Editing Vector field (section)

===Coordinate transformation law===
In physics, a [[Euclidean vector|vector]] is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system.  The [[Euclidean vector#Vectors, pseudovectors, and transformations|transformation properties of vectors]] distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a [[covector]].

Thus, suppose that {{math|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} is a choice of Cartesian coordinates, in terms of which the components of the vector {{mvar|V}} are
<math display="block">V_x = (V_{1,x}, \dots, V_{n,x})</math>
and suppose that (''y''<sub>1</sub>,...,''y''<sub>''n''</sub>) are ''n'' functions of the ''x''<sub>''i''</sub> defining a different coordinate system.  Then the components of the vector ''V'' in the new coordinates are required to satisfy the transformation law
{{NumBlk||<math display="block">V_{i,y} = \sum_{j=1}^n \frac{\partial y_i}{\partial x_j} V_{j,x}.</math>|{{EquationRef|1}}}}

Such a transformation law is called [[covariance and contravariance of vectors|contravariant]].  A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of ''n'' functions in each coordinate system subject to the transformation law ({{EquationNote|1}}) relating the different coordinate systems.

Vector fields are thus contrasted with [[scalar field]]s, which associate a number or ''scalar'' to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.