Problem 10 - Calculus Volume 3

Question

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=3 \mathbf{i}+x \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field shows constant rightward vectors (3 units) with direction varying vertically by the \( x \)-coordinate.

Step 1: Understand the Vector Field

The given vector field is \( \mathbf{F}(x, y) = 3 \mathbf{i} + x \mathbf{j} \). This indicates that every vector in the field has a horizontal component of 3 (no matter what) and a vertical component that depends on the \( x \)-coordinate of the point.

Step 2: Determine Sample Points

Choose a few sample points in the \( xy \)-plane to evaluate the vector field. Typical choices could include points like (0,0), (1,0), (-1,0), (0,1), (0,-1), and so on. These provide a basic representation of the field.

Step 3: Evaluate Vectors at Sample Points

Evaluate \( \mathbf{F}(x, y) \) at your sample points: - At \( (0, 0) \), \( \mathbf{F}(0, 0) = 3 \mathbf{i} + 0 \mathbf{j} = \langle 3, 0 \rangle \).- At \( (1, 0) \), \( \mathbf{F}(1, 0) = 3 \mathbf{i} + 1 \mathbf{j} = \langle 3, 1 \rangle \).- At \( (-1, 0) \), \( \mathbf{F}(-1, 0) = 3 \mathbf{i} - 1 \mathbf{j} = \langle 3, -1 \rangle \).- At \( (0, 1) \), \( \mathbf{F}(0, 1) = 3 \mathbf{i} + 0 \mathbf{j} = \langle 3, 0 \rangle \).- At \( (0, -1) \), \( \mathbf{F}(0, -1) = 3 \mathbf{i} + 0 \mathbf{j} = \langle 3, 0 \rangle \).

Step 4: Draw the Vectors

On a grid, plot the vectors calculated in Step 3 using arrows. Starting from each point, draw the arrow in the direction and magnitude given by the vector components. For example, at point (0,0), draw an arrow 3 units to the right, indicating \( \langle 3, 0 \rangle \), and at point (1,0), draw an arrow 3 units to the right and 1 unit up.

Step 5: Interpret the Vector Field

Observe that at each point the vectors have a component of 3 units in the positive \( x \)-direction. The vertical component varies with \( x \), showing an increasing upwards vector component as \( x \) increases positively or negatively, proportionally with \( x \). This indicates a consistent rightward flow, with an upward or downward shift depending on the point's \( x \)-value.

Key Concepts

Vector Components

Vector components are the building blocks of vector fields. These components help us understand how a vector behaves in a given plane, usually through a combination of horizontal and vertical motions.
In our example vector field, \( \mathbf{F}(x, y) = 3 \mathbf{i} + x \mathbf{j} \), each vector has two parts: the horizontal component \( 3 \mathbf{i} \) and the vertical component \( x \mathbf{j} \).

• Horizontal Component (\( 3 \mathbf{i} \)): This means every vector in this field moves 3 units right on the x-axis, unchanged by other values.
• Vertical Component (\( x \mathbf{j} \)): This depends on the x-coordinate. For each x-value, the vertical component shifts the vector up (if x is positive) or down (if x is negative). When x is zero, it means there's no vertical shift.

Understanding vector components allows us to predict how vectors behave across different points in the field, helping us visualize the overall pattern and flow of the field.

Vector Field Representation

Vector field representation involves illustrating the vectors at different points in the plane to get a sense of the direction and magnitude of the field they create. Our vector field, \( \mathbf{F}(x, y) = 3\mathbf{i} + x\mathbf{j} \), illustrates a dynamic behavior over the plane.
To represent this, we select sample points such as (0,0), (1,0), and others, compute the vectors at these points, and then plot them on a graph. For example:

• At \( (0,0) \): The vector is \( \langle 3, 0 \rangle \). This shows a full 3-unit push to the right with no vertical shift.
• At \( (1,0) \): The vector is \( \langle 3, 1 \rangle \). Here, the vector moves 3 units right and 1 unit up.
• At \( (-1,0) \): The vector is \( \langle 3, -1 \rangle \). This indicates a 3 units right shift and 1 unit down.

By plotting several such vectors, we can effectively map out how the field behaves across the plane—showing a steady rightward flow with variable vertical shifts.

Evaluating Vector Fields

Evaluating vector fields involves calculating vectors at specific points to understand how the field operates over its domain. By assessing these vectors step-by-step, you can see how both components come together.
Considerations for evaluating our vector field \( \mathbf{F}(x, y) = 3 \mathbf{i} + x \mathbf{j} \):

• Sample Point Selection: Begin by choosing various points on the x-y plane. This includes both positive and negative values, as well as points on both axes.
• Computing Vectors: Insert each x-value from chosen points into the formula. Multiply by the respective unit vectors to find your specific vector. For example, at \( (2,5) \), \( \mathbf{F}(2,5) = 3\mathbf{i} + 2\mathbf{j} = \langle 3, 2 \rangle \).
• Interpreting Results: Observe how the horizontal component remains constant and the vertical component varies with x. This shows consistent lateral movement with x-dependent vertical adjustments.

By systematically evaluating the field at various points, you gain a comprehensive view of how the vector field behaves, critical for understanding both its local and global properties.