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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0answers
20 views

Definition of higher order Fréchet derivative

My lecturer has recommended to us that we check that the obvious two candidates for the $k^{th}$ order Fréchet derivative are the same. That is defining the $k^{th}$ order Fréchet derivative ...
4
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2answers
76 views

How to solve simultaneous inequalities?

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1 ≤ 2x+y ≤ 2$$ $$0 ≤ ...
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0answers
17 views

The average value of f on region R (double integrals)

If I am given f(x,y) and I calculate the average of the region given by the formula in this image should the average always = (min of f(x,y) +max of f(x,y))/2. I think this is only the case if the ...
3
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3answers
550 views

How do I solve a double integral that has an absolute value?

First off, I apologize for any English mistakes. I've come across a double integral problem that I haven't been able to solve: Find $\int_0^\pi \int_0^\pi \left |\cos(x+y) \right| \,dx\,dy$ ...
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1answer
28 views

Problem with integration limits using spherical substitution

Good night, i have a problem with this integral, please help me with the integration limits. \begin{align} ...
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1answer
17 views

Problem with integration limits with cylindric cordinates.

Good night, i have a problem when i go to verify the integration limit $0\leq\theta\leq\varPi/2$ because i think the integration limit go to $0\leq\theta\leq\varPi$ because is an half a circle. ...
2
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1answer
60 views

Product of two uniform random variables/ expectation of the products

Suppose I want the expectation, $E\Phi(X-\mu)\Phi(\mu-X)$, where $\Phi(.)$ represents the Normal CDF, and X is $Normal(\beta,1)$. Consequently $\Phi(.)$'s are uniform[0,1] and at the same time two ...
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1answer
37 views

Problem solving an multiple integral with integration limits.

Good night, i have a serious problem solving this integral. $\int_{0}^{2}\int_{0}^{\sqrt{2x-x^{2}}}\int_{0}^{a}z\sqrt{x^{2}+y^{2}}dzdydx$ I make a change of cylindrical coordinates, and when i make ...
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1answer
25 views

Bounding a gradient of a function

Defining an infinitely differentiable fucntion $\phi$ as $$\phi(x) = \left(\frac d2 \right)^{-n} \int_{\mathbb R^n} \psi\left( \frac{y-x}{d/2} \right) \, dy,$$ I need to show that $$|\text{grad} \, ...
1
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0answers
35 views

Finding the Maximum and Minimum values w/constraint [duplicate]

I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is. Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ ...
1
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1answer
31 views

How to find the derivative of matrix conjugation for unitary matrices at a point where the matrices commute?

Let $\text{SU}(2)$ denote the special group of $2 \times 2$ unitary matrices, that is, unitary matrices with determinant $1$. Define $f : \text{SU}(2) \times \text{SU}(2) \to \text{SU}(2) \times ...
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1answer
24 views

Finding Volume of Revolution with Multivariate Calculus

For some function $f\left(x\right)$ it is possible to rotate it along the x-axis and find the area using $$\intop_{a}^{b}\pi\left(f\left(x\right)\right)^{2}dx$$ I'm curious how to do this with ...
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1answer
20 views

How to determine $\varphi$ in spherical coordinates

Assume that I would like to integrate some continuous a.e. function $f(x,y,z)$ over the following set: $ a^2_1 \le x^2 + y^2 +z^2 \le a^2_2$, and $z\ge c^2(x^2+y^2)^{1/2}$. So, in a case I would like ...
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1answer
54 views

Issue with substituting a new variable

I have a function of the form $$u(x,t)=\int_{0}^{t}{\frac{u_0\,x\exp[-h\tau-(x^2/4k\tau)]}{2\sqrt{\pi k\tau^3}}}d\tau$$ Now substituting $\eta=\frac{x}{2\sqrt{k\tau}}$ in the above equation, I get ...
-1
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1answer
49 views

Double integral converges [closed]

Find the range value of number k for which the following integral converges $$\iint_D \frac{1}{\left(x^2+y^2\right)^k}\ \mathrm dx\ \mathrm dy$$ Where $$D = \left\{(x,y) \in \mathbb{R}^2 | ...
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1answer
32 views

Calculating points where a function is not invertible

Let $f: \mathbb{R^3} \rightarrow \mathbb{R^3}$ be the function $f(x,y,z)=(x+y+z, x^2+y^2+z^2, x^3+y^3+z^3).$ For which $a \in \mathbb{R^3}$ is $f$ not invertible?
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1answer
18 views

Proper way to find the critical points of a 2 variable function

I want to find the critical points of $g(x,y) = x^3 +y^3+3xy$ Do I need to find the points in which $\dfrac{\delta f}{\delta x} = 0 $ AND $\dfrac{\delta f}{\delta y} = 0$ or do I need to find the ...
-2
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1answer
43 views

How do I minimize this function $f(x,y,z)$ using lagrange multipliers

$$f(x,y,z) = ln(x^2+1)+ln(y^2+1)+ln(z^2+1)$$ subject to $x^2+y^2+z^2=12$ There is both a max and min. I found the max to be ...
0
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1answer
54 views

