Applied Exterior Calculus (1985) by Dominic G. B. Edelen

By Dominic G. B. Edelen

This booklet offers an utilized creation to external calculus for top department undergraduates and starting graduate scholars. improvement is operational with an emphasis on computation talent and basic geometric notions. attention is proscribed to neighborhood questions. The publication additionally beneficial properties absolutely labored out examples and issues of solutions.

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9 u = x 2 + y2. 10 u = X2 _ y2. This representation by means of rectangular coordinates has, however, two disadvantages. First, geometric visualization fails us whenever we have to deal with three or more independent variables. Second, even for two independent variables it is often more convenient to confine the discussion to the x,y-plane alone, since in the plane we can sketch and can perform geometrical constructions without difficulty. er geometrical representation of a function of two variables, by means of contour lines, is sometimes preferable.

If then we define the extended function {*(x. y) by {*(x, y) = {(x, y) =e-x 2 II/ for y > 0 and all x, and by {*(x,O) =0 for x *" O. the function {* will be continuous in its domain R* where R* is the closed upper half-plane y ~ 0 with the exception of the point (0, 0). At the origin {* does not have a limit, and hence it is not possible to define {*(O, 0) in such a way that the extension is continuous at the origin. Indeed, for (x. y) on the parabola y = kX2, we have Functions of Several Variables and Their Derivatives 21 f(x, y) = e- lik • Approaching the origin along different parabolas leads to different limiting values, so that there exists no single limit of f(x, y) for (x, y) ~o.

It is, in fact, sufficient to assume that in addition to the first partial derivatives fx and (y, only one mixed partial derivative, say fyx, exists and that this derivative is continuous at the point in question. To prove this, we return to equation (11), divide by hk, and then let k alone tend to o. Then the right· hand side has a limit, and there· fore the left·hand side also has a limit, and r ~ _ (y(x + h,y) - (y(x, y) k~~ kh h . Further, it was proved above with the sole assumption that fyx exists that h~ = fllx(x + 9h, y + 9'k).

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