- #1
Rasalhague
- 1,387
- 2
"f(x) dx = f"
Two related questions, or bundles of questions.
First, I read in Spivak, Calculus, p. 244, Ch. 13: (Here [itex]f:\mathbb{R} \rightarrow \mathbb{R}[/itex].)
Any comments welcome! It seems like a useful thing to have a symbol for, and it would be handy to have one that could be used in any context. But would I be right in thinking this particular definition, f(x) dx = f, is incompatible with the language of differential forms, even in this very context of integration for which they're designed? For example, the exterior derivative df would then be d(f(x) dx) = d(f(x)) dx + f(x) d(dx) = d(f(x)) + 0. But f(x) is a number, and d takes a function as its input, so the expression is a syntax error.
Second. Let F map Rm to Rn. Let F* be the pushforward under F. F* (that's meant to be a subscript star) is sometimes denoted dF, sometimes DF, and called the differential of F, and represented by the Jacobian matrix of F. Is this usage incompatible with the language of differential forms, where d is defined as the exterior derivative, and acts on a covariant alternating tensor field (CAT field). I think it is incompatible in the sense that the domain of D includes functions which are not CAT fields. But are there instances where it's actually ambiguous? If I've got this right, there are occasions where the two notations dovetail nicely, e.g. if df is a differential 1-form (1-CAT field) on Rn, and c is a parametrization, mapping J, a closed interval of R into Rn, then, assuming all necessary conditions of well-behavedness, I get this expression, which looks reassuringly like the single variable substitution rule, the chain rule in reverse:
[tex]\int_{c(J)} \mathrm{d}f = \int_J c^*(\mathrm{d}f) = \int_J ((\mathrm{d}f)\circ c)\circ c_*[/tex]
[tex]= \int_J ((\mathrm{d}f)\circ c)\circ \mathrm{d} c = \int_J \mathrm{d}(f \circ c) = \int_{\partial J} f \circ c.[/tex]
(...where "partial J" denotes the oriented boundary of J, in this case the set {b,-a}.) I presume the definitions of pullback and pushforward were designed to reduce to the single variable substitution rule, so it's not surprising they also produce something similar in the slightly more complicated case of a differential 1-form integrated over a paramatrized curve. If this is right, is it something that generalizes to cases where the domain of c is Rm, with m not equal to 1? Or is the resemblance to the single variable substitution rule beguiling, and the notations for exterior derivative and regular derivative (differential, pushforward) best kept distinct from the outset.
If there is an overlap, where does the distinction first manifest: do the two d notations become incompatible, in this context, for functions between real tuple spaces of more variables, or do they only appear when we move to the more general theory of manifolds not necessarily equal to a real tuple space, Rn?
Two related questions, or bundles of questions.
First, I read in Spivak, Calculus, p. 244, Ch. 13: (Here [itex]f:\mathbb{R} \rightarrow \mathbb{R}[/itex].)
[tex]\int_a^b f(x) \, dx[/tex]
means precisely the same as
[tex]\int_a^b f.[/tex]
Notice that, as in the notation
[tex]\lim_{x \rightarrow a} f(x),[/tex]
the symbol x can be replaced by any other letter. [...] The symbol dx has no meaning in isolation, any more than the symbol [itex]x \rightarrow[/itex] has any meaning, except in the context
[tex]\lim_{x \rightarrow a} f(x).[/tex]
In the equation
[tex]\int_a^b x^2 \, dx = \frac{b^3}{3} - \frac{a^3}{3}[/tex]
the entire symbol x2 dx may be regarded as an abreviation for the function f such that f(x) = x2 for all x.
Any comments welcome! It seems like a useful thing to have a symbol for, and it would be handy to have one that could be used in any context. But would I be right in thinking this particular definition, f(x) dx = f, is incompatible with the language of differential forms, even in this very context of integration for which they're designed? For example, the exterior derivative df would then be d(f(x) dx) = d(f(x)) dx + f(x) d(dx) = d(f(x)) + 0. But f(x) is a number, and d takes a function as its input, so the expression is a syntax error.
Second. Let F map Rm to Rn. Let F* be the pushforward under F. F* (that's meant to be a subscript star) is sometimes denoted dF, sometimes DF, and called the differential of F, and represented by the Jacobian matrix of F. Is this usage incompatible with the language of differential forms, where d is defined as the exterior derivative, and acts on a covariant alternating tensor field (CAT field). I think it is incompatible in the sense that the domain of D includes functions which are not CAT fields. But are there instances where it's actually ambiguous? If I've got this right, there are occasions where the two notations dovetail nicely, e.g. if df is a differential 1-form (1-CAT field) on Rn, and c is a parametrization, mapping J, a closed interval of R into Rn, then, assuming all necessary conditions of well-behavedness, I get this expression, which looks reassuringly like the single variable substitution rule, the chain rule in reverse:
[tex]\int_{c(J)} \mathrm{d}f = \int_J c^*(\mathrm{d}f) = \int_J ((\mathrm{d}f)\circ c)\circ c_*[/tex]
[tex]= \int_J ((\mathrm{d}f)\circ c)\circ \mathrm{d} c = \int_J \mathrm{d}(f \circ c) = \int_{\partial J} f \circ c.[/tex]
(...where "partial J" denotes the oriented boundary of J, in this case the set {b,-a}.) I presume the definitions of pullback and pushforward were designed to reduce to the single variable substitution rule, so it's not surprising they also produce something similar in the slightly more complicated case of a differential 1-form integrated over a paramatrized curve. If this is right, is it something that generalizes to cases where the domain of c is Rm, with m not equal to 1? Or is the resemblance to the single variable substitution rule beguiling, and the notations for exterior derivative and regular derivative (differential, pushforward) best kept distinct from the outset.
If there is an overlap, where does the distinction first manifest: do the two d notations become incompatible, in this context, for functions between real tuple spaces of more variables, or do they only appear when we move to the more general theory of manifolds not necessarily equal to a real tuple space, Rn?