∇ ⋅ F

Definition

The divergence of a vector field $\mathbf{F}$ is a scalar field that describes the density of the outward flux of a vector field from an infinitesimal volume around a given point. It is a measure of how much the vector field is 'spreading out' from that point.

5 Must Know Facts For Your Next Test

1. The divergence of a vector field $\mathbf{F}$ is denoted by $\nabla \cdot \mathbf{F}$, where $\nabla$ is the del or nabla operator.
2. The divergence theorem, also known as Gauss's theorem, relates the divergence of a vector field to the flux of that field through the boundary of a region.
3. A vector field $\mathbf{F}$ is said to be divergence-free or solenoidal if $\nabla \cdot \mathbf{F} = 0$ at every point in the domain.
4. The divergence of a gradient field is always zero, i.e., $\nabla \cdot (\nabla f) = 0$ for any scalar field $f$.
5. The divergence of a curl field is also always zero, i.e., $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ for any vector field $\mathbf{F}$.

Review Questions

• Explain the physical interpretation of the divergence of a vector field $\mathbf{F}$.
  • The divergence of a vector field $\mathbf{F}$, denoted as $\nabla \cdot \mathbf{F}$, represents the density of the outward flux of $\mathbf{F}$ from an infinitesimal volume around a given point. In other words, it measures how much the vector field is 'spreading out' or 'diverging' from that point. A positive divergence indicates that the field is diverging, while a negative divergence indicates that the field is converging.
• Describe the relationship between the divergence theorem and the divergence of a vector field.
  • The divergence theorem, also known as Gauss's theorem, states that the integral of the divergence of a vector field $\mathbf{F}$ over a region $V$ is equal to the flux of $\mathbf{F}$ through the boundary $\partial V$ of that region. Mathematically, this can be expressed as $\int_V \nabla \cdot \mathbf{F} \, dV = \int_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dA$, where $\mathbf{n}$ is the outward-pointing normal vector to the surface $\partial V$. This theorem provides a powerful tool for relating the local properties of a vector field (its divergence) to its global properties (its flux through a surface).
• Analyze the properties of a vector field $\mathbf{F}$ with a divergence of zero (i.e., $\nabla \cdot \mathbf{F} = 0$).
  • If a vector field $\mathbf{F}$ has a divergence of zero (i.e., $\nabla \cdot \mathbf{F} = 0$) at every point in its domain, then it is said to be a divergence-free or solenoidal field. Such fields have the property that the net flux of $\mathbf{F}$ through any closed surface is zero, as per the divergence theorem. Additionally, any gradient field (i.e., a field that can be expressed as the gradient of a scalar function) is automatically divergence-free, and any curl field (i.e., a field that can be expressed as the curl of another vector field) is also divergence-free.

Related terms

Vector Field: A vector field is a function that assigns a vector to every point in a given space, such as the 3D Euclidean space.

Gradient: The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the rate of change in that direction.

Curl: The curl of a vector field $\mathbf{F}$ is a vector field that describes the infinitesimal rotation of the field about a given point. It measures the amount of 'circulation' or 'spinning' of the vector field.