The command VectorField(...) is a constructor method for creating a VectorField object. Once a valid VectorField object is created, it has access to various methods which allow it to be manipulated and its contents queried. see Overview of VectorField object for more detail.

A vector field $X$ is an expression of the form $X\=\sum _{i\=0}^n{\mathrm{\xi }}^i\left(x_1\,x_2\,\dots \,x_n\right)$ living on a space with coordinates $\left(x_1\,x_2\,\dots \,x_n\right)$. The ${\mathrm{\xi }}^i$ are referred to as components, and $x_1\,x_2\,\dots \,x_n$  are referred to as (coordinates of) space.   $\frac{\partial }{\partial x_i}\left({\mathrm{\xi }}^i\right)\left(x_1\,x_2\,\mathrm{..}\,x_n\right)$

The VectorField command first validates the user input arguments and then constructs a VectorField object. A valid VectorField object consists of two data attributes: components ${\mathrm{\xi }}^1\,{\mathrm{\xi }}^2\,\dots \,{\mathrm{\xi }}^n$ and space variables $x_1\,x_2\,\dots \,x_n$.

In the first calling sequence, both arguments components = compList, space=varList are required. These two lists must be of the same length.

The second calling sequence is a textual representation of the usual appearance of a vector field.  The space = varList argument is optional; if present, its specification of the space overrides the space [x1, x2,..., xn] implied by DExpr.

In the third calling sequence, the space =varList argument is optional; if present, its specification of the space overrides the space [x1, x2,..., xn] implied by listOfPairs.

The fourth calling sequence is a special constructor for the zero vector field on the specified space; the space = varList argument is required.

This command is part of the VectorField package. For more detail, see Overview of the VectorField package.

This command can be used in the form VectorField(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-VectorField(...).

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$X≔\mathrm{VectorField}\left(\mathrm{components}=\left[x^2\,xy\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$X≔x^2\left(\frac{\ⅆ}{\ⅆx}\right)+xy\left(\frac{\ⅆ}{\ⅆy}\right)$

$X≔\mathrm{VectorField}\left(x^2\mathrm{D}\left[x\right]+xy\mathrm{D}\left[y\right]\right)$

$X≔x^2\left(\frac{\ⅆ}{\ⅆx}\right)+xy\left(\frac{\ⅆ}{\ⅆy}\right)$

$X≔\mathrm{VectorField}\left(\left[\left[x^2\,x\right]\,\left[xy\,y\right]\right]\right)$

$X≔x^2\left(\frac{\ⅆ}{\ⅆx}\right)+xy\left(\frac{\ⅆ}{\ⅆy}\right)$

$Z≔\mathrm{VectorField}\left(0\,\mathrm{space}=\left[x\,y\right]\right)$

$Z≔0$

$\mathrm{Tx}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\,t\right]\right)$

$\mathrm{Tx}≔\frac{\ⅆ}{\ⅆx}$

$\mathrm{GetComponents}\left(\mathrm{Tx}\right),\mathrm{GetSpace}\left(\mathrm{Tx}\right)$

$\left[1\,0\,0\,0\right],\left[x\,y\,z\,t\right]$