Definition:Vector Field

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Definition

Let $F$ be a field which acts on a region of space $S$.

Let the point-function giving rise to $F$ be a vector quantity.


Then $F$ is a vector field.


Vector Field on Smooth Manifold

Let $M$ be a smooth manifold.

Let $TM$ be the tangent bundle of $M$.

Let $T_p M$ be the tangent space at $p \in M$.


Then by a vector field on $M$ we mean a continuous mapping $X : M \to TM$ such that:

$\forall p \in M : X_p \in T_p M$

where $X_p$ denotes the value of $X$ at $p$.


Classification

Conservative Vector Field

$\mathbf V$ is a conservative vector field if and only if its curl is everywhere zero:

$\curl \mathbf V = \bszero$


Solenoidal Vector Field

$\mathbf V$ is defined as being solenoidal if and only if its divergence is everywhere zero:

$\operatorname {div} \mathbf V = 0$


Examples

Velocity of Fluid

In a moving fluid, the velocity $\mathbf v$ of the fluid is an example of a vector field.

That is, the velocity $\mathbf v$ at a point $P$ in the fluid is the velocity of the particle which is situated at $P$ at a given instant.


Electric Field Strength

Let $R$ be a region of space in which there exists an electric field.

The electric field strength in $R$ gives rise to a vector field over $R$.


Magnetic Field Strength

Let $R$ be a region of space in which there exists an magnetic field.

The magnetic field strength in $R$ gives rise to a vector field over $R$.


Also see

  • Results about vector fields can be found here.


Sources

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