Articles of calculus

Why does the tangent vector measure the rate of change of the angle which neighboring tangent make with the tangent?

I’m reading the classical Manfredo’s differential geometry book and I couldn’t prove formally the following statement written on page 16: Why does the tangent vector measure the rate of change of the angle which neighboring tangent make with the tangent at $s$?

Need help with an integral

need help with calculating this: $$\int_{0}^{2\pi}\frac{1-x\cos\phi}{(1+x^2-2x\cos\phi)^{\frac{3}{2}}}d\phi$$ Thanks in advance!

Prove $\lim_{x\to-2}\frac{x+8}{x+3}=6 $ using $\epsilon-\delta$

Prove: $$\lim_{x\to-2}\frac{x+8}{x+3}=6 $$ I started with: $ |\frac{x+8}{x+3} – 6| < \epsilon => |\frac{x+8-6x-18}{x+3} | < \epsilon => |\frac{-5x-10}{x+3} | < \epsilon =>| \frac{-5(x+2)}{x+3} | < \epsilon $ From the definition of limit we know that: $ |x+2| <\delta $ Just don’t know how to continue now… any suggestions?

Question about supremum(inequality)

I am reading a proof and in one part there is: $\sup(f+g)\le \sup f +\sup g $ where $f$ and $g$ are functions. Why is this true?I can’t see why(even though it might be obvious)

A question regarding a proof about limit and continuity

I am trying to understand a part of the following proof. Prove that $(1)$ and $(2)$ are equivalent: $(1)$ $\lim_{x \to c}f(x)=f(c)$ $(2)$ $f$ is continuous at $c$. I understood the proof of $(1) \Rightarrow (2)$. My question is about a part of the proof of $(2) \Rightarrow (1)$. Definition Definition of continuity I am […]

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are independent or not, we always have that $\bf{z}$ also follows from the Gaussian distribution $N([u_x,u_y]^T, Cov([x,y]^T))$, where $Cov$ means the covariance. The above claim is reformulated […]

Prove that $x+\sin x$ is strictly increasing

I have a function $f(x)=x+\sin x$ and I want to prove that it is strictly increasing. A natural thing to do would be examine $f(x+\epsilon)$ for $\epsilon > 0$, and it is equal to $(x+\epsilon)+\sin(x+\epsilon)=x+\epsilon+\sin x\cos \epsilon + \sin \epsilon \cos x$. Now all I need to prove is that $x+\epsilon+\sin x\cos \epsilon + \sin […]

Convergence of improper double integral.

Please help me to determine $\alpha$ and $p$, such that the integral $$ I = \iint_G \frac{1}{(x^{\alpha}+y^3)^p} \ dx dy $$ converges, where $G = {x>0, y >0, x+y <1}$ and $\alpha >0, p>0$. I am comfortable with proper double integrals. I am also comfortable with improper double integrals when $f$ is continuous in $G$ […]

Function is defined on the whole real line and $|f(x) -f(y)| \leq |x-y|^\alpha$, then…

Given: $f(x)$ is defined on $\mathbb{R}$ and $|f(x) -f(y)| \le |x-y|^\alpha$. Which of the following statements are true? I. If $\alpha > 1$, then $f(x)$ is constant. II. If $\alpha = 1$, then $f(x)$ is differentiable. III. $0 < \alpha < 1$, then $f(x)$ is continuous. Answer: I $-$ true, II $-$ false, III $-$ […]

How do I show that a directional derivative is defined in every direction (for every vector) for a function $f: \mathbb{R}^2 \to \mathbb{R}$?

I’m working through some analysis books, and while working through the section on directional derivatives, I searched here and found this answer, which states Let $f: R^2 \to R$ be defined by $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$. This function is continuous; its directional derivative is defined at each point in […]