Differentials and 1-forms
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Differentials in one dimension
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Introduction: In order to understand 1-forms it is necessary first to understand differentials. The concept of a 1-form is a generalization of the notion of a differential. Differentials in one-dimension: So let us begin with a step-by-step reconstruction of how differentials arise as objects of interest to mathematicians. Calculus originates in the idea of finding the instantaneous rate of change of a function [Equation goes here - download the original to see it.]. The geometric interpretation of this problem arises from the graph of f. The instantaneous rate of change of f at a given point x on the graph of [Equation goes here - download the original to see it.] will be the gradient to the graph at that point. [Diagram goes here - download the original to see it.] The diagram above shows how the derivative of the function f is the limit of the gradient of the cord joining the points [Equation goes here - download the original to see it.] on the graph of [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] represent a small change in x and the corresponding small change in y. Hence [Equation goes here - download the original to see it.]. However, as the gradient of the tangent to the graph the derivative [Equation goes here - download the original to see it.] is also given as the ratio of a small increase in y to the small increase in x that produces it. Let us denote the small increases in x and y by dx and dy respectively, and call these differentials. Then we can write [Equation goes here - download the original to see it.]. The previous diagram makes the meaning of this clear; dx and dy stand for increases in the x and y directions, therefore are real number values, and are connected by the equation [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.]. Note, the term “small” in the phrase “small increase in x” is not really necessary. The rate of change (gradient or tangent of the angle) of the function f is the same no matter how big the “small” increase in x is, provided that the corresponding “small” increase in y is also given. [Diagram goes here - download the original to see it.] The term “small” is introduced as a prelude to the idea of taking a limit, for we think of [Equation goes here - download the original to see it.] as arising as the limit Equation goes here - download the original to see it.] In this context we may think of [Equation goes here - download the original to see it. as the instantaneous rate of change of y with respect to x. In the diagram [Diagram goes here - download the original to see it.] we have also indicated that we may think of the tangent as a vector, and the differentials as the coordinate functions of this vector. The tangent is any vector lying on the tangent, and if v is such a tangent vector, then [Equation goes here - download the original to see it.]. This equation makes it clear that the differentials dx and dy are related to any tangent vector v that has the same direction as the tangent to the curve [Equation goes here - download the original to see it.] at the point [Equation goes here - download the original to see it.]. This makes dx and dy not numbers as such, but functions whose values are real numbers. The function [Equation goes here - download the original to see it.] is a mapping from the tangent vector v to its first coordinate lying in the x-direction. Letting [Equation goes here - download the original to see it.] stand for the set of all tangent vectors at the point p [Equation goes here - download the original to see it.] So the symbol dx is in fact ambiguous. Strictly it stands for the function dx that maps the vector [Equation goes here - download the original to see it.] to its coordinate in the x-direction; but it also represents the value that this function takes at that point. Furthermore, as the diagram above makes clear, the function dx operates on a position vector, that is a vector v at a point p, so strictly we should make this clear, as follows [Equation goes here - download the original to see it.] The expression [Equation goes here - download the original to see it.] means “the first coordinate of the tangent vector v at p”. It is and does not depend on p. We need to do a little work on the notion of the tangent space [Equation goes here - download the original to see it.] as well. [Diagram goes here - download the original to see it.] There are an infinite number of tangent vectors at a given point p. They are all tangent to the curve [Equation goes here - download the original to see it.] and therefore have the same direction and are all members of the set [Equation goes here - download the original to see it.]. If [Equation goes here - download the original to see it.] are two such tangent vectors, then [Equation goes here - download the original to see it.] or more ambiguously (as before) [Equation goes here - download the original to see it.]. Furthermore, the story does not stop there. By the fundamental theorem of calculus differentiation is the inverse operation of integration. Given the equation [Equation goes here - download the original to see it.] then, by the fundamental theorem of calculus [Equation goes here - download the original to see it.] It is this that allows us to separate the differentials in [Equation goes here - download the original to see it.] and then integrate them according to the rule [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] .Finally, the curve [Equation goes here - download the original to see it.] may also be given a parametric representation [Equation goes here - download the original to see it. Then the differential dx stands for the rate of change of [Equation goes here - download the original to see it.] in the x-direction. That is [Equation goes here - download the original to see it.] . Likewise [Equation goes here - download the original to see it.] . We are applying the chain rule here. If the parameter in [Equation goes here - download the original to see it.] is t standing for time, then r stands for the position of a particle P as it moves along the curve [Equation goes here - download the original to see it.] and dx stands for the rate of change of this position in the x-direction.
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Contents of
Differentials and 1-forms
1 Differentials in one dimension
2 Summary – differentials in one dimension
3 Differentials in two-dimensions
4 Example: differential forms and directional derivatives
5 Differential forms: three or more dimensions
6 1-forms
7 Definition of the differential of a function f
8 Duals
9 Withdrawn
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