Question

Design your own vector field Specify the component functions of a vector field \(\mathbf{F}\) in \(\mathbb{R}^{2}\) with the following properties. Solutions are not unique. At all points except \((0,0), \mathbf{F}\) has unit magnitude and points away from the origin along radial lines.

Short answer

The constructed vector field is given by: $$ \mathbf{F}(x,y) = \begin{cases}\phantom{-} x\frac{\mathbf{i}}{\sqrt{x^2+y^2}} + y\frac{\mathbf{j}}{\sqrt{x^2+y^2}}, & (x, y) \neq (0,0) \\ \mathbf{0}, & (x, y) = (0,0)\end{cases}. $$

Step-by-step solution

Convert to polar coordinates

We are working in \(\mathbb{R}^{2}\), so we can represent the points using polar coordinates (\(r\), \(\theta\)), with \(r \geq 0\) and \(0 \leq \theta < 2\pi\). The position vector \(\mathbf{r}\) in polar coordinates can be written as: $$\mathbf{r} = r\cos(\theta)\mathbf{i} + r\sin(\theta)\mathbf{j}.$$

Create the radial field

We want the vector field to point away from the origin along radial lines. A simple way to achieve this is by making the vector field parallel to the position vector \(\mathbf{r}\). Therefore, we can write the vector field \(\mathbf{F}\) as: $$\mathbf{F} = f_r(r, \theta)\mathbf{i} + f_\theta(r, \theta)\mathbf{j},$$ where \(f_r(r, \theta)\) and \(f_\theta(r, \theta)\) are the component functions in the radial and angular directions, respectively. The radial direction is given by the unit vector \(\mathbf{e_r} = \cos(\theta)\mathbf{i} + \sin(\theta)\mathbf{j}\). To ensure that \(\mathbf{F}\) points away from the origin, we can make the radial component function proportional to the radial coordinate \(r\). Moreover, since we want unit magnitude at all points except \((0,0)\), we can write the radial component function as \(f_r(r, \theta) = C r\) with \(C = 1/r\). The angular component function should be zero, as the vector field must point along radial lines and not have any tangential component. Therefore, we can write \(f_\theta(r, \theta)=0\).

Write the final vector field

Now, we have the component functions for the vector field \(\mathbf{F}\) in \(\mathbb{R}^{2}\): $$ f_r(r, \theta) = \frac{r}{r} = 1, $$ and $$ f_\theta(r, \theta) = 0. $$ Converting back to Cartesian coordinates (\(x\), \(y\)), we need to replace \(r = \sqrt{x^2+y^2}\) and \(\cos(\theta) = x/r\), \(\sin(\theta) = y/r\). Therefore, the vector field \(\mathbf{F}\) in Cartesian coordinates is: $$ \mathbf{F} = \phantom{-} x\frac{\mathbf{i}}{\sqrt{x^2+y^2}} + y\frac{\mathbf{j}}{\sqrt{x^2+y^2}}. $$ However, this vector field is undefined at the origin \((0,0)\). Since the solutions are not unique, we can simply define the vector field \(\mathbf{F}(0,0) = \mathbf{0}\), meaning it has zero magnitude at the origin. Thus, the final vector field is: $$ \mathbf{F}(x,y) = \begin{cases}\phantom{-} x\frac{\mathbf{i}}{\sqrt{x^2+y^2}} + y\frac{\mathbf{j}}{\sqrt{x^2+y^2}}, & (x, y) \neq (0,0) \\ \mathbf{0}, & (x, y) = (0,0)\end{cases}. $$

Key concepts

Polar Coordinates

In mathematics, polar coordinates offer a unique and intuitive way to represent points in a plane using a pair - a radius and an angle. Unlike Cartesian coordinates that rely on horizontal and vertical distances (x, y), polar coordinates ( r, \theta) use:

• r - The distance from the origin to the point, essentially the radius.
• \theta - The angle measured from the positive x-axis counterclockwise to the point in the plane.

This system is particularly useful when dealing with circular or rotational symmetries. For example, in the case of vector fields, polar coordinates simplify calculations that involve radial patterns, as these naturally align with the established radial direction.
When working with polar coordinates, converting back to Cartesian coordinates involves using the relationships: \[x = r\cos(\theta), \quad y = r\sin(\theta)\]This allows us to easily translate our understanding from a radial to a Cartesian context.

Unit Magnitude

A vector field with unit magnitude means that regardless of the position in the field, the length or magnitude of the vectors remains equal to one unit (except at points specially defined otherwise, like the origin in this problem). This type of field is important because:

• Unit magnitude ensures uniform strength; every point exhibits the same intensity.
• This property simplifies mathematical modeling where dimensions or scaling factors aren't a consideration.

For the vector field described, ensuring unit magnitude means the vectors will appear as arrows pointing outward, each having a consistent length. In polar coordinates, this is achieved by setting the radial component proportional to 1 over the radius, \[ f_r(r, \theta) = \frac{r}{r} = 1\], indicating a perpetual maintenance of unit length as distance changes.

Radial Lines

Radial lines extend from the center (often the origin) directly outward in all directions like the spokes of a wheel. Understanding radial lines is crucial when designing vector fields that exhibit patterns radiating outwards. Properties of these lines include:

• They do not curve; they remain straight as they extend from the origin.
• The directionality of radial lines is solely dependent on the angle \theta in polar coordinates.

In the context of our vector field, the essence of radial lines is captured by ensuring the vectors point along these lines, away from the origin. The radial direction in polar coordinates,\[\mathbf{e_r} = \cos(\theta)\mathbf{i} + \sin(\theta)\mathbf{j}\]encourages outward movement along these straight paths. Radial lines naturally align with the vector fields aiming to express such a pattern.

Cartesian Coordinates

Cartesian coordinates form the foundation of representing points in a two-dimensional plane using orthogonal axes (x, y). Unlike polar coordinates, they rely on:

• x - The horizontal axis, representing horizontal distance.
• y - The vertical axis, representing vertical distance.

For mathematical modeling and problem-solving, transitioning between polar and Cartesian coordinates is essential. This conversion allows the expression of radial patterns in a familiar grid-like structure. For the vector field transformation:\[\mathbf{F}(x,y) = x\frac{\mathbf{i}}{\sqrt{x^2+y^2}} + y\frac{\mathbf{j}}{\sqrt{x^2+y^2}}\]is the derived formula that portrays the radial concept within a Cartesian framework. It ensures that each vector points directly outward, parallel to radial lines.
While the field is undefined at the origin, adjustments such as setting the magnitude to zero at that point ensure continuity and maintain its pattern across the plane.