS = dx \, dy
\]
And the total area \(S\) is:
\[
S = a \times b = 2 \times 2 = 4
\]

### Step 3: Calculate the flux
Since the dot product is constant over the entire surface:
\[
\text{Flux} = \iint_S \vec{v} \cdot \hat{n} \, dS = (-2) \times \text{Area of the surface}
\]
\[
\text{Flux} = -2 \times 4 = -8
\]

So, the flux of the vector field through the rectangular region is \(\boxed{-8}\).

Would you like more details on the steps? If not, here are some related questions for further exploration:

1. How would the flux change if the surface were oriented downward instead?
2. What is the physical significance of a negative flux value?
3. How would you compute the flux if the vector field were not constant?
4. What if the surface were tilted instead of lying flat in the \(xy\)-plane?
5. How does the flux computation change for a non-rectangular surface?

**Tip:** Always pay attention to the orientation of the surface when calculating flux, as it affects the direction of the normal vector and thus the sign of the flux.

Learn how to compute the flux of the vector field \( \vec{v} = 5\hat{i} - 5\hat{j} - 2\hat{k} \) through a rectangular region. Understand the concept of flux in vector fields, using surface integrals and differential area elements. Suitable for college-level mathematics.

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