91Ó°ÊÓ

A vector field is a mathematical construct where a vector is associated with every point in space. In our exercise, the vector field is defined as \( \mathbf{F}(x, y, z) = (y^2 - 2y)\mathbf{i} + 0\mathbf{j} + 0\mathbf{k} \). This means every vector in the field has a component only in the \( x \)-direction, which depends on the \( y \)-value, while the other components are zero.
In simpler terms, for every point along the \( y \)-axis from 0 to 2, the vector points along the \( x \)-direction, either positively or negatively, based on the value of \( y \). A vector field can be visualized as arrows pointing in the direction of the vector at each point, allowing us to interpret physical phenomena like fluid flow or electric force fields.
Understanding the vector field involves not only the direction but also the magnitude of vectors, which in this case is \( y^2 - 2y \). It demonstrates how vectors can change over a specified region based on vector functions of coordinates.

The curl of a vector field is a measure of its tendency to rotate around a point. Given the formula for curl, \( abla \times \mathbf{F} \) allows us to compute the vector describing such rotation. For the vector field \( \mathbf{F}(x,y,z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is given by:

• \( \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} \)
• \( \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} \)
• \( \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \)

For our vector field with \( P = y^2 - 2y \), \( Q = 0 \), and \( R = 0 \), the curl simplifies to \( -(2y - 2)\mathbf{k} \). This calculation shows that the curl only has a non-zero component in the \( z \)-direction (or \( k \)-axis), indicating rotation about the \( z \)-axis.
The presence of a non-zero curl implies a directional rotation rather than mere translational motion in the field. The expression \( -(2y - 2) \) tells us how the magnitude of this rotational behavior changes with \( y \).

Rotational properties of a vector field help determine how the vectors 'twist' around an axis. Based on the curl given by \( abla \times \mathbf{F} = -(2y - 2)\mathbf{k} \), we know the rotation axis is along the \( z \)-direction. As the scalar value of \( -(2y - 2) \) changes with \( y \), it denotes varying magnitudes of rotational behavior, signifying the twist strength at different \( y \)-coordinates.
This expression achieves maximum absolute value at \( y = 0 \) and \( y = 2 \), where rotational effects are strongest, specifically \( -2 \) at \( y = 0 \) and \( 2 \) at \( y = 2 \). This indicates a reversal in direction at the endpoints of our interval. In physical terms, such rotation might correspond to swirling currents or other circular motions within the field.
The understanding of these properties is crucial for fields like fluid dynamics, where you need to know the twisting effects inside a fluid flow to predict behavior.

Sketching field vectors visually represents the vector field's behavior. In our scenario, the vector field \( \mathbf{F}(x, y, z) \) only has components in the \( x \)-direction depending on \( y \). The task involves plotting these vectors along the range \( 0 \leq y \leq 2 \).
Here are some calculated points:

• At \( y = 0 \), the vector \( F = 0 \), showing no magnitude at the origin.
• At \( y = 1 \), the vector \( F = -1\mathbf{i} \), pointing to the negative \( x \)-direction.
• At \( y = 2 \), the vector \( F = 0 \), returning to zero magnitude.

The plotted vectors illustrate a change in direction from positive to negative \( x \), back to positive, embodying a classic rotational behavior in the \( xy \)-plane. These sketches offer insights into the nature of the vector field, especially in how it influences movement or transfer of quantities, such as force or velocity, across the field.