So many notations for derivatives - what's the difference?

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In summary, the conversation discusses two related questions about the notation and usage of derivatives in calculus and differential forms. It is noted that the symbol dx has no meaning in isolation and is only used in the context of integration, and there is a discussion about the compatibility of the notation for derivatives and differential forms. The conversation ends with a mention of some useful resources for further reading on the topic.
  • #1
Rasalhague
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"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].)

[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?
 
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  • #2
Notations of derivatives in the differentiation calculus vary a lot. There is always something missing: the function, the direction, the evaluation point, the flow, the manifold, or whatever. The clue is to know where you are, will say from which perspective you look at it. I counted 10 possibilities
(see https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/)
without mention something as slope. Some useful reads:

https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
 

1. What does "F(x) dx = f" mean?

"F(x) dx = f" is a notation used in mathematics to represent the integral of a function. The integral is a mathematical operation that finds the area under a curve, and it is represented by the symbol ∫. The function that is being integrated is denoted by f(x) and the variable of integration is represented by dx. The result of the integral is the function F(x), also known as the antiderivative or indefinite integral of f(x).

2. How is the integral related to the derivative?

The integral and the derivative are inverse operations of each other. The derivative of a function represents its rate of change, while the integral of a function represents the accumulation of its rate of change. In other words, the derivative tells us how fast the function is changing at a specific point, while the integral tells us the total change of the function over a given interval.

3. What is the purpose of using the integral in mathematics?

The integral is a fundamental concept in mathematics and is used to solve various problems in different fields, including physics, economics, and engineering. It allows us to find the area under a curve, calculate volumes and areas of irregular shapes, and solve optimization problems. It also plays a crucial role in the development of other mathematical concepts, such as differential equations and Fourier series.

4. How do you solve an integral?

To solve an integral, you can use different techniques, depending on the type of integral and the function being integrated. Some common methods include substitution, integration by parts, and partial fractions. It is also essential to have a good understanding of the rules of integration and the properties of various functions, such as polynomials, trigonometric functions, and logarithms.

5. Are there any practical applications of the integral?

Yes, the integral has numerous practical applications in various fields. For example, in physics, it is used to calculate the work done by a force, the velocity of an object, and the position of an object over time. In economics, it is used to calculate the total revenue and the cost of production. In engineering, it is used to determine the stress and strain of a material, the centroid of a shape, and the volume of a solid. These are just a few examples, and the applications of the integral are endless.

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