Reddit reviews Introduction to Calculus and Analysis, Vol. 1 (Classics in Mathematics)

Historically, mathematicians had a goal of obtaining all integrals of rational integrands as rational expressions, rational expressions that would be given explicitly (or in closed form) in terms of elementary expressions. However, it was realized eventually that such a goal is a hopeless goal, one that is not possible in the traditional sense, and that the traditional artificial restrictions imposed on elementary analysis are thus unjustified.

They were brought to this realization most popularly by the elliptic integrals, integrals of rational expressions (rational expressions with the square root of a polynomial of the 4th or 3rd degree as an argument) which does not resolve itself into an explicit elementary expression by the methods of substitution or integration by parts.

Instead, due to greater rigor as gifted to us by the field of mathematical analysis, we were thus able to justify processes of approximations with a level of confidence and certainty that was not offered before.

As an elementary example, from Richard Courant's and Fritz John's "Introduction to Calculus and Analysis I", page 410 - 411, an integral expression for the time period of an ordinary pendulum was obtained, it is an elliptic integral, which means, we cannot proceed by way of a simple transformation of the independent variable ("method of substitution") or by breaking the integral apart into smaller parts by way of integration by parts and still hope to obtain a simple explicit elementary expression.

So, instead, there is an expression in the integrand, being, 1/√[1 - u^2 sin^2 (𝜃/2)]. For sufficiently small values of 𝜃, we find that the expression is arbitrarily close to the value 1, and therefore, this entire expression in the integrand was reduced to the factor one, allowing us to approximate the elliptic integral in sufficiently small intervals of 𝜃.

Noting that, the margin of error must be calculated (as was done so in the book). At least the physicists now have an expression for the time period of an ordinary pendulum - an imperfect approximation.

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Admitting defeat:

Over the decades and eventually, centuries, mathematicians decided to allow functions as integral expressions without requiring always that they must be solved explicitly in terms of elementary expressions due to the convenience offered, in fact, the famous Gamma function is exactly the example, a function that is usually expressed as an integral (and, of course, don't forget about the elliptic integrals).

In the end, not all integrals are meant to be solved the same way that an integral of an elementary polynomial is, this philosophy is not merely isolated to that of integral calculus, as its analog can be found in differential calculus as differential equations.

Square one...

You should have a solid base in math:

Introduction to Calculus and Analysis, Vol. 1 by Courant and John. Gotta have some basic knowledge of calculus.

Mathematical Methods in the Physical Sciences by Mary Boas. This is pretty high-level applied math, but it's the kind of stuff you deal with in serious physics/astrophysics.

You should have a solid base in physics:

They Feynman Lectures on Physics. Might be worth checking out. I think they're available free online.

You should have a solid base in astronomy/astrophysics:

The Physical Universe: An Introduction to Astronomy by Frank Shu. A bit outdated but a good textbook.

An Introduction to Modern Astrophysics by Carroll and Ostlie.

Astrophysics: A Very Short Introduction by James Binney. I haven't read this and there are no reviews, I think it was very recently published, but it looks promising.

It also might be worth checking out something like Coursera. They have free classes on math, physics, astrophysics, etc.

Here is the mobile version of your link

You may be thinking of Courant's original Differential and Integral Calculus in two volumes. What I have is the solutions to problems from the updated and expanded to about twice the size, Introduction to Calculus and Analysis, in three volumes.

introduction to calculus and analysis (3 book set) by courant and john:

volume 1

volume 2 book 1

volume 2 book 2