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 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: $$\lim_{x\to-2}\frac{x+8}{x+3}=6 $$ 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?

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)

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 […]

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 […]

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 […]

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 $G$ […]

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 $-$ […]

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 […]

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