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Chapter 14: Problem 3

What does it mean if the divergence of a vector field is zero throughout a region?

Short Answer

Expert verified
Answer: A zero divergence in a vector field signifies that the vector field is incompressible, has no sources or sinks, and has constant divergence in the region. The net flux across any closed surface in that region is also zero.

Step by step solution

01

Understand the concept of divergence

Divergence is a measure of how a vector field spreads out. Mathematically, the divergence of a vector field F with components (F_x, F_y, F_z) is given by: \[ \nabla \cdot \boldsymbol{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]
02

Interpretation of a zero divergence

If the divergence of a vector field is zero throughout a region, it means that the vector field is neither shrinking nor expanding. The rate at which the field lines diverge remains constant everywhere in that region.
03

Connection to sources and sinks

A zero divergence means that there are no sources or sinks in the region under consideration. A source is a place where a field line originates, and a sink is a place where a field line terminates. In a vector field with zero divergence, the number of field lines entering a small volume element is equal to the number of field lines leaving the volume element.
04

Flux across a closed surface in a region with zero divergence

In a region with zero divergence, the net flux through a closed surface is also zero. Flux is a measure of the field lines passing through a surface. Mathematically, the net flux across a closed surface S is: \[ \oint_S \boldsymbol{F} \cdot d\boldsymbol{A} = 0 \] where F is the vector field and dA is the differential area vector. In conclusion, if the divergence of a vector field is zero throughout a region, it means the vector field is incompressible, has no sources or sinks, and has constant divergence in that region. Furthermore, the net flux across any closed surface in that region is also zero.

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Most popular questions from this chapter

a. For what values of \(a, b, c,\) and \(d\) is the field \(\mathbf{F}=\langle a x+b y, c x+d y\rangle\) conservative? b. For what values of \(a, b,\) and \(c\) is the field \(\mathbf{F}=\left\langle a x^{2}-b y^{2}, c x y\right\rangle\) conservative?

Consider the potential function \(\varphi(x, y, z)=G(\rho),\) where \(G\) is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}} ;\) therefore, \(G\) depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\) b. Let \(S\) be the sphere of radius \(a\) centered at the origin and let \(D\) be the region enclosed by \(S\). Show that the flux of \(\mathbf{F}\) across \(S\) is $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2} G^{\prime}(a) $$ c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use part (c) to show that the flux across \(S\) (as given in part (b)) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.)

Use Stokes' Theorem to find the circulation of the following vector fields around any simple closed smooth curve \(C\). $$\mathbf{F}=\langle 2 x,-2 y, 2 z\rangle$$

The French physicist Andr茅-Marie Amp猫re \((1775-1836)\) discovered that an electrical current \(I\) in a wire produces a magnetic field \(\mathbf{B} .\) A special case of Amp猫re's Law relates the current to the magnetic field through the equation \(\oint_{C} \mathbf{B} \cdot d \mathbf{r}=\mu I,\) where \(C\) is any closed curve through which the wire passes and \(\mu\) is a physical constant. Assume that the current \(I\) is given in terms of the current density \(\mathbf{J}\) as \(I=\iint_{S} \mathbf{J} \cdot \mathbf{n} d S\) where \(S\) is an oriented surface with \(C\) as a boundary. Use Stokes' Theorem to show that an equivalent form of Amp猫re's Law is \(\nabla \times \mathbf{B}=\mu \mathbf{J}\)

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and parametric description of the sphere.
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