Editing Vector calculus (section)

=== Different 3-manifolds ===
Vector calculus is initially defined for [[Euclidean space|Euclidean 3-space]], <math>\mathbb{R}^3,</math> which has additional structure beyond simply being a 3-dimensional real vector space, namely: a [[norm (mathematics)|norm]] (giving a notion of length) defined via an [[inner product]] (the [[dot product]]), which in turn gives a notion of angle, and an [[orientability|orientation]], which gives a notion of left-handed and right-handed. These structures give rise to a [[volume form]], and also the [[cross product]], which is used pervasively in vector calculus.

The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the [[coordinate system]] to be taken into account (see ''{{slink|Cross product#Handedness}}'' for more detail).

Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric [[nondegenerate form]]) and an orientation; this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the [[special orthogonal group]] {{math|SO(3)}}).

More generally, vector calculus can be defined on any 3-dimensional oriented [[Riemannian manifold]], or more generally [[pseudo-Riemannian manifold]]. This structure simply means that the [[tangent space]] at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate [[metric tensor]] and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.