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Is the essential difference between one-variable calculus and multi-variable calculus exterior differential or something else?

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Matrices and vectors often do not behave like scalars. –  J. M. Nov 25 '10 at 11:53
The number of variables? :-) –  ShreevatsaR Nov 25 '10 at 12:38
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4 Answers

up vote 5 down vote accepted

There's essentially no difference in the concept of the derivative. It is still the best linear approximation to your function at each point. I.e. $f$ differentiable at $x_0$ still means that $$ f(x_0+h) = f(x_0) + f'(x_0)x_0 + \varphi(h), $$ for $h$ in some nbhd. of $0$ and $\varphi$ a continuous function that tends to $0$ at $0$ faster than linear. Except now $f'(x_0)$ is an $m \times n$ matrix called the Jacobian matrix and much more complicated to work with.

The first new thing we get when we go to higher dimensions is the notion of directional derivative, i.e. "how much does $f$ change in the direction of $v$?" We actually already have that on $\mathbb{R}$, it's just that there is only one direction (actually two, but it just comes down to a sign change), so it's never really looked at that way. The notion of directional derivative is exactly what you want when you want to generalize further to smooth manifolds, except you have to be a bit clever since you don't have an ambient space in which to have tangent vectors and instead use derivations (of e.g. smooth functions).

Integrals in multiple variables are much more complicated that the usual Riemann integral. Even when the functions are continuous. As demonstrated by Fubini's theorem.

Tangentially calculus-related (really it's more analysis, but those are related anyway): The fact that $\mathbb{R}$ has an ordering allows one to define things like the Henstock–Kurzweil integral. Such an extension of the Lebesgue integral is (AFAIK) not possible in $\mathbb{R}^n$ for $n > 1$.

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While I agree with the remarks on derivatives, replacing the Riemann integral by the mathematically more sound Lebesgue integral, Fubini's theorem holds under minimal conditions. Riesz and Sz.-Nagy point out in their book, that there is not that much of a difference between the one- and multi-dimensional integrals. –  Guido Kanschat Jul 27 '13 at 11:02
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It seems to me that an important difference is that while in one-variable calculus one only deals with one derivative, in multi-variable calculus there are infinitely many derivatives, the directional derivatives, a particular case of which are the partial derivatives. There are still the total derivatives for functions whose variables depend on another variable. Though all these derivatives are a generalization of the derivative of a single variable function.

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The topology of $R^n$ is much more complicated than the topology of $R$. For example, a simply connected subset of $R$ is just an interval. A simply connected subset in $R^n$ can be very complicated. This matters because simple connectivity enters as a hypothesis in several theorems.

The boundary of an interval in $R$ is simply a pair of points (or 1 or 0 points if the interval is unbounded). But the boundary of an open set in $R^n$ can be a manifold or something more complicated. Since the fundamental theorem of calculus requires integrating over a boundary, the theorem is more complicated in several variables. Integration over the boundary of an interval in $R$ is simply evaluating a function at two points. Integration over a manifold is more subtle and requires a large amount of machinery to do formally.

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The question is too wide - the answer depends on who you ask.

Essential in analysis is to look at limits.

(1) In one real variable there are two ways/directions to narrow a point. In two variables there are infinitely lines through a point, hence there are infinitely many directions to narrow a point and moreover you might get different results on different other kind of curves. (I think this is what Américo Tavares meant by infinitely many derivatives - directional derivatives).

(2) Integration over a 2-dimensional set might be done in at least three ways - a double integral or two iterated integrations - which if the conditions of Fubini-Tonelli's theorem is fulfilled has the same result.

(3) The fundamental theorem of calculus is a bit harder - Stoke's theorem.

(4) Partial differential equations are, generally speaking, harder than ordinary differential equations.

(5) Also, several variables makes one variable easier, as an example you will learn an easy way to calculate $$\int_{-\infty}^\infty e^{-x^2}dx$$

(6) ...

(7) ...

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I think you've got the essence of a fundamental difference which is the approach to limits. One thing I've noticed is that having a norm seems to be essential to do any analysis. In addition, suprema and infima are part of the extension of limits to higher dimensional spaces. Finally, limits of convergent sequences play a role in meaninfully extending the limit to higher dimensional spaces. –  Todd Wilcox May 15 '13 at 20:56
@ToddWilcox Norms are good to have, sometimes we do not have norms on spaces (which happens only for infinite dimensional vector spaces) e.g. $$C(0,1)\ni f \mapsto \|f\|_p =\left(\int_0^1|f(x)|^p dx\right)^{1/p} $$ is a norm for fixed $p\geq1$. For $0<p<1$ this is not the case, but $f\mapsto\|f\|_p^p$ is a metric. –  AD. May 16 '13 at 17:42
There are even interesting spaces in analysis without metrics too... –  AD. May 16 '13 at 17:49
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