Problem 36 - Calculus Early Transcendentals

Question

Determine whether the vector field F(x, y, z) is free of sources and sinks. If it is not, locate them. $$ \mathbf{F}(x, y, z)=\left(x^{3}-x\right) \mathbf{i}+\left(y^{3}-y\right) \mathbf{j}+\left(z^{3}-z\right) \mathbf{k} $$

Step-by-Step Solution

Answer

The field has sources and sinks outside the sphere of radius 1 centered at the origin.

Step 1: Understanding the Problem

The problem asks us to determine if the vector field \( \mathbf{F}(x, y, z) = (x^3 - x) \mathbf{i} + (y^3 - y) \mathbf{j} + (z^3 - z) \mathbf{k} \) is free of sources and sinks. A vector field has sources and sinks if its divergence is non-zero at any point; if the divergence is zero everywhere, the field is free of sources and sinks.

Step 2: Calculate the Divergence of the Vector Field

The divergence of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). For this vector field:- \( P = x^3 - x \)- \( Q = y^3 - y \)- \( R = z^3 - z \) Calculate each partial derivative:- \( \frac{\partial P}{\partial x} = 3x^2 - 1 \)- \( \frac{\partial Q}{\partial y} = 3y^2 - 1 \)- \( \frac{\partial R}{\partial z} = 3z^2 - 1 \).Substitute these into the divergence formula: \( abla \cdot \mathbf{F} = (3x^2 - 1) + (3y^2 - 1) + (3z^2 - 1) \).

Step 3: Simplify the Divergence Expression

We simplify the divergence expression obtained: \( abla \cdot \mathbf{F} = (3x^2 - 1) + (3y^2 - 1) + (3z^2 - 1) = 3x^2 + 3y^2 + 3z^2 - 3 \). This can be further simplified as: \( abla \cdot \mathbf{F} = 3(x^2 + y^2 + z^2 - 1) \).

Step 4: Analyze the Divergence Result

The divergence \( abla \cdot \mathbf{F} = 3(x^2 + y^2 + z^2 - 1) \) is zero when \( x^2 + y^2 + z^2 = 1 \). This indicates that there are no sources or sinks on the surface of a sphere centered at the origin with a radius of 1. However, at points where \( x^2 + y^2 + z^2 eq 1 \), the field has sources or sinks.

Key Concepts

Divergence

Divergence is a crucial concept when analyzing vector fields. In essence, it measures the 'spread' of a vector field at a particular point. If the vectors in a field are moving away from a point, the divergence at that point is positive, indicating a source. Conversely, if the vectors are converging towards a point, the divergence is negative, indicating a sink.

For a given vector field \( \mathbf{F} \), the divergence is mathematically expressed as:

• \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \)

where \( P, Q, \) and \( R \) are the components of the vector field.

In the original problem, the divergence \( abla \cdot \mathbf{F} \) was found to be \( 3(x^2 + y^2 + z^2 - 1) \). This indicates that the vector field is divergence-free, meaning there are no sources or sinks, only on a sphere of radius 1 centered at the origin.

Sources and Sinks

Sources and sinks in a vector field represent points where vectors emanate from or converge into, respectively. Understanding whether a vector field has these characteristics involves analyzing its divergence.

If the divergence at a point is positive, the point acts as a source, sending out vectors. If it's negative, the point behaves as a sink, drawing in vectors.

In the exercise, we determined that \( abla \cdot \mathbf{F} = 3(x^2 + y^2 + z^2 - 1) \). Hence:

• The vector field has no sources or sinks on the surface of a unit sphere \((x^2 + y^2 + z^2 = 1)\).
• Outside this sphere, since the divergence is non-zero, the field exhibits sources and sinks.

Recognizing sources and sinks is fundamental in visualizing how vector fields behave in space.

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiation and integration of vector fields. It extends calculus to functions of vector variables.

It encompasses several operations, including divergence, curl, and gradient, which help in analyzing vector fields like the one in the exercise. Here's what each operation signifies:

• Divergence: Measures the rate at which a vector field 'spreads out' from a point.
• Curl: Signifies the twisting or rotational effect at a point within a vector field.
• Gradient: Provides a vector pointing in the direction of the greatest rate of increase of a scalar field.

In the problem you worked on, vector calculus enabled the direct computation of divergence to assess the presence of sources and sinks across the vector field, highlighting its role in understanding spatial phenomena.