or vector fields: 0-forms and 3-forms with scalar fields, 1-forms and 2-forms with vector fields. Thus for example the curl naturally takes as input a vector field or 1-form, but naturally has as output a 2-vector field or 2-form (hence pseudovector field), which is then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in the curl of a vector field in higher dimensions not having as output a vector field. See also Real-valued function Function of a real variable Function of several real variables Vector calculus identities Vector algebra relations Del in cylindrical and spherical coordinates Directional derivative Conservative vector field Solenoidal vector field Laplacian vector field Helmholtz decomposition Orthogonal coordinates Skew coordinates Curvilinear coordinates Tensor Geometric calculus References Citations Sources Sandro Caparrini (2002) "The discovery of the vector representation of moments and angular velocity", Archive for History of Exact Sciences 56:151–81. Barry Spain (1965) Vector Analysis, 2nd edition, link from Internet Archive. Chen-To Tai (1995). A historical study of vector analysis. Technical Report RL 915, Radiation Laboratory, University of Michigan. External links The Feynman Lectures on Physics Vol. II Ch. 2: Differential Calculus of Vector Fields A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen-To Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of Willard Gibbs) by Edwin Bidwell Wilson, published 1902. Mathematical physics