# Tagged Questions

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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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### 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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### Expression for volume without changing variables

My question is the same as this: Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables However, my solution, while makes perfect sense to me, is slightly ...
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### Surface integral over d-sphere for $|x-y|^{(2-d)}$

I am looking for $\int_{S_{r}(0)}|x-y|^{2-d}dS_{y}$ for $x\neq 0$. The parametrization is hard to work with and the integrand is not rotationally symmetric. I will post any updates. any ideas thank ...
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### Finding extremal values on a set

Let $f(x,y)=(x-1)^2+y^2+xy$. Find the maximal and minimal values of $f$ on the set $M=\{(x,y):|x|+|y|\leq4\}$. Attempt: By taking partial derivatives and solving the homogenous algebraic system we ...
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### Swapping limit with inner product.

The exercise is: Given $\xi : U \subset \Bbb R^m \rightarrow \Bbb R$, $U$ open, given by $\xi (x) = \langle f(x), g(x) \rangle$, where $f,g: U \rightarrow \Bbb R^p$ are differentiable functions, ...
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### Does boundedness imply continuity?

the partial derivatives of a given function f is bounded. does this implies that the given function is continous ? how can we prove that a function is continous when we only knw that its partial ...
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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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### Let U be open and $f: U \rightarrow \mathbb{R}$ be partial differentiable.

The Assignment: Let $U \subset \mathbb{R}^n$ be open and $f : U \rightarrow \mathbb{R}$ be partial differentiable and let all partial directional derivatives be continous function on $U$. Show ...
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### Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
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 52 views ### Using Stokes's theorem to calculate a value of integral Use Stokes's theorem to calculate the integral $$I= \int_\Gamma (x^2+2y)dx+(y+z)dy+(z^2+x^2)dz$$ where$\Gamma$is the boundary of $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\}$$ ... 1answer 21 views ### Problems regarding multivariable calculus Let$f:\Omega\to \mathbb R$be differentiable at$x_0\in \Omega$($\Omega$is a nonempty open subset of$\mathbb R^n$), let$f(x_0)=0$and let$g:\Omega\to \mathbb R$be continuous at$x_0$. We want ... 1answer 40 views ### Multiple Integration: where's the mistake in my process? Evaluate: $$\iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where D is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ... 2answers 97 views ### Computing the inverse Fourier transform of \frac{1}{1+|\xi|^2} for \xi \in \mathbb{R}^n. I'm trying to compute the integral$$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$I know that for an integral like$$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ... 0answers 27 views ### Evaluating a triple integral explained step by step Evaluate: $$\iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where$D$is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ... 1answer 46 views ### Smoothness of f(x)/(1+|f(x)|) where f \in C^1(E) for E an open subset of \mathbb{R}^n (a) Show that if E is an open subset of \mathbb{R} and f \in C^1(E) then the function$$F(x) = \frac{f(x)}{1+|f(x)|}$$satisfies$F \in C^1(E)$. (b) Extend the results of part (a) to$f \in ...
So in this question I am assuming that $f$ is of bound variation on $\mathbb{R}^{n}$ so $\nabla f$ is a vector valued measure and $|\nabla f|$ is its total variation but you can assume that $f$ is ...