To solve this problem, we need to calculate the line integral of the vector field $\mathbf{F}(x, y, z) = \langle e^z, e^{x - y}, e^y \rangle$ along a piecewise path from $P = (0, 0, 0)$ to $Q = (-5, 5, 5)$, passing through points $R = (0, 0, 5)$ and $S = (0, 5, 5)$.

The line integral over a vector field $\mathbf{F}$ along a path $C$ parameterized by $\mathbf{r}(t)$ is given by: $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt$ To compute this, we can break the integral into three segments:

1. From $P$ to $R$,
2. From $R$ to $S$,
3. From $S$ to $Q$.

Step 1: Parametrize Each Path Segment

1. Path from $P$ to $R$:

   • $P = (0, 0, 0)$ to $R = (0, 0, 5)$
   • Parameterize with $z = t$, $0 \leq t \leq 5$, so $\mathbf{r}_1(t) = (0, 0, t)$.

2. Path from $R$ to $S$:

   • $R = (0, 0, 5)$ to $S = (0, 5, 5)$
   • Parameterize with $y = t$, $0 \leq t \leq 5$, so $\mathbf{r}_2(t) = (0, t, 5)$.

3. Path from $S$ to $Q$:

   • $S = (0, 5, 5)$ to $Q = (-5, 5, 5)$
   • Parameterize with $x = -t$, $0 \leq t \leq 5$, so $\mathbf{r}_3(t) = (-t, 5, 5)$.

Step 2: Evaluate the Line Integral Along Each Path Segment

For each path segment, compute $\mathbf{F} \cdot d\mathbf{r}$ and integrate over the parameter $t$.

1. Integral from $P$ to $R$:

   • $\mathbf{F}(\mathbf{r}_1(t)) = \langle e^t, 1, 1 \rangle$
   • $d\mathbf{r}_1 = \langle 0, 0, 1 \rangle \, dt$
   • $\mathbf{F} \cdot d\mathbf{r}_1 = 1 \cdot dt$
   • $\int_{P \to R} \mathbf{F} \cdot d\mathbf{r} = \int_0^5 1 \, dt = 5$

2. Integral from $R$ to $S$:

   • $\mathbf{F}(\mathbf{r}_2(t)) = \langle e^5, e^{-t}, e^t \rangle$
   • $d\mathbf{r}_2 = \langle 0, 1, 0 \rangle \, dt$
   • $\mathbf{F} \cdot d\mathbf{r}_2 = e^{-t} \cdot dt$
   • $\int_{R \to S} \mathbf{F} \cdot d\mathbf{r} = \int_0^5 e^{-t} \, dt = \left[ -e^{-t} \right]_0^5 = -e^{-5} + 1$

3. Integral from $S$ to $Q$:

   • $\mathbf{F}(\mathbf{r}_3(t)) = \langle e^5, e^{-5 - t}, e^5 \rangle$
   • $d\mathbf{r}_3 = \langle -1, 0, 0 \rangle \, dt$
   • $\mathbf{F} \cdot d\mathbf{r}_3 = -e^5 \cdot dt$
   • $\int_{S \to Q} \mathbf{F} \cdot d\mathbf{r} = \int_0^5 -e^5 \, dt = -5e^5$

Step 3: Sum the Results

Now, we add up the integrals from each segment: $\int_C \mathbf{F} \cdot d\mathbf{r} = 5 + (1 - e^{-5}) - 5e^5$

Thus, the final answer is: $\int_C \mathbf{F} \cdot d\mathbf{r} = 6 - e^{-5} - 5e^5$

Would you like further details or have any questions?

Related Questions

1. How do you find the line integral for vector fields over more complex paths?
2. What are alternative methods to evaluate line integrals, such as using Green's Theorem?
3. How does parametrization affect the calculation of line integrals?
4. How do we approach line integrals in cases where the vector field is conservative?
5. What are practical applications of line integrals in physics or engineering?

Tip

When working with piecewise paths, break down the integral over each segment carefully and ensure the parametrization is consistent with the direction of movement along the path.