To compute the work done by the force $\mathbf{F} = (\sin(x + y), xy, x^2 z)$ in moving an object along a given trajectory, we can break down the trajectory into segments, calculate the work done on each segment, and then sum them up.

The work done by a force $\mathbf{F}$ along a path $C$ is given by:

$W = \int_C \mathbf{F} \cdot d\mathbf{r}$

where $d\mathbf{r}$ is the differential displacement vector along the path $C$.

Step 1: Work Done on the First Segment

The first segment goes from $(1, 1, 1)$ to $(2, 2, 2)$.

1. Parameterize the line segment from $(1, 1, 1)$ to $(2, 2, 2)$: $\mathbf{r}_1(t) = (1 + t, 1 + t, 1 + t), \quad t \in [0, 1]$

   This parametrization implies $x = 1 + t$, $y = 1 + t$, and $z = 1 + t$.

2. Calculate $d\mathbf{r}_1$: $d\mathbf{r}_1 = \frac{d\mathbf{r}_1}{dt} \, dt = (1, 1, 1) \, dt$

3. Evaluate $\mathbf{F}(\mathbf{r}_1(t))$:

   • $F_x = \sin((1 + t) + (1 + t)) = \sin(2 + 2t)$
   • $F_y = (1 + t)(1 + t) = (1 + t)^2$
   • $F_z = (1 + t)^2(1 + t) = (1 + t)^3$

   So, $\mathbf{F}(\mathbf{r}_1(t)) = \left( \sin(2 + 2t), (1 + t)^2, (1 + t)^3 \right)$

4. Compute $\mathbf{F} \cdot d\mathbf{r}_1$: $\mathbf{F} \cdot d\mathbf{r}_1 = \sin(2 + 2t) \cdot 1 + (1 + t)^2 \cdot 1 + (1 + t)^3 \cdot 1$ $= \sin(2 + 2t) + (1 + t)^2 + (1 + t)^3$

5. Integrate over $t$ from 0 to 1: $W_1 = \int_0^1 \left( \sin(2 + 2t) + (1 + t)^2 + (1 + t)^3 \right) dt$

Step 2: Work Done on the Second Segment

The second segment goes from $(2, 2, 2)$ to $(-3, 6, 4)$.

1. Parameterize the line segment from $(2, 2, 2)$ to $(-3, 6, 4)$: $\mathbf{r}_2(t) = (2 - 5t, 2 + 4t, 2 + 2t), \quad t \in [0, 1]$

2. Calculate $d\mathbf{r}_2$: $d\mathbf{r}_2 = \frac{d\mathbf{r}_2}{dt} \, dt = (-5, 4, 2) \, dt$

3. Evaluate $\mathbf{F}(\mathbf{r}_2(t))$:

   • $x = 2 - 5t$
   • $y = 2 + 4t$
   • $z = 2 + 2t$

   Then,

   • $F_x = \sin((2 - 5t) + (2 + 4t)) = \sin(4 - t)$
   • $F_y = (2 - 5t)(2 + 4t)$
   • $F_z = (2 - 5t)^2 (2 + 2t)$

   So, $\mathbf{F}(\mathbf{r}_2(t)) = \left( \sin(4 - t), (2 - 5t)(2 + 4t), (2 - 5t)^2 (2 + 2t) \right)$

4. Compute $\mathbf{F} \cdot d\mathbf{r}_2$: $\mathbf{F} \cdot d\mathbf{r}_2 = \sin(4 - t)(-5) + (2 - 5t)(2 + 4t) \cdot 4 + (2 - 5t)^2 (2 + 2t) \cdot 2$

5. Integrate over $t$ from 0 to 1: $W_2 = \int_0^1 \left( -5 \sin(4 - t) + 4(2 - 5t)(2 + 4t) + 2(2 - 5t)^2(2 + 2t) \right) dt$

Step 3: Total Work Done

The total work done $W$ is the sum of the work done on each segment:

$W = W_1 + W_2$

This will yield the total work done by the force $\mathbf{F}$ along the given trajectory. You can evaluate these integrals either by hand (if you wish to simplify further) or using a computational tool.

Would you like to see the detailed evaluation of these integrals?

Related Questions

1. How would the result change if the force field was conservative?
2. What is the physical interpretation of work done in vector fields?
3. How does the parameterization of a line segment affect the calculation of work done?
4. Can work be negative, and under what conditions does this occur?
5. How would you approach this problem if the trajectory were curved instead of piecewise linear?

Tip:

Breaking down vector calculations into each component can help simplify complex integrals.