For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
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Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= (\frac{\partial}{\partial t}\hat{\Phi}_t) \hat{\Phi}_t^{-1}) \\ &= (\frac{\partial}{\partial t}\hat{\Phi}_t) ...
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Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Also, what are the actual ...
4
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2answers
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Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$

Given: $x,y\in (-\sqrt2;\sqrt2)$ and $x^4+y^4+4=\dfrac{6}{xy}$ Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ?