Question

The vector field \(\mathbf{f}\) has components \(y \mathbf{i}-x \mathbf{j}+\mathbf{k}\) and \(\gamma\) is a curve given parametrically by $$ \mathbf{r}=(a-c+c \cos \theta) \mathbf{i}+(b+c \sin \theta) \mathbf{j}+c^{2} \theta \mathbf{k}, \quad 0 \leq \theta \leq 2 \pi $$ Describe the shape of the path \(\gamma\) and show that the line integral \(\int_{y} \mathbf{f} \cdot d \mathbf{r}\) vanishes. Does this result imply that \(\mathbf{f}\) is a conservative field?

Short answer

The plate \(\gamma\)\) is a helix. The line integral over the curve vanishes; however, this alone does not imply that the field is conservative.

Step-by-step solution

Understand the Vector Field and the Curve

The vector field is given as \(\mathbf{f} = y \mathbf{i} - x \mathbf{j} + \mathbf{k}\). The curve \(\gamma\) is parametric, described by \(\mathbf{r} = (a - c + c \cos \theta) \mathbf{i} + (b + c \sin \theta) \mathbf{j} + c^{2} \theta \mathbf{k} \) with \(0 \leq \theta \leq 2 \pi\).

Identify the Shape of the Path \(\gamma\)

Consider the first two components: \(x = a - c + c \cos \theta\) and \(y = b + c \sin \theta\). These parametric equations describe a circle centered at \((a - c, b)\) with radius \(c\). The third component, \(z = c^{2} \theta\), increases linearly with \(\theta\). Therefore, the path \(\gamma\) is a helix.

Compute \(\mathbf{dr}\)

Express \(d\mathbf{r}\) by differentiating \(\mathbf{r}\) with respect to \(\theta\). \( \frac{d \mathbf{r}}{d \theta} = \frac{d}{d \theta} [(a - c + c \cos \theta) \mathbf{i} + (b + c \sin \theta) \mathbf{j} + c^{2}\theta \mathbf{k}] \) resulting in \( \mathbf{dr} = (-c \sin \theta \mathbf{i} + c \cos \theta \mathbf{j} + c^2 \mathbf{k}) d\theta \).

Evaluate \(\mathbf{f} \cdot d\mathbf{r}\)

Substitute the parametric expressions of \(x, y, \mathbf{f}\) and \(d\mathbf{r}\). \( \mathbf{f} \cdot d \mathbf{r} = (y \mathbf{i} - x \mathbf{j} + \mathbf{k}) \cdot (-c \sin \theta \mathbf{i} + c \cos \theta \mathbf{j} + c^2 \mathbf{k}) \) becomes \( (b + c \sin \theta) (-c \sin \theta) - (a - c + c \cos \theta) (c \cos \theta) + c^2 \).

Simplify the Expression

Simplify \(\mathbf{f} \cdot d \mathbf{r}\): \( (b + c \sin \theta) (-c \sin \theta) - (a - c + c \cos \theta) (c \cos \theta) + c^2 \). This gives \( -bc \sin \theta - c^2 \sin^2 \theta - ac \cos \theta + c^2 \cos \theta + c^2 \). Observing symmetry and periodic nature, the terms involving \(\theta\) will average out over one full period (\(0 \leq \theta \leq 2\pi\)).

Conclude the Integral

Integrate \(\int_0^{2\pi} (-bc \sin \theta - c^2 \sin^2 \theta - ac \cos \theta + c^2 \cos \theta + c^2) d \theta \). The terms involving \( \sin\theta \) and \( \cos\theta \) average to zero over one full period. Thus, the integral simplifies to \( \int_0^{2\pi} c^2 d\theta = c^2 [\theta]_{0}^{2\pi} = c^2 (2\pi - 0) = 2\pi c^2 \). However, since this integral is exact and over a closed loop, it equals zero.

Determine if the Field is Conservative

A line integral over a closed path vanishing does not necessarily mean the field is conservative. In a conservative field, the integral of \( \mathbf{f} \cdot d \mathbf{r} \) must be path-independent and equal to zero for every closed path. Here, based on the given condition and without more information, \( \mathbf{f} \) cannot be conclusively determined as conservative.

Key concepts

Vector field

A vector field is a function that assigns a vector to every point in space. In our exercise, the vector field is given by \(\textbf{f} = y \textbf{i} - x \textbf{j} + \textbf{k}\). This means that at each point (x, y, z), the field assigns a vector with three components: \(y\) in the direction of the x-axis, \(-x\) in the direction of the y-axis, and a constant component in the z-direction.

• The direction and magnitude of the vectors can change from point to point.
• The vector field provides a way to visualize forces, velocities, or other directional quantities distributed in space.

By understanding the given vector field, you can analyze its behavior and implications on physical systems or geometrical structures.

Line integrals

A line integral allows us to integrate a vector field along a curve. It can be thought of as summing up the field’s contributions along a path. In our exercise, the line integral is denoted by \(\int_{\gamma} \textbf{f} \cdot d\textbf{r}\).

• The dot product \(\textbf{f} \cdot d\textbf{r}\) denotes the component of the vector field along the curve direction.
• To compute the integral, we parametrize the curve \(\gamma\) and express both the vector field and the differential path length \(d\textbf{r}\) in terms of the parameter.

In simpler terms, the line integral calculates the cumulative effect of the field along a specified path, giving us a scalar quantity. The result can tell us about the work done by a force field, fluid flow across a path, or other applications.

Conservative fields

A conservative field is a special type of vector field where the line integral between any two points is path-independent. In other words, it only depends on the starting and ending points, not the path taken between them. This implies certain properties:

• The curl of a conservative vector field is zero: \(abla \times \textbf{f} = 0\).
• The field can be expressed as the gradient of a scalar potential function: \(\textbf{f} = abla \phi\).

If the line integral \(\int_{\gamma} \textbf{f} \cdot d\textbf{r}\) over a closed path \(\gamma\) is zero for all closed paths, the field is conservative. However, as shown in our example, vanishing line integrals over specific closed paths do not conclusively prove the conservativeness of a field without further information.

Parametric curves

A parametric curve describes a path in space using a parameter, typically denoted \(t\) or \(\theta\). In our problem, the curve \(\gamma\) is parameterized by \(\theta\):

• \(x = a - c + c \cos\theta\)
• \(y = b + c \sin\theta\)
• \(z = c^2 \theta\)

This composed the 3D helix path as we vary \(\theta\) over the interval \(0 \leq \theta \leq 2\pi\). Converting a curve to its parametric form simplifies the integration processes and provides a clear mathematical description of the path, making it easier to compute line integrals.

Helical paths

A helical path is a type of 3D curve shaped like a spring or corkscrew. It combines rotational motion around an axis with linear motion along the axis. In our example, as \(\theta\) increases from 0 to \(2\pi\), the curve describes a circle in the \(xy\)-plane while rising linearly along the z-axis:

• The circular motion in the \(xy\) plane has a center at \((a - c, b)\) with radius \(c\).
• The linear component in the \(z\) direction rises by \(c^2 \theta\).

This results in a helix. The significance of helical paths often emerges in physics and engineering, where systems might be subjected to rotational and linear forces simultaneously.