can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let . Then Stokes' theorem Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ: Divergence theorem Suppose is a subset of (in the case of represents a volume in 3D space) which is compact and has a piecewise smooth boundary (also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then the divergence theorem says: The left side is a volume integral over the volume , the right side is the surface integral over the boundary of the volume . The closed manifold is quite generally the boundary of oriented by outward-pointing normals, and is the outward pointing unit normal field of the boundary . ( may be used as a shorthand for .) In topology Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a knot in a piece of string. In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble . In finite geometry Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. For example, any three skew lines in PG(3,q) are contained in exactly one regulus. See also Dimensional analysis Distance from a point to a plane Four-dimensional space Three-dimensional graph Solid geometry Two-dimensional space Notes References Arfken, George B. and Hans J. Weber. Mathematical Methods For Physicists, Academic Press; 6 edition (June 21, 2005). . External links Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry Keith Matthews from University of Queensland, 1991 Analytic geometry Multi-dimensional geometry Three-dimensional coordinate systems 3 (number)