Question

Vector fields in polar coordinates A vector field in polar coordinates has the form \(\mathbf{F}(r, \theta)=f(r, \theta) \mathbf{u}_{r}+g(r, \theta) \mathbf{u}_{\theta},\) where the unit vectors are defined in Exercise \(56 .\) Sketch the following vector fields and express them in Cartesian coordinates. $$\mathbf{F}=r \mathbf{u}_{\theta}$$

Short answer

Question: Express the given vector field in Cartesian coordinates and describe the sketch of the vector field: $$\mathbf{F}=r \mathbf{u}_{\theta}$$ Answer: The vector field, expressed in Cartesian coordinates is: $$\mathbf{F}=-r\sin\theta \mathbf{i} + r\cos\theta \mathbf{j}$$. When sketching the vector field, we see vectors tangential to concentric circles centered at the origin with the direction (counter-clockwise or clockwise) based on positive or negative values of 'r'. The magnitude of the vectors increases as we move away from the origin.

Step-by-step solution

Understand the given vector field in polar coordinates

The given vector field is \(\mathbf{F}=r \mathbf{u}_{\theta}\). This means that the magnitude of the vector field is proportional to the radial distance 'r' and the direction is in the theta direction.

Express unit vectors in Cartesian coordinates

To express the given vector field in Cartesian coordinates, we'll need to convert the unit vectors \(\mathbf{u}_{r}\) and \(\mathbf{u}_{\theta}\) into Cartesian coordinates. From the definition of polar coordinates and using trigonometry, we have: $$\mathbf{u}_{r} = \cos\theta \mathbf{i} + \sin\theta \mathbf{j} \quad \text{and}\quad \mathbf{u}_{\theta}= -\sin\theta \mathbf{i} + \cos\theta \mathbf{j}$$

Express the given vector field in Cartesian coordinates

Now we can express the given vector field in Cartesian coordinates by replacing the polar unit vector. We have: $$ \mathbf{F} = r \mathbf{u}_{\theta} = r(-\sin\theta \mathbf{i}+\cos\theta \mathbf{j}) = -r\sin\theta \mathbf{i} + r\cos\theta \mathbf{j}$$ Hence the vector field \(\mathbf{F}\) is expressed as: $$\mathbf{F}=-r\sin\theta \mathbf{i} + r\cos\theta \mathbf{j}$$ in Cartesian coordinates.

Sketching the vector field

To sketch the vector field first, note that the magnitude is determined by the radial distance 'r'. The direction of the vectors is along the \(\mathbf{u}_\theta\) direction, i.e., tangential to the circle at any given point. 1. Start by drawing a set of concentric circles centered at the origin to represent different radial distances. 2. At each point on the circles, draw a tangent such that it is perpendicular to the radius at that point. 3. The length of the tangent should be proportional to the radial distance from the origin. 4. Positive values of 'r' imply that the tangents will be in the counter-clockwise direction, while negative values indicate clockwise direction tangents. Following these points, the sketch of the vector field will show vectors tangential to the circles and increasing in magnitude as we move away from the origin.

Key concepts

Polar Coordinates

Polar coordinates provide a unique way of representing a point in a plane through the use of an angle and a distance from a fixed central point, often referred to as the origin or pole. Unlike the familiar Cartesian coordinate system that defines a location using an x-coordinate (horizontal position) and a y-coordinate (vertical position), polar coordinates use a radial distance, denoted as 'r', and an angular coordinate, known as '\theta' (theta).

Polar coordinates are particularly useful when dealing with systems that have a natural circular or radial symmetry, such as the field lines around a magnetic or electric charge, or the ripples from a stone thrown in water. In the given exercise, understanding polar coordinates helps in visualizing and sketching vector fields that naturally align with circular patterns.

Another aspect is the conversion of coordinates. To relate polar coordinates to Cartesian, you can use the formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). This relationship is crucial for translating vector fields from polar to Cartesian form for further analysis or visual representation.

Cartesian Coordinates

On the other hand, Cartesian coordinates are organized as a grid, often used as the default system for representing points in both two and three-dimensional spaces. Each point has its unique address defined by orthogonal axes, typically labeled 'x', 'y', (and 'z' for three dimensions).

Cartesian coordinates allow for straightforward algebraic manipulations and are widely adopted in most mathematical formulations, including calculus and algebraic geometry. When dealing with vector fields, converting the vector components from polar to Cartesian coordinates makes it easier to perform such algebraic operations and apply them in contexts like physics and engineering, where Cartesian systems are standard.

Unit Vectors

Unit vectors are fundamental elements in vector calculus and physics, serving as building blocks for representing directionality in space. They are vectors of length one and are typically used to specify directions of vectors in any coordinate system. Each unit vector points along the direction of one of the coordinate axes and has no dimensions associated with it.

In polar coordinates, the unit vectors are \( \mathbf{u}_r \) pointing radially outwards from the origin and \( \mathbf{u}_\theta \) pointing in the direction of increasing \theta, tangential to the radius at any given point on the plane. It's important to fully comprehend the role of these unit vectors when analyzing vector fields, as their orientation at different points reflects the local behavior of the field. Understanding unit vectors and their transformations between coordinate systems is crucial for tasks such as sketching vector fields accurately, as seen in the exercise solution.

Vector Field Magnitude and Direction

A vector field represents a distribution of vectors within a region of space, with each vector having a magnitude and direction that can vary from point to point. The magnitude of a vector at any point reflects the strength or intensity of the field at that location, whereas the direction indicates the way the field is oriented.

In the context of the given exercise, the vector field magnitude is proportional to 'r', implicating that the field's intensity increases with distance from the origin. The direction, dictated by \( \mathbf{u}_\theta \), means that vectors are tangential to the radial lines emanating from the origin, consistent with a rotational or swirling pattern. The skill to visualize and understand the changes in both magnitude and direction of a vector field across different points is fundamental to grasping complex physical phenomena described by fields such as gravity, electromagnetism, or fluid flow. The step-by-step solution further reinforces this understanding by sketching the vector field, thus bringing a deeper comprehension of the abstract concepts.