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 Starting off with part A, the straight line path for C1, we're going to start off with the integ

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

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

Key Concepts

Line Integral of a Vector Field

A line integral of a vector field computes the cumulative effect of a field along a curve. It is formulated as the integral of the dot product between the vector field and the differential tangent vector of the path. This concept is key in connecting the field’s behavior along a trajectory, and it arises in various applications from work done by a force to fluid flow along a path.

Curve Parameterization

Parameterization of a curve involves expressing the coordinates of points on the curve as functions of a single variable. This approach simplifies the computation of line integrals by providing a clear prescription for traversing the curve and determining the associated differential element, allowing conversion of a path integral into a standard integral with respect to the parameter.

Dot Product in Line Integrals

The dot product is a crucial operation in the evaluation of line integrals as it projects the vector field onto the direction of the path. This projection captures the component of the field that contributes along the tangent direction to the curve, making the dot product essential for computing quantities like work or circulation along the path.

Non-conservative Vector Fields and Path Dependence

A vector field is non-conservative if it does not have a potential function, which means that its line integrals depend on the specific path taken between the endpoints. In such cases, the work done or the net effect of the field cannot be simply determined by the values of a potential function at the endpoints, and each distinct path must be evaluated individually.

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