Problem 8 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

Vector field \( \mathbf{F}(x, y) \) has identical vectors \( \langle 1, 1 \rangle \) at every point, pointing uniformly in the northeast direction.

Step 1: Recognize the vector field components

The vector field is given by \( \mathbf{F}(x, y) = \mathbf{i} + \mathbf{j} \). This means that for any point \((x, y)\), the vector has components \(1\) in the \(x\)-direction and \(1\) in the \(y\)-direction.

Step 2: Determine the vectors at various points

Since the vector components are constant, the same vector applies everywhere. Specifically, at each point \((x, y)\), the vector is \(\langle 1, 1 \rangle\).

Step 3: Visualize vector consistency

Draw a few vectors on a grid to illustrate the vector field. For example:- At \((0, 0)\), draw a vector starting at the origin that points to \((1, 1)\).- At \((1, 0)\), draw a vector starting at \((1, 0)\) pointing to \((2, 1)\).- At \((0, 1)\), draw a vector starting at \((0, 1)\) pointing to \((1, 2)\).The vectors are consistent across the plane, parallel and equally spaced, each indicating a shift of \(1\) unit right and \(1\) unit up.

Step 4: Conclude the vector field description

The vector field \( \mathbf{F}(x, y) \) consists of uniform vectors with magnitude \(\sqrt{2}\), each pointing diagonally upwards to the "+45 degree" or "northeast" direction, parallel to every other vector in the field.

Key Concepts

Vector Components

When we talk about vector components, we're breaking down vectors into simpler parts that describe movement in different directions. Each vector in a field, like \( \mathbf{F}(x, y) = \mathbf{i} + \mathbf{j} \), can be broken into two main components:

• The \( x \)-component, which is aligned with the horizontal axis, marked by the vector \( \mathbf{i} \).
• The \( y \)-component, which follows the vertical axis, represented by the vector \( \mathbf{j} \).

In our example vector field, the components are always \(1\) in the \(x\)-direction and \(1\) in the \(y\)-direction no matter where you are in the plane.
This means every vector is essentially the same, resulting in a uniform field. These components help us understand the direction and magnitude of vectors wherever we analyze them.

Visualization of Vectors

Visualizing vectors helps in understanding the behavior and orientation of vectors in space. In our example, this involves drawing vectors on a grid to see how they behave across different points.

• Pick a point on the grid, like \((0,0)\). From there, draw a vector to \((1,1)\), following the direction of \( \mathbf{i} + \mathbf{j} \).
• Repeat this at other points, such as \((1,0)\) directing to \((2,1)\), and \((0,1)\) pointing to \((1,2)\).

Every vector you draw will be diagonal and consistent in orientation and length, indicating a steady pattern.
By drawing these vectors, you visually establish a field that is uniform and cohesive, helping students accurately interpret how the field functions at any given point.

Uniform Vector Field

A uniform vector field is characterized by vectors that maintain the same magnitude and direction throughout the space. In the field \( \mathbf{F}(x, y) = \mathbf{i} + \mathbf{j} \), every vector is identical.

• The magnitude is consistent, calculated as \(\sqrt{1^2 + 1^2} = \sqrt{2}\).
• Direction is always towards the northeast, creating a seamless flow throughout the grid.

This uniformity means that no matter where you are on the grid, the vector retains its direction and pace.
In practical terms, this type of field illustrates uniform motion such as wind or water current that's steady and unchanged across a plane.
Understanding this concept simplifies other complex vector fields and aids in analyzing steady forces in physics and engineering.