Definition:Vector Field
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 the vector field on $M$ we mean the continuous map $X : M \to TM$ such that:
- $\forall p \in M : X_p \in T_p M$
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
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $5$. Scalar and Vector Fields
- 1961: D.R. Bland: Solutions of Laplace's Equation ... (next): Chapter $1$: Occurrence and Derivation of Laplace's Equation: $1$. Situations in which Laplace's equation arises.
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.2$ Rotation of Coordinates
- 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.2$ The Electric Field