Find the line integrals of 𝐅 from (0,0,0) to (1,1,1) over each of the following paths in the acc

step 1 Okay, so starting off with part A, the straight line path for C1. So we start off with the equat

Find the line integrals of 𝐅 from (0,0,0) to (1,1,1) over...

Find the line integrals of $\mathbf{F}$ from $(0,0,0)$ to $(1,1,1)$ over each of the following paths in the accompanying figure. $$ \begin{array}{l}{\text { a. The straight-line path } C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { b. The curved path } C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { c. The path } C_{3} \cup C_{4} \text { consisting of the line segment from }(0,0,0)} \\ {\text { to }(1,1,0) \text { followed by the segment from }(1,1,0) \text { to }(1,1,1)}\end{array} $$ $$ \mathbf{F}=(y+z) \mathbf{i}+(z+x) \mathbf{j}+(x+y) \mathbf{k} $$

Find the line integrals of $\mathbf{F}$ from $(0,0,0)$ to $(1,1,1)$
over each of the following paths in the accompanying figure.
$$
\begin{array}{l}{\text { a. The straight-line path } C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { b. The curved path } C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { c. The path } C_{3} \cup C_{4} \text { consisting of the line segment from }(0,0,0)} \\ {\text { to }(1,1,0) \text { followed by the segment from }(1,1,0) \text { to }(1,1,1)}\end{array}
$$
$$
\mathbf{F}=(y+z) \mathbf{i}+(z+x) \mathbf{j}+(x+y) \mathbf{k}
$$

Key Concepts

Path Independence

Path independence is a property of conservative vector fields where the value of the line integral depends only on the initial and final points, and not on the specific path taken between them. This principle allows the evaluation of line integrals by merely finding a potential function or by using the endpoints, making calculations much more manageable when dealing with conservative fields.

Potential Functions

A potential function is a scalar function whose gradient equals the given vector field. For conservative fields, the line integral along any path from one point to another is simply the difference in the potential function's values at those endpoints. This concept simplifies the computation of line integrals by bypassing the need for explicit integration along the path.

Line Integral of a Vector Field

A line integral of a vector field is the process of integrating a vector field along a curve. It involves computing the dot product of the field and an infinitesimal vector element of the curve, then summing this contribution over the curve. This quantifies the cumulative effect of the field along a chosen path, often representing physical quantities such as work done by a force field along a trajectory.

Parameterization of Paths

Parameterizing a path involves expressing the coordinates of points on the path as functions of a single parameter, usually denoted by t. This technique simplifies the evaluation of line integrals by transforming the path into a parametric form, allowing the use of calculus tools such as differentiation and integration with respect to the parameter.

Conservative Vector Fields

A conservative vector field is one in which the line integral between any two points is independent of the path taken; rather, it depends solely on the endpoints. This occurs when the field can be expressed as the gradient of some scalar potential function. Checking for conservativeness often involves verifying that the curl of the field is zero over the domain of interest.

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