91Ó°ÊÓ

Parametric equations are a way to describe a line segment using a parameter, often denoted as \(t\), that varies over a certain interval. In this context, we use them to define the path from one point to another in a three-dimensional space. For the line segment from the point \((1, 1, 1)\) to \((2, 3, -1)\), the parametric equations are created such that the parameter \(t\) moves from 0 to 1.

These equations are: \(x = 1 + t\), \(y = 1 + 2t\), and \(z = 1 - 2t\). Here's why they work:

• At \(t = 0\), the equations give us the initial point \((1, 1, 1)\).
• At \(t = 1\), they provide the terminal point \((2, 3, -1)\).

This way, the trajectory of the line segment is captured entirely by the variable \(t\). This helps in calculating other essential quantities along the path, like the line integral, by simplifying our expressions into a single-variable form.

A vector field assigns a vector to every point in space. It is a function that maps each location \( (x, y, z) \) to a vector \( \mathbf{F}(x, y, z) \). In this exercise, the specific vector field is given by \( \mathbf{F} = y\mathbf{i} + x\mathbf{j} + 4\mathbf{k} \).

In simpler terms, this means at every point in space, the vector has a:

• Component of length \(y\) in the direction of the x-axis (\(\mathbf{i}\)).
• Component of length \(x\) in the direction of the y-axis (\(\mathbf{j}\)).
• Fixed component of 4 in the direction of the z-axis (\(\mathbf{k}\)).

This vector field guides how any quantities, such as potentials or flows, might vary as you move through space. Understanding the structure of \(\mathbf{F}\) allows one to set up an integral across a path through the field, by estimating how the field behaves at every point along that path.

A conservative vector field is one where the line integral is path-independent. That means that the value of the integral depends only on the start and end points and not on the particular path taken between them.

In such fields, every closed loop integral is zero, and there exists a potential function \(\phi\) such that \(\mathbf{F} = abla \phi\). If you can express \(\mathbf{F}\) as the gradient of a function, the field is conservative.

• For example, any change in potential \(\phi\) between two points can be directly calculated from \(\mathbf{F}\).
• In this exercise, the vector field \(\mathbf{F} = y\mathbf{i} + x\mathbf{j} + 4\mathbf{k}\) is confirmed as conservative, so the integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\) is path-independent.

This concept simplifies the integration process since, regardless of the path chosen, the result remains consistent, found directly from the potential difference.

Differential elements, in the context of parametric equations and line integrals, are tiny changes in the coordinates \((x, y, z)\) as you move along the path. They represent how much the coordinates change with an infinitesimal change in \(t\).

Given the parametric equations \(x = 1 + t\), \(y = 1 + 2t\), and \(z = 1 - 2t\), the differential elements can be calculated:

• The derivative \(dx = dt\) shows how \(x\) changes with \(t\).
• The derivative \(dy = 2dt\) indicates the rate of change of \(y\) with respect to \(t\).
• Similarly, \(dz = -2dt\) tells us about the change in \(z\).

For evaluating a line integral, these differential elements substitute into the integral to measure the contribution of \(\mathbf{F}\) over the path, essentially giving us an accurate sum of influences the field exerts along each segment.