Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch a plot of the vector field $$ \mathbf{F}=-\frac{x}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \mathbf{i}-\frac{y}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \mathbf{j} $$ for \((x, y)\) in the rectangle \(-1 \leq x \leq 1,-1 \leq y \leq 1\). From the plot, use the marginal box that describes the interpretation of div and curl to determine whether div is positive, negative, or zero at the origin, and whether a paddle wheel placed at the origin would rotate clockwise, counterclockwise, or not at all. (For the curl, think of \(\mathbf{F}\) as being a vector field in 3-space with \(z\)-component equal to 0.)

Step-by-Step Solution

Answer

Divergence is zero at the origin; the paddle wheel does not rotate.

Step 1: Understanding the Vector Field

The given vector field is \( \mathbf{F} = -\frac{x}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \mathbf{i} -\frac{y}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \mathbf{j} \). This represents a 2-dimensional vector field where the magnitude and direction of \( \mathbf{F} \) depend upon the coordinates \( (x, y) \). The components \(-\frac{x}{\left(1+x^{2}+y^{2}\right)^{3 / 2}}\) and \(-\frac{y}{\left(1+x^{2}+y^{2}\right)^{3 / 2}}\) decrease with increasing distance from the origin, due to the negative sign and the denominator's growth.

Step 2: Sketching the Vector Field

To sketch the vector field in the region \(-1 \leq x \leq 1, -1 \leq y \leq 1\), substitute different \( (x,y) \) values into the components of \( \mathbf{F} \). Notice that near the origin, vectors point toward the origin, indicating an attractive vector field. As \( x^2 + y^2 \) increases, the length of the vectors decreases. Sketch arrows at various grid points within the specified region, showing the direction and relative size of \( \mathbf{F} \).

Step 3: Interpreting Divergence at the Origin

The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is given by \( \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). Calculate \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x} \left(-\frac{x}{\left(1+x^{2}+y^{2}\right)^{3 / 2}}\right) \) and \( \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y} \left(-\frac{y}{\left(1+x^{2}+y^{2}\right)^{3 / 2}}\right) \). After computation, both terms exactly cancel out at the origin, hence the divergence is zero at the origin.

Step 4: Interpreting Curl Based on the Vector Field

The curl in 2D settings for field \( \,\mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is given by \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \) treated as the k-component. Compute \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} \left(-\frac{y}{\left(1+x^{2}+y^{2}\right)^{3 / 2}}\right) \) and \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y} \left(-\frac{x}{\left(1+x^{2}+y^{2}\right)^{3 / 2}}\right) \). These components also cancel out at the origin, which means the curl is zero, indicating no rotation.

Key Concepts

Divergence

Divergence is a measure of how much a vector field spreads out from a point. In simpler terms, it tells us whether a region is a source (positive divergence) or a sink (negative divergence) of vector fields. When we talk about vector fields in two dimensions with components \( P \) and \( Q \), the divergence is calculated as \( \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). This formula helps determine if there is any net 'outflow' from a small region surrounding a point, or an 'inflow'.

• If the divergence is positive, it means the vector field is acting as a source, pushing vectors out.
• If it's negative, vectors are being pulled inward, acting like a sink.
• At the origin for our specific vector field, the divergence is zero, suggesting no net movement in or out.

Understanding divergence supports us in grasping the nature of fluid or force fields, particularly how points behave as sources or sinks.

Curl

Curl is a concept used to describe the rotation or twisting of a vector field around a point. In 2D fields, the curl is particularly important in determining whether the vector field induces any rotation around a point when imagining a small paddle wheel placed at that spot. When a vector field is given as \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \), the curl expression simplifies to \( \text{curl} \ = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).

• A positive curl indicates a counterclockwise rotation.
• A negative curl implies a clockwise spin.
• If the curl is zero, as it is for our field at the origin, it means no rotation occurs.

Recognizing curl is essential in fields like fluid dynamics and electromagnetism, where it's vital to understand the local field rotation characteristics.

2D Fields

Two-dimensional vector fields simplify the complexity of analysis by reducing it to a plane, yet convey significant information about vector directions and magnitudes that change over a two-dimensional space. They are defined by vectors that have only two components, \( P \mathbf{i} + Q \mathbf{j} \). Understanding these fields involves:

• Identifying the direction of vectors using components \( P \) and \( Q \).
• Observing how vectors point inwards, outwards, or rotate, depending on their position relative to the origin.
• Visualizing how vector lengths change according to their positions, as seen in the given field where vector length decreases with distance.

2D vector fields are prevalent in visual representations, like graphics programming and physical simulations, where they effectively illustrate force, velocity, or other directional properties on a plane.

Calculus

Calculus is the mathematical framework that enables the analysis of vector fields through differentiation and integration. It is instrumental in finding the divergence and curl, two key features that define the behavior of vector fields. The calculus of vector fields involves partial derivatives, as seen in the calculations for both divergence and curl:

• Partial derivatives of the components \( P \) and \( Q \) are needed to understand how they change in one direction while keeping the other constant.
• These derivatives help compute the attributes like divergence \( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \) and curl \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).

Calculus provides the toolkit for analyzing changes and interactions in continuous systems, bridging the abstract concepts of mathematics with their practical applications in physics and engineering.