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The four-vector style can be used: , which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. . Most of the 3-space vector rules have analogues in four-vector mathematics.
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The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a ...
The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds. In particular, suppose ω is a form defined on a contractible domain, and the integral of ω over any closed manifold is zero. Then there exists a form ψ such that ω = dψ.
The most basic non-trivial differential one-form is the "change in angle" form . This is defined as the derivative of the angle "function" θ ( x , y ) {\\displaystyle \\theta (x,y)} (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function.
The conservative vector fields correspond to the exact-forms, that is, to the forms which are the exterior derivative of a function (scalar field) on . The irrotational vector fields correspond to the closed 1 {\displaystyle 1} -forms , that is, to the 1 {\displaystyle 1} -forms ω {\displaystyle \omega } such that d ω = 0 {\displaystyle d ...