∇ × F

Definition

The curl of a vector field $\mathbf{F}$, denoted as $\nabla \times \mathbf{F}$, is a vector field that describes the infinitesimal rotation of the vector field $\mathbf{F}$ around a given point. It represents the circulation or tendency of the field to rotate about an axis perpendicular to the plane of the field.

5 Must Know Facts For Your Next Test

1. The curl of a vector field $\mathbf{F}$ is a vector field that describes the infinitesimal rotation of $\mathbf{F}$ around a given point.
2. The curl of a vector field $\mathbf{F}$ is denoted as $\nabla \times \mathbf{F}$.
3. The curl of a vector field is defined as the limit of the circulation of $\mathbf{F}$ around a small closed curve divided by the area of the curve as the curve shrinks to a point.
4. The curl of a vector field is related to the line integral of the vector field around a closed curve by Green's Theorem.
5. Stokes' Theorem relates the curl of a vector field $\mathbf{F}$ over a surface $S$ to the line integral of $\mathbf{F}$ around the boundary $\partial S$ of that surface.

Review Questions

• Explain the physical interpretation of the curl of a vector field $\mathbf{F}$.
  • The curl of a vector field $\mathbf{F}$, denoted as $\nabla \times \mathbf{F}$, represents the infinitesimal rotation or circulation of the vector field around a given point. It describes the tendency of the field to rotate about an axis perpendicular to the plane of the field. Physically, the curl of $\mathbf{F}$ can be interpreted as the density of the rotational component of the field, which is important in understanding phenomena such as fluid flow and electromagnetism.
• Describe the relationship between the curl of a vector field $\mathbf{F}$ and Green's Theorem.
  • Green's Theorem establishes a connection between the curl of a vector field $\mathbf{F}$ and the line integral of $\mathbf{F}$ around a closed curve. Specifically, Green's Theorem states that the line integral of $\mathbf{F}$ around a closed curve $C$ is equal to the double integral of the curl of $\mathbf{F}$ over the region $R$ bounded by $C$. This relationship allows for the calculation of line integrals using the curl of the vector field, which is often easier to compute.
• Explain how the curl of a vector field $\mathbf{F}$ is related to Stokes' Theorem and its applications.
  • Stokes' Theorem relates the curl of a vector field $\mathbf{F}$ over a surface $S$ to the line integral of $\mathbf{F}$ around the boundary $\partial S$ of that surface. Mathematically, Stokes' Theorem states that the surface integral of the curl of $\mathbf{F}$ over $S$ is equal to the line integral of $\mathbf{F}$ around the boundary $\partial S$. This powerful theorem has numerous applications in various fields, such as electromagnetism, fluid dynamics, and differential geometry, where the curl of a vector field plays a crucial role in understanding and analyzing physical phenomena.

Related terms

Vector Field: A vector field is a function that assigns a vector to every point in a given space, often used to represent physical quantities like force, velocity, or electric field.

Divergence: The divergence of a vector field $\mathbf{F}$, denoted as $\nabla \cdot \mathbf{F}$, is a scalar field that describes the density of the outward flux of a vector field from an infinitesimal volume around a given point.

Stokes' Theorem: Stokes' Theorem relates the curl of a vector field $\mathbf{F}$ over a surface $S$ to the line integral of $\mathbf{F}$ around the boundary $\partial S$ of that surface.