Articles of calculus

Problem with $Pullback$ Calculation

Let $$\omega=-xdx \wedge dy-3dy\wedge dz$$ and $$\phi:\mathbb R^2 \to \mathbb R^3, (u,v) \to(uv,u^2,3u+v)$$ I tried to compute the pullback $\phi^*(\omega)$, but was not able to solve it. $$\phi^*(\omega)=(-uv)(vdu+udv)-3(2udu)\wedge(3du+dv)=-uv^2du-u^2vdv-6udu\wedge(3du+dv)=uv^2du-u^2vdv-6udu\wedge3du-6udu\wedge dv$$ Could someone tell me, what I am doing wrong?

Generalization of the mean value theorem for the space $C$

Does there exist the following generalization of the mean value theorem for the space $C[a,b]$ ? Let $f:\mathbb{R}\to\mathbb{R}$ be a continuously differentiable on $\mathbb{R}$ function, and $y,z\in C[a,b]$. Then there exists $\xi\in C[a,b]$ such that $$ f(y(x))-f(z(x))=f'(\xi(x))(y(x)-z(x)) \quad \forall x\in[a,b]. $$

$FTC$ problem $\frac d{dx}\int_1^\sqrt xt^tdt$

I stumbled upon a sneaky example while reading about the Fundamental Theorem of Calculus here $$\frac d{dx}\int_1^\sqrt xt^tdt$$ and it makes me question my whole understanding. I know that the FTC says $$\frac d{dx}\int_a^x f(t)dt\,=\, f(x)$$ so I would say that $f(x)=\sqrt x^\sqrt x$ but this is incorrect I should get $$f(x)=\frac 12 x^{\frac {\sqrt […]

Describing the bended regions of a four-parameter logistic function

I’m working with the four-parameter logistic function. $y = a + \frac{b-a}{1+e^{c(d-x)}}$ There are two points on the curve at which the oblique portion of the curve meets the lower and upper plateaus of the function. What methods would be recommended for characterizing these two points? That is, how to characterize the points on the […]

How to show $f,g$ are equal up to n'th order at $a$.

I need help with this problem. Let two real-valued functions $f,g$ be equal to the $n’$th order at $a$ if $$\lim_{h\rightarrow 0}\frac{f(a+h)-g(a+h)}{h^n} = 0 $$ If $f'(a),\ldots,f^n(a)$ exist, show that $f$ and the function $g$ be given by $$g(x) = \sum_{i=0}^n \frac{f^{(i)}(a)}{i!}(x-a)^i$$ are equal up to the $n$’th order at $a$. I’m also given the […]

How do I solve a double integral with an absolute value?

Given the following integral $$\int_{y=0}^1 \int_{x=0}^1|x-y|(6x^2y) \, dx \, dy$$ how do I change the limits of integration? According to my textbook, it is $$\int_{y=0}^1 \int_{x=y}^{1}(x-y)(6x^2y) \, dx \, dy +\int_{y=0}^1 \int_{x=0}^y (y-x)(6x^2y)\,dx\,dy$$ How did the textbook get the new limits of integration? Can someone explain to me he steps I should use?

Limit of $n!/n^n$ as $n$ tends to infinity

This question already has an answer here: What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$? 2 answers

Derivative of the Frobenius norm with respect to a vector

I am trying to calculate the derivative of an energy function with respect to a vector $\mathbf{x}$. The energy is given by: $\psi(\mathbf{F}) = \lVert\mathbf{F}-\mathbf{I}\rVert_F^2.$ Where $\mathbf{F}$ is a square matrix that is a function of $\mathbf{x}$ (a column vector): $\mathbf{F(x)} = (\mathbf{x}\cdot\mathbf{u^T})\mathbf{A}$ $\mathbf{u^T}$ is a constant row vector and $\mathbf{A}$ is a constant square […]

Integrating the inverse of a function.

$f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral, $$\int_0^a f^{-1}(x) dx.$$ Reasoning The interval $[0,a]$ is “reflected” across the line $y=x$ onto the interval $0 \leq y \leq a.$ $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = […]

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, $\delta$ is the small radius. We consider $\displaystyle f(z) = \frac{\log^2(z)}{z^2 + 1}$ where $z = x+ iy$ How can we prove: $$\oint_{\Gamma} f(z) dz \to 0 \space \text{when} […]