Problem 11 - Calculus Volume 3

Question

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=y \mathbf{i}+\sin x \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field's x-component scales with \( y \), and its y-component oscillates sinusoidally with \( x \), forming an oscillating pattern as a graph.

Step 1: Understand the Components of the Vector Field

The vector field \( \mathbf{F}(x, y) = y \mathbf{i} + \sin x \mathbf{j} \) consists of two components. The first component, \( y \mathbf{i} \), means that the x-component of the vectors varies with \( y \), essentially increasing or decreasing based on the y-coordinate. The second component, \( \sin x \mathbf{j} \), implies that the y-component of the vectors depends on the sine of the x-coordinate.

Step 2: Evaluate Vectors at Specific Points

To understand the vector field, evaluate vectors at specific points. For example, at point \( (0, 0) \), the vector is \( 0 \mathbf{i} + \sin 0 \mathbf{j} = \mathbf{0} \). At point \( (\pi/2, 1) \), it becomes \( 1 \mathbf{i} + \sin(\pi/2) \mathbf{j} = \mathbf{i} + \mathbf{j} \). At point \( (\pi, -2) \), it will be \(-2 \mathbf{i} + \sin(\pi) \mathbf{j} = -2 \mathbf{i} \).

Step 3: Draw Vectors on a Coordinate Plane

Plot these vectors within a coordinate grid, using their x and y components. At each of these points, draw an arrow starting from the origin of the point (e.g., at \( (0,0) \) just a dot since it's \( \mathbf{0} \), at \( (\pi/2, 1) \) a diagonal vector going one unit \( \mathbf{i} \) and one unit \( \mathbf{j} \), and at \( (\pi, -2) \) a vector going two units in the negative \( \mathbf{i} \) direction).

Step 4: Consider the Behavior Across the Grid

Notice that for a given \( y \) value, the x-component of the vectors increases linearly with \( y \). For a given \( x \) value, the y-component oscillates with \( \sin x \), meaning vectors oscillate up and down as \( x \) changes. This results in the vector field having a sinusoidal pattern in the vertical component while the horizontal component scales with \( y \).

Step 5: Combine Components to Visualize the Vector Field

By combining both patterns, the vector field can be visualized as a set of arrows that oscillate in their y-component based on \( \sin(x) \), while their x-component increases with \( y \). This creates an overall field where arrows curve in and out vertically as they move horizontally, with longer arrows further away from the x-axis.

Key Concepts

Vector Components

Understanding vector components is crucial to analyzing vector fields. Vectors are mathematical entities that have both magnitude and direction. In the given vector field \( \mathbf{F}(x, y) = y \mathbf{i} + \sin x \mathbf{j} \), we have two primary components:

• The \( y \mathbf{i} \) component: This represents the vector's horizontal component or its x-component. It scales directly with the value of \( y \), meaning its magnitude increases or decreases in proportion to \( y \).
• The \( \sin x \mathbf{j} \) component: This is the vector's vertical component or its y-component. Here, the sine function, which oscillates between -1 and 1, determines the direction and magnitude of this component.

Combining these two components, the vectors exhibit unique behavior across the coordinate plane, with their direction and length changing according to both the y-coordinate and the sine of the x-coordinate.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which vectors are represented. It's made up of two axes, usually labeled as x and y. In our vector field \( \mathbf{F}(x, y) = y \mathbf{i} + \sin x \mathbf{j} \), each vector can be visualized as an arrow rooted at a specific point on this plane.

A key aspect of interpreting vector fields like this one is understanding how vectors change as you move across the plane:

• At any point \( (x, y) \), the horizontal (or x) component of the vector is \( y \), and the vertical (or y) component is \( \sin x \).
• By plotting vectors at various points, we can depict the flow and behavior of the field across the plane.

The coordinate plane thus serves as a visual framework to explore and represent how vector fields behave at different points, helping to understand the directional tendencies and intensities of forces described by the vectors.

Oscillating Functions

Oscillating functions are functions that periodically vary between maximum and minimum values. A common example of these are sine and cosine functions. In the vector field \( \mathbf{F}(x, y) = y \mathbf{i} + \sin x \mathbf{j} \), the oscillating nature of the \( \sin x \) term plays a significant role.

• The \( \sin x \) function varies smoothly, creating a periodic swinging pattern between -1 and 1 as x changes.
• As a result, the y-component of our vectors oscillates vertically from negative to positive, influencing the direction of vectors at different x-values.

This periodic behavior is essential for modeling natural phenomena, such as waves, vibrations, and other cyclic patterns, allowing us to see how the vectors oscillate across the coordinate plane.

Graphical Representation of Vectors

Graphical representation of vectors involves drawing vectors on the coordinate plane to visually convey the properties of a vector field. This helps in understanding the direction and magnitude of vectors easily.

To graphically represent vectors in a field like \( \mathbf{F}(x, y) = y \mathbf{i} + \sin x \mathbf{j} \):

• Draw arrows at various points \( (x, y) \), where each arrow represents a vector from that point.
• The length (or magnitude) and direction of each arrow are determined by the vector's components \( (y, \sin x) \).
• This results in a visual map that shows how vectors change across the plane, revealing patterns such as the lengthening of vectors with increased \( y \) values and the oscillation due to \( \sin x \).

Such graphical tools are invaluable in illustrating complex vector systems, enabling a clearer understanding of the mathematical relationships and behaviors within vector fields.