Problem 4 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=x \mathbf{i}+2 y \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \) directs vectors based on point coordinates, with the x-component as the point's x-coordinate and the y-component twice the y-coordinate.

Step 1: Identify Vector Field Component Functions

The given vector field is \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \). This means the horizontal component (i.e., in the direction of \( \mathbf{i} \)) is \( x \), and the vertical component (i.e., in the direction of \( \mathbf{j} \)) is \( 2y \).

Step 2: Select Points for Vector Representation

Choose points in the Cartesian plane to plot vectors. Common choices can be \( (0,0), (1,0), (0,1), (1,1), (-1,0), \) and \( (0,-1) \). These points should be sufficient to understand the characteristics of the vector field.

Step 3: Compute Vector Values at Selected Points

Calculate the vector \( \mathbf{F} \) at the chosen points:- At \( (0, 0) \), \( \mathbf{F}(0, 0) = 0 \mathbf{i} + 0 \mathbf{j} = (0, 0) \)- At \( (1, 0) \), \( \mathbf{F}(1, 0) = 1 \mathbf{i} + 0 \mathbf{j} = (1, 0) \)- At \( (0, 1) \), \( \mathbf{F}(0, 1) = 0 \mathbf{i} + 2 \mathbf{j} = (0, 2) \)- At \( (1, 1) \), \( \mathbf{F}(1, 1) = 1 \mathbf{i} + 2 \mathbf{j} = (1, 2) \)- At \( (-1, 0) \), \( \mathbf{F}(-1, 0) = -1 \mathbf{i} + 0 \mathbf{j} = (-1, 0) \)- At \( (0, -1) \), \( \mathbf{F}(0, -1) = 0 \mathbf{i} - 2 \mathbf{j} = (0, -2) \)

Step 4: Graph Vectors on the Cartesian Plane

Draw the vectors starting from their corresponding points:1. At \( (0, 0) \), draw a vector \( (0, 0) \) which means no vector is drawn (a zero vector).2. At \( (1, 0) \), draw a vector of length 1 along the positive x-axis.3. At \( (0, 1) \), draw a vector of length 2 along the positive y-axis.4. At \( (1, 1) \), draw a vector pointing towards \( (1, 2) \).5. At \( (-1, 0) \), draw a vector of length 1 along the negative x-axis.6. At \( (0, -1) \), draw a vector of length 2 along the negative y-axis. This visualization helps in understanding the vector field behavior across the selected points.

Step 5: Analyze the Vector Field

From the graph, observe that vectors originating from the points move in the direction determined by their components. The field has vectors spread such that the x-component corresponds directly with x-position, and the y-component is scaled by 2 times the y-position.

Key Concepts

Vector Components

Understanding vector components is crucial in the study of vector fields. Each vector in a vector field can be thought of as having two independent parts: the horizontal component and the vertical component.
In a typical Cartesian coordinate system, we often denote these components as related to the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \):

• \( \mathbf{i} \) represents the horizontal component, or movement along the x-axis.
• \( \mathbf{j} \) signifies the vertical component, or movement along the y-axis.

Therefore, any vector can be expressed in terms of these components. For instance, the vector field \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \) is composed of the vector components \( x \mathbf{i} \) and \( 2y \mathbf{j} \).
The vector components indicate how much the vector's influence spreads in the horizontal and vertical directions, respectively. By plotting the vectors at different points, we assess how these components affect the field’s overall behavior.

Cartesian Plane

The Cartesian plane is the stage on which vector fields perform. It is a two-dimensional surface defined by the perpendicular intersections of the x-axis and y-axis.
It's an essential tool in graphing and analyzing vector fields due to the clarity and precision it offers:

• The x-axis is the horizontal line extending from left to right, usually representing numbers from negative to positive.
• The y-axis is the vertical line extending from bottom to top, also usually numbered negatively to positively.

The intersection point of these axes is the origin, denoted as \((0,0)\). Each position on the plane is identified by a pair of numbers \((x, y)\), which determine the position's distance and direction from the origin.
Understanding the Cartesian plane is crucial when plotting vectors from a vector field, as it clearly shows vector magnitudes and directions at various points in the plane.

Graphing Vectors

Graphing vectors on a Cartesian plane helps visualize the vector field. Each vector is drawn based on the given vector field's components and the corresponding point in the plane.
Here are basic steps to graph vectors effectively:

• Choose relevant points on the plane, such as \((0, 0), (1, 0), (0, 1), (1, 1), (-1, 0), (0, -1)\).
• Calculate the vector values using the vector field equation \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \).
• Draw each vector from its respective point.

For example, from the point \((0, 1)\), the vector calculated in this field is \((0, 2)\), resulting in a vertical line of length 2, extending alongside the y-axis.
By graphing these vectors, one observes how they vary and the direction in which they are pointing, offering insight into the field’s structure.

Vector Field Analysis

Vector field analysis involves interpreting the patterns and behaviors of the field once graphed. This process is essential to understand the influence each vector component has across different points in the plane.
In the example of \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \), this analysis reveals:

• The x-component of the vectors is directly proportional to the x-coordinate of the point where the vector is located.
• The y-component is twice any given y-coordinate, making the vertical component more pronounced.
• Vectors at every point exhibit a specific direction and magnitude, creating a pattern of movement across the plane.

Such analysis helps identify areas of stronger or weaker vector influence and can inform further mathematical calculations or real-world applications. Recognizing these vector behaviors can provide clarity on how particular systems or phenomena can be modeled in a two-dimensional space.