calculus

I was reading Gelfand&Fomin Calculus of Variations book, in which I found this Lemma 2 If $\alpha(x)$ is continuous in $[a, b]$, and if $$\int_a^b \alpha(x) h'(x)\, dx = 0$$ for every function ...

f[n_Integer] := Nest[-D[#, x]/2 &, Cos[x], n] Sum[f[n], {n, 0, Infinity}](*does not work*) The sum of that series can be manually found as $$s(x) = \frac{4}{5} \cos x + \frac{2}{5} \sin x$$ Is ...

The previous post looks at the nonlinear pendulum equation and what difference it makes to the solutions if you linearize the equation. If the initial displacement is small enough, you can simply replace sin θ with θ. If the initial displacement is larger, you can improve the accuracy quite a bit by solving the linearized […] The post Closed-form solution to the nonlinear pendulum equation first …

There’s a nice formula for the nth derivative of a product. It looks a lot like the binomial theorem. There is also a formula for the nth derivative of a quotient, but it’s more complicated and less known. We start by writing the quotient rule in an unusual way. Applying the quotient rule twice gives the following. […] The post nth derivative of a quotient first appeared on John D. Cook .

I wanted to try calculating some tangent vectors for a picture, just for fun. This requires taking partial derivatives. It will be some time before I can automate partial derivatives algebraically for ...

Fundamental Theorem of Calculus Although the notion of area is intuitive, its mathematical treatment requires a rigorous definition. This post introduces the Riemann integral, and proves the fundamental theorem of calculus—a beautiful result that connects integrals and derivatives. Riemann integral § Given a bounded1 1 Note that continuity is not required here; boundedness alone ensures the subin…

I have to solve this triple integral ∭ x y|z|³/(1+ (x²+y²)⁴)  dx  dy dz, with a domain T={(x, y, z) ∈ ℝ³: x ≤ 0, y ≥ 0, z² ≤ x²+y² ≤ 1}. Plotting with DESMOS 3D I see this: Actually I am not able to ...

I have been converting double integrals to polar coordinates and I found a wrong way to convert to polar. Can someone explain to me the error in doing it? $$\iint _Df(x,y)\text dx\text dy$$Let ...

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…

In Michael Tabor's "Chaos and Integrability in Nonlinear Dynamics", 1989, page 12, the nonlinear pendulum differential equation is integrated by introducing an invariant $E'$ which yields ...

The extreme value theorem says that any continuous function on a bounded, closed interval has a maximum and a minimum value. I came up with the following construction of a continuous function on a ...

Multivariable calculus helps explain how things work when more than one variable interacts at once, connecting with ideas such as architecture, nature, and most significantly football. In the latter, the ellipsoid equation has a significant effect on the aerodynamics of the ball, particularly the spiral motion when thrown. A football has three main dimensions: length, width, and height. Changing …

What you'll learn: - Understanding differentiation, integration, and their applications in engineering. - Solving systems of matrix operations, vector spaces and linear equations. - Studying various types of differential equations and their solutions. - Applying statistical methods to engineering data analysis and uncertainty assessment. - Using computational techniques to solve engineering probl…

What you'll learn: - Exploring advanced topics like calculus, linear algebra. - Understanding the principles of probability and statistical analysis. - Applying mathematical concepts to financial calculations. - Learning methods to optimize functions and solve optimization problems. - Studying topics like graph theory, combinatorics, and discrete structures. - Using numerical algorithms to solve …

log|x| + C revisited Posted by Mike Shulman A while ago on this blog, Tom posted a question about teaching calculus: what do you tell students the value of is? The standard answer is , with an “arbitrary constant”. But that’s wrong if means (as we also usually tell students it does) the “most general antiderivative”, since is a more general antiderivative, for two arbitrary constants and . (I’m w…

I saw an integral posted online that came from this year’s MIT Integration Bee. My thoughts on seeing this were, in order: It looks like a beta function. The answer is a small number. You can evaluate the integral using the substitution u = 1 − x2025. I imagine most students’ reactions would be roughly […] The post Riff on an integration bee integral first appeared on John D. Cook .

In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics. The post The Jagged, Monstrous Function That Broke Calculus first appeared on Quanta Magazine

As a student, I often made the mistake of thinking that if I knew a more powerful theorem, I didn’t need to learn a less powerful theorem. The reason this is a mistake is that the more powerful theorem may be better by one obvious criterion but not be better by other less-obvious criteria. The […] The post Integrals involving secants and tangents first appeared on John D. Cook .

Something that seems like an isolated trick may turn out to be much more important. This is the case with a change of variables discovered by Karl Weierstrass. Every calculus student learns a handful of standard techniques: u-substitutions, partial fractions, integration by parts, and trig substitutions. Then there is one more technique that is like […] The post The world’s sneakiest substitution…

Paddy MacMahon calculates tangents and turning points without calculus The post Who needs differentiation? appeared first on Chalkdust .