# Tagged Questions

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### $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$be continuous and define$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ by $g(x)=|f(x)|$ prove g is continuous

This was assigned as a practice problem for my multivariable calculus class, and its really not making sense to me, can someone help me out Let $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a ...
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### line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
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### How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}.$$ Then how to evaluate the double integral $$\int_C \int (x^2+ y^2) dx dy?$$ My ...
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### Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
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### Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
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### Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
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### Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
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### Lipschitz function proof

Statement Let $F(t,X)=A(t)X+b(t)$ with $A(t) \in \mathbb R^{n\times n}$ and $b(t) \in \mathbb R^n$. If the coefficients $a_{ij}(t)$ and $b_i(t)$ are continuous functions of the variable $t$ in a ...
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### Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
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how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$=2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ... 0answers 31 views ### Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors? We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors?? 1answer 141 views ### Show that L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0 Let u be a function such that$$ \Delta u + \lambda u = 0 $$for some \lambda \in \mathbb R, also let w be a function such that$$ \Delta w + \beta w < 0. $$for some \beta \in \mathbb R. ... 2answers 28 views ### Question on derivation of vector identites and using some symbolic manipulations Let f,g : \mathbb R^n \to \mathbb R, then for the gradient we have the product rule$$ \nabla(fg) = (\nabla f) \cdot g + f \cdot (\nabla g). $$And by \Delta(f) = \mbox{div}(\nabla(f)) = \nabla ... 2answers 66 views ### Showing a two-variable function is continuous The problem asks to show that$$f(x,y) = \left\{ \begin{align} \frac{x^3y^2}{x^4+y^4}, & (x,y) \neq (0,0), \\ 0, & (x,y) = (0,0), \end{align} \right.$$is continuous at the origin, however it ... 1answer 38 views ### How to prove that F(x,y)=(f(x)h(y),g(y)) is a diffeomorphism? Let F:\mathbb{R}^2\to\mathbb{R}^2 be given by F(x,y)=(f(x)h(y),g(y)), where h:\mathbb{R}\to\mathbb{R} is a diferentiable function and f,g:\mathbb{R}\to\mathbb{R} are diffeomorphisms. ... 1answer 49 views ### How to show that \varphi(x,y)=(x+f(y),f(x)+y) is bijective? Let f:\mathbb{R}\to\mathbb{R} be a C^1 function such that |f'(t)|\leq k<1 for all t\in \mathbb{R}. Let \varphi:\mathbb{R}^2\to\mathbb{R}^2 be the function given by ... 1answer 36 views ### a question about multivariable integral! If \lfloor x \rfloor denotes the greatest integer in x, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}. This ... 1answer 47 views ### Evaluate integral \int\int xe^{xy} dx dy, strange result after rearranging I have to compute the following integral$$ \int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy It exists according to WolframAlpha. Now I want to evaluate it, let \varepsilon > 0, then \begin{align*} ... 1answer 60 views ### Calculate area enclosed by curve Calculate the area of the bounded surface enclosed by the curve (x+y)^4 = x^2y with the help of the coordinate transformation x = r\cos^2 t, y = r\sin^2 t. As I see it the area is unbounded, so ... 1answer 29 views ### Verification of Stokes Theorem I want to verify Stokes Theorem for the surface \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$and the vector field F(x,y,z) := (y,z,x). For this I use the ... 0answers 19 views ### Integral invariant under parametrization Consider a continuous function F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R} and the functional$$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$Prove that ... 1answer 67 views ### Characterization of differentiable functions from \mathbb{R}^m to \mathbb{R}^n. Let U\subset\mathbb{R}^m be an open set. Consider a function f:U\to\mathbb{R}^n and a point a\in U. I need help to prove that the following sentences are equivalents. (a) There exists a ... 3answers 168 views ### Is speed a function of position? Let x be a smooth function from [0,\infty) to \mathbb{R}^n satisfying the following differential equation x''(t) = f(x(t)), where f is a smooth function from \mathbb{R}^n to itself. Then ... 2answers 51 views ### If a is a limit point of f^{-1}(b), then the linear mapping f'(a) is not injective. Let U\subset\mathbb{R}^m be an open set and f:U\to\mathbb{R}^n a differentiable function. Suppose that there exists b\in\mathbb{R}^n and a\in U such that a is an accumulation point of ... 1answer 42 views ### f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R} of class C^2 for which f_x(x,y)=\frac{y}{x^2+y^2} and f_y(x,y)=\frac{-x}{x^2+y^2} Is there exists f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R} of class C^2 for which f_x(x,y)=\frac{y}{x^2+y^2} and f_y(x,y)=\frac{-x}{x^2+y^2} for all ... 0answers 32 views ### A question about the existence of a smooth function [duplicate] Does there exists a smooth function f: R^2 \rightarrow R, such that f(x,y)\ge0, for any (x,y) \in R^2, and f has exactly two critical points (x_1,y_1), (x_2, y_2) \in R^2 with ... 1answer 40 views ### surface integral (curl F n ds) Let F be a vector field and let n be normal vector of the closed surface S. Then show that$$\iint_S \mathrm{curl} \ F \cdot n\ ds=0. $$I need help on this exercise. 3answers 31 views ### a simple multivariable limit \lim_{(x,y)\to(0,0)}\frac{-x}{\sqrt{x^2+y^2}} I get confused finding this limit. I approach with lines y=mx and i get \lim_{x\to 0}\frac{-x}{\sqrt{x^2+m^2x^2}}. How can i ended that this limit ... 2answers 22 views ### Prove that F is constant on S. If F'(x;y)=0 for every x in an open convex set S and for every y in \mathbb{R^n}, prove that F is a constant on S, where S\subset\mathbb{R^n}. Somewhere I need to define a function ... 1answer 37 views ### surface area of the graph of a convex function I started out with the following question: Say \Omega is a nice bounded domain in \mathbb{R}^{n-1}. (One can imagine it being a unit ball in \mathbb{R}^{n-1}.) Let f:\Omega\rightarrow ... 2answers 42 views ### showing that f(x,y) is continuous at (0,0) Let$$f(x,y) = \begin{cases} 0, & \text{if $y \le 0$, $y \ge x^2$ } \\[2ex] 1, & \text{if $0 \lt y \lt x^2$ } \\ \end{cases}$$Show that f(x,y) \to 0 as (x,y) \to (0,0) along any ... 2answers 51 views ### Does the set of Differentiable functions change if we change our norm? This may be a naive question. I am reading the definition of differetiablity of a function f:\mathbb{R^n}\rightarrow \mathbb{R^m} in the book Calculus Manifolds. I already know that all norms on ... 2answers 68 views ### a question about how to prove mutivariable integral, I am struggling about it! If f(x) is Riemann integrable in [a,b], and then how to prove$$\int_{a}^{b} f(x_1) \, dx_1 \int_{a}^{x_1}f(x_2) \, dx_2 \cdots \int_{a}^{x_{n-1}}f(x_n) \, dx_n={1\over n!} \left[\int_a^b f(x) \, ...
Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...