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Chapter 20: Problem 40

In Problems \(38-46,\) is the statement true or false? Assume \(\vec{F}\) and \(\vec{G}\) are smooth vector fields and \(f\) is a smooth function on 3 -space. Explain. If \(\vec{F}\) is a vector ficld with \(\operatorname{div} \vec{F}=0\) and \(\operatorname{curl} \vec{F}=\overrightarrow{0}\) then \(\overrightarrow{\boldsymbol{F}}=\overrightarrow{\boldsymbol{0}}\)

Short Answer

Expert verified
False. A vector field could be a non-zero constant vector.

Step by step solution

01

Understand the Problem

We are given a vector field \( \vec{F} \) with properties \( \operatorname{div} \vec{F} = 0 \) and \( \operatorname{curl} \vec{F} = \overrightarrow{0} \). We need to determine if these conditions imply \( \vec{F} = \overrightarrow{0} \). The divergence is zero, indicating the field is incompressible, and the curl is zero, indicating the field is irrotational.
02

Analyze the Implications of Divergence and Curl

If a vector field \( \vec{F} \) has zero divergence, it might mean it is constant or has no sources/sinks. Zero curl suggests it is a gradient field, \( \vec{F} = abla f \), for some scalar function \( f \). However, both conditions together (zero divergence and zero curl) do not necessarily imply the vector field is zero.
03

Use Helmholtz's Theorem

Helmholtz's Theorem states that a vector field is determined by its divergence and curl to within a gradient of a harmonic scalar function. Since \( \operatorname{div} \vec{F} = 0 \) and \( \operatorname{curl} \vec{F} = \overrightarrow{0} \), \( \vec{F} \) could still be a non-zero constant vector, satisfying both zero divergence and zero curl conditions.
04

Consider a Counterexample

Consider a constant vector field \( \vec{F} = c \vec{i} \), where \( c \) is a constant. Both \( \operatorname{div} \vec{F} \) and \( \operatorname{curl} \vec{F} \) are zero, but \( \vec{F} eq \overrightarrow{0} \). This shows that \( \vec{F} = \overrightarrow{0} \) is not generally true under the given conditions.
05

Conclusion

The statement is false. A vector field with zero divergence and zero curl is not necessarily the zero vector field; it can be a non-zero constant vector as demonstrated by the counterexample.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divergence
Divergence is a fundamental concept in vector calculus that measures the magnitude of a vector field's source or sink at a given point. It essentially evaluates how much a vector field spreads out from or converges into a point. Think of divergence as assessing if the vector field is 鈥榖reathing in鈥� or 鈥榦ut鈥�.

If the divergence of a vector field is zero throughout the field, it indicates that the field is incompressible.
  • Zero divergence: The vector field is neither creating nor destroying volume.
  • Divergence formula: For a vector field \( \vec{F} = P \hat{i} + Q \hat{j} + R \hat{k} \), the divergence is \( abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
When you apply these properties, knowing when a field is truly incompressible helps in various fields like fluid dynamics, to determine how a fluid's velocity field behaves.
Curl
Curl quantifies the rotation of a vector field around a point. It's like visualizing small whirlpools or how things spin around each point in the field. When you determine the curl of a vector field, you are essentially looking for its tendency to rotate.

For a vector field with zero curl, or irrotational field, there is no net rotation around any point in the field.
  • Zero curl: No local rotation in the vector field.
  • Curl formula: Given \( \vec{F} = P \hat{i} + Q \hat{j} + R \hat{k} \), the curl is \( abla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \hat{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \hat{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \hat{k} \).
Understanding curl is important in electromagnetism and fluid mechanics, where the rotation of fields or eddies within fluids is crucial to the system's dynamics.
Helmholtz's Theorem
Helmholtz's Theorem is a powerful result in vector calculus which states that any vector field in a domain that decreases fast enough at infinity can be decomposed uniquely into its divergence and curl components. This means any vector field can be expressed as the sum of an irrotational field and a solenoidal field.

Knowing Helmholtz's Theorem is crucial for comprehending how vector fields behave and are structured.
  • Application: Decomposition of vector fields into sum of a gradient of a scalar potential and a curl of a vector potential.
  • The implication: Even if both divergence and curl are zero, Helmholtz's Theorem indicates that the vector field might not be zero. It can still exist as a constant vector, highlighting its subtlety and importance.
This theorem is especially relevant in fields like fluid dynamics, electromagnetism, and acoustics, where comprehensive understanding of field behavior is required.
Incompressible Vector Field
An incompressible vector field is one with zero divergence across the entire field. This means that at every point within the field, there is no net volume change; the fluid or airflow isn鈥檛 expanding or compressing.

Incompressible vector fields are significant in studying fluid dynamics and aerodynamics.
  • No sources or sinks within the field.
  • Volume preservation: The field behaves consistently like water flowing through a pipe without any leakage.
Absolutely mastering this concept is key in physics and engineering, where knowing how and why a fluid maintains its volume is vital to designing systems like pipes and ducts. Understanding incompressible fields also assists in handling more complex flow scenarios.
Irrotational Vector Field
An irrotational vector field is a field with zero curl everywhere within the field. This means that there are no whirlpools or cycling motions at any point; it is free from rotational movement.

These fields are significant in various contexts, such as gravitational or electrostatic fields, which inherently lack rotational characteristics.
  • Can be represented as a gradient: \( \vec{F} = abla f \), where \( f \) is a scalar potential.
  • Absence of local spinning: The field appears straight and non-circular at any level.
Understanding irrotational fields unlocks deeper insights into potential fields where forces act along a direction without spinning, and it is especially useful in fields like thermodynamics and electrostatics.

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Most popular questions from this chapter

Is the statement true or false? Give a reason for your answer. If \(\vec{F}\) is a gradient field, then \(\int_{S} \operatorname{curl} \vec{F} \cdot d \vec{A}=0,\) for any smooth oricnted surface, \(S,\) in 3 -space.

calculate the circulation, \(\int_{C} \vec{F} \cdot d \vec{r},\) in two ways, directly and using Stokes" Theorem. \(\vec{F}=x y \vec{i}+y z \vec{j}+x z \vec{k}\) and \(C\) is the boundary of \(S\) the surface \(z=1-x^{2}\) for \(0 \leq x \leq 1\) and \(-2 \leq y \leq 2\) oriented upward. Sketch \(S\) and \(C\)

Is the vector field a gradient field? $$\vec{F}=y z \vec{i}+\left(x z+z^{2}\right) \vec{j}+(x y+2 y z) \vec{k}$$

In Problems \(38-46,\) is the statement true or false? Assume \(\vec{F}\) and \(\vec{G}\) are smooth vector fields and \(f\) is a smooth function on 3 -space. Explain. If \(\vec{F}\) and \(\vec{G}\) are vector fields, then \(\operatorname{curl}(\vec{F}+\vec{G})=\) \(\operatorname{curl} \vec{F}+\operatorname{curl} \vec{G}\)

In Problems \(36-37,\) give an example of: A vector field \(\vec{F}(x, y, z)\) such that \(\operatorname{curl} \vec{F}=\overrightarrow{0}\)
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