Tagged Questions

Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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Fréchet differentiability of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$?

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ I want to determine ...
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higher derivatives of $R^m \to R^n$ [duplicate]

What's a good source (paper, book, website,...) where I can learn more about higher derivatives of functions $R^m \to R^n$ as multilinear functions or tensors? Thanks.
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When is the line integral independent of parameterization?

Let $\alpha: [a,b] \rightarrow \mathbb{R}^2$ be a smooth path (i.e. $\alpha'$ is continuous on $[a,b]$), and let $f$ be a continuous vector field. The line integral of $f$ along $\alpha$ is defined as ...
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Is this map an immersion?

Let $g:\mathbb{R}^2\to \mathbb{R}^4,\ (x,y)\mapsto ((2+3\cos(2\pi x))\cos(4\pi y),\ (2+3\cos(2\pi x))\sin (4\pi x),\ 3\sin(2\pi x)\cos(2\pi y),\ 3\sin(2\pi x)\sin(2\pi y))$ I have to prove that for ...
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Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
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Computing $\underset{x^2+y^2+(z-2)^2\le 1}{\int\int\int}{1\over x^2+y^2+z^2}dxdydz$ in Spherical Coordinates

Compute: $\underset{x^2+y^2+(z-2)^2\le 1}{\int\int\int}{1\over x^2+y^2+z^2}dxdydz$. Hint given: show that $\cos \theta> {r^2+3\over 4r}$ $1<r<3$ What I already did: I shift the unit ...
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Calculating Gradient of a $2$ Variable Function

Let $f(x,y)$ have continuous partial derivatives at every point. We know that $$\nabla f(0,3)=5 {\bf{i}} - {\bf{j}}$$ Then we define $g(x,y)=f(x^2-y^2, 3x^2y)$. I am not sure what I should do to ...
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Supremum and infimum of function of two variables

Consider $D = \left \{ x \in \mathbb{R} : x_1^2 + 44x_2^2 \leqslant 5 \right \}$ and function $f: D \rightarrow \mathbb{R}$, $f(x) = 13x_1 - 22x_2$. Find supremum and infimum of $f$. For both of them ...
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Inverse image of a function in multivariable calculus?

Let $f: R^2 \rightarrow R^2$ defined by $f(x,y) = (x+y,xy).$ Claim : Inverse image of each point in $R^2$ under f has at most two elements. My Claim : Suppose $f(x,y) = (x+y,xy)= (p,q).$ We have ...
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Find $\iint (\nabla \times F)\cdot dS$ if S the surface of the sphere $x^2+y^2+z^2=a^2$

Find $\iint (\nabla \times F)\cdot dS$ if $F= y i+(x-2xz)j- (xy) k$ and S the surface of the sphere $x^2+y^2+z^2=a^2$ above of the $xy-$plane I do not know if I must use the stokes theorem or try ...
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Find the area between the cylinder $z^2+y^2=r^2$ and two planes

I'm having trouble with this problem: Find the surface area between the top of $z^2+y^2=r^2$ between $z=ax$ and $z=bx$ (consider $a \gt b \gt 0$). I think I must find the area between the ...
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I am trying to evaluate the following integral $$I = \int_1^t u^{2 \kappa -1} \exp\left(-\frac{1}{2} b \omega^2 u^{2 \kappa} + a \omega u \right) du.$$ Upon integrating by parts we have $$I = u ^{... 1answer 36 views Local minimums and maximums of function of three variables I got such function: f(x_1, x_2, x_3) = x_1 x_2 x_3(4-x_1-x_2-x_3) I need to find all local minimums and maximums of this function. I calculated partial derivatives and I got that the only points ... 1answer 33 views To solve large systems of multivariate polynomial equations Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (... 0answers 29 views Explaination of the proof of directional derivative formula Hi guys :) i read that proof of the formula of the directional derivative and i didn't understand the sceond step and what is h(o) and where does it come from? Where does the dot product between the ... 0answers 26 views C alculating flux using the divergence theorem when the divergence is 0 I calculated the divergence of my vector field \langle x^2 + y^2, y^2 + z^2, 1 − 2xz − 2yz\rangle to be 0. The flux is meant to be over the unit hemisphere. If I do use the divergence theorem, ... 0answers 29 views On inequalities related with f(s):=-(1-\frac{2}{2^s})^{-1} My Question. a) How can you prove easily that the multivariable function in LHS is positive on x^2+y^2<1$$2^{1-x}\cos(y\log 2)-1>0?$$b) Let s=\sigma+it the complex variable, ... 0answers 22 views integrating a function of a dirichelet random vector on a subset Consider a random vector \bar{X} of n many variables X_i \in (0,1) such that \sum_{i=1}^{n}X_i = 1 and let the sequence of x_i be a random realization of \bar{X}. The distribution of \bar{X}... 0answers 36 views Trying to show that this is a 2-dimensional surface I have been working on some problems within multivariate analysis, and I am trying to go about proving that given the set$$B = \{(u,v) \in [0,1] \times [0,1] | u < v\}$$and the set$$Q = \{(1+uv,...
The equation of an ellipsoid is $$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$ The ellipsoid is arbitrary rotated and the orientation angle are given as θ, β and Ѱ and the center is at (x',y',z')....
Let A an open set of ℝⁿ and ƒ:A→ℝᵐ of class C'(A). If for some P₀∈A, ƒ'(P₀) is surjective, prove that exists δ>0 such that B$_{δ}$(P₀)⊂A and for all open subset Ω⊂B$_{δ}$(P₀) it holds that ƒ(Ω) is ...