on tangent spaces for details). A differential -form on is defined as a function that assigns to every a -covector on the tangent space of at , usually denoted . In brief, a differential -form is a -covector field. The space of -forms on is usually denoted ; thus if is a differential -form, we write . By convention, a continuous function on is a differential 0-form: . We first construct differential 1-forms from 0-forms and deduce some of their basic properties. To simplify the discussion below, we will only consider smooth differential forms constructed from smooth () functions. Let be a smooth function. We define the 1-form on for and by , where is the total derivative of at . (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions) , defined by , where is the ith standard coordinate of . The 1-forms are known as the basic 1-forms; they are conventionally denoted . If the standard coordinates of are , then application of the definition of yields , so that , where is the Kronecker delta. Thus, as the dual of the standard basis for , forms a basis for . As a consequence, if is a 1-form on , then can be written as for smooth functions . Furthermore, we can derive an expression for that coincides with the classical expression for a total differential: [Comments on notation: In this article, we follow the convention from tensor calculus and differential geometry in which multivectors and multicovectors are written with lower and upper indices, respectively. Since differential forms are multicovector fields, upper indices are employed to index them. The opposite rule applies to the components of multivectors and multicovectors, which instead are written with upper and lower indices, respectively. For instance, we represent the standard coordinates of vector as , so that in terms of the standard basis . In addition, superscripts appearing in the denominator of an expression (as in ) are treated as lower indices in this convention. When indices are applied and interpreted in this manner, the number of upper indices minus the number of lower indices in each term of an expression is conserved, both within the sum and across an equal sign, a feature that serves as a useful mnemonic device and helps pinpoint errors made during manual computation.] Basic operations