In Exercises 7-12, find the work done by force 𝐅 from (0,0,0) to (1,1,1) over each of the follow

step 1 find the work done therefore the parameters are given and the integration become integral 0 to 1

In Exercises 7-12, find the work done by force 𝐅 from...

In Exercises $7-12,$ find the work done by force $\mathbf{F}$ from $(0,0,0)$ to $(1,1,1)$ over each of the following paths (Figure 16.21$)$ : a. The straight-line path $C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1$ b. The curved path $C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1$ c. The path $C_{3} \cup C_{4}$ consisting of the line segment from $(0,0,0)$ to $(1,1,0)$ followed by the segment from $(1,1,0)$ to $(1,1,1)$ $$ \mathbf{F}=\sqrt{z} \mathbf{i}-2 x \mathbf{j}+\sqrt{y} \mathbf{k} $$

In Exercises $7-12,$ find the work done by force $\mathbf{F}$ from $(0,0,0)$ to $(1,1,1)$ over each of the following paths (Figure 16.21$)$ :
a. The straight-line path $C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1$
b. The curved path $C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1$
c. The path $C_{3} \cup C_{4}$ consisting of the line segment from $(0,0,0)$ to $(1,1,0)$ followed by the segment from $(1,1,0)$ to $(1,1,1)$
$$
\mathbf{F}=\sqrt{z} \mathbf{i}-2 x \mathbf{j}+\sqrt{y} \mathbf{k}
$$

Key Concepts

Line Integrals

A line integral of a vector field computes the cumulative effect (such as work) of a force along a given curve. It is calculated by taking the dot product of the vector field with the differential element of the curve, then integrating over the path. This concept is fundamental in evaluating how a vector field acts along various trajectories.

Parameterization of Curves

Parameterization involves representing a path by expressing its coordinates as functions of a single parameter. This process allows the transformation of a curve integral into a standard integral, where the derivative of the parameterized position vector provides the differential element needed for integration.

Work Done by a Force Field

The work done by a force field along a path is determined by integrating the dot product of the force and the displacement vectors along the curve. This quantifies the energy transferred by the force when moving an object along that path, and is crucial in both physical applications and theoretical analysis.

Path Dependence in Vector Fields

The concept of path dependence addresses whether the work done by a vector field between two points depends on the trajectory taken. In conservative fields, the work is path independent; however, for non-conservative fields, the amount of work can vary significantly with the chosen path. Recognizing this difference is essential for correctly applying line integrals in various scenarios.

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