Directional Derivative Existence question

Let $f:\mathbb{R}^2\to\mathbb{R}$ and $(p,q)\in\mathbb{R}^2$ such that both $f_x$ and $f_y$ exist at $(p,q)$. Suppose $f_x$ is continuous at $(p,q)$, show that the directional derivative $D_vf$ at ...
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2answers
17 views

“Change of variable” in Limit

Let $f:\mathbb{R}^2\to\mathbb{R}$. Let $(p,q), (v_1,v_2)\in\mathbb{R}^2$. Are the following limits the same? 1) $$\lim_{h\to 0}\frac{f(p+hv_1,q+hv_2)-f(p,q+hv_2)}{h}$$ 2) $$\lim_{\epsilon\to ...
3
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3answers
87 views

Differentiation under the integral sign for $\int_{0}^{1}\frac{\arctan x}{x\sqrt{1-x^2}}\,dx$

Hello I have a problem that is: $$\int_0^1\frac{\arctan(x)}{x\sqrt{1-x^2}}dx$$ I try use the following integral $$ \int_0^1\frac{dy}{1+x^2y^2}= \frac{\arctan(x)}{x}$$ My question: if I can do ...
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1answer
17 views

Taking second partial derivatives of a function composition

Let $H(u,v)$, $\, u:=u(x,y)$, and $\, v:=v(x,y)$ be real valued functions. I am having trouble taking the second partial derivatives $H_{xx}$ and $H_{yy}$ of this function composition using the chain ...
2
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2answers
31 views

Constructing a dense subset $ C \subseteq [0,1] \times [0,1] $ with a special property.

Construct a subset $ C \subseteq [0,1] \times [0,1] $, dense in $[0,1] \times [0,1] $ that has the property of every horizontal or vertical line has at most one point of $C$. I think if I guarantee ...
-1
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1answer
19 views

Function of product of two uniform random variables [closed]

If X and Y are uniform(0,1) then what is the distribution of $X^kY^m$ for some integers k and m?
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1answer
42 views

Is there a matrix that converts the gradient of any function to gradient of other function?

The study of hamiltonian mechanics brought me to the following question. Let $n$ be a natural number ($n>1$). Let $A(\mathbf{x})$ be a $n\times n$ matrix consisting of functions ...
0
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2answers
40 views

Derivative as a continuous mapping of a subset of $R^n$ into the set of all invertible elements

I have been studying the proof of the inverse function theorem in the Rudin's book (Principles of mathematical analysis). Near the end of the proof, he makes the following statement "(...) observe ...
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1answer
18 views

To calculate the flux of water through a parabolic cylinder

If velocity vector is given as $\mathbf F=y\mathbf i +2 \mathbf j+\mathbf k$ , then find the flux of water through the parabolic cylinder $y=x^2$, $0\le x\le 3$, $0\le z \le 3$. For this ...
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4answers
92 views
+50

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
0
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1answer
31 views

Evaluate $\oint \mathbf{F}\mathbf \cdot\mathbf{n} ds \quad\text{ where }\quad \mathbf F=y \mathbf i+x\mathbf j$

$$\oint \mathbf{F}\mathbf \cdot\mathbf{n} ds \quad\text{ where }\quad \mathbf F=y \mathbf i+x\mathbf j$$ Can I say that $$\oint \mathbf{F}\mathbf \cdot\mathbf{n} ds=\oint xdx+ydy$$ ? I don't ...
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3answers
37 views

How to prove that $\frac{1}{x^4+y^4} e^{-\frac{1}{x^2+y^2}}\to0$ when $(x,y)\to (0,0) $?

How can I calculate the limit $$ \lim_{(x,y)\to (0,0)} \frac{1}{x^4+y^4} e^{-\frac{1}{x^2+y^2}} $$ and show that it is zero? When switching to polar coordinates, I get: $$ \lim_{(x,y)\to ...
0
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4answers
25 views

Evaluate $\oint x\,dx$ over a particular curve $C$

$$\oint x\,dx\qquad C:\{x=0,y=0,y=-x+1 \} $$ My attempt: $$\oint x\,dx=\int_{\uparrow}x\,dx+\int_{\nwarrow}x\,dx+\int_{\rightarrow}x\,dx$$ I don't know what should I do now, all the integral ...
0
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1answer
30 views

Find the area bounded by a curve by changing variables

Calculate the area bounded by the following formula: $$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2} \right)^2 = \frac{xy}{c^2}$$ where $a,b,c>0.$ I have used changing variable of $x=au$ and $y=vb$ to ...
0
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4answers
72 views

Where is $|xy|$ function differentiable

I'm trying to solve this problem: Let $f(x,y) = |xy|$. Find the sets of all points $(x,y) \in \Bbb R$ where $f$ is differentiable and compute the differential in those points. Can someone explain ...
2
votes
4answers
105 views

How does one parameterize $x^2 + xy + y^2 = \frac{1}{2}$?

Parameterize the curve $C$ that intersects the surface $x^2+y^2+z^2=1$ and the plane $x+y+z=0$. I have this replacing equations: $$ x^2+y^2+(-x-y)^2=1$$ and clearing have the following: $$ ...
5
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3answers
75 views

Can the directional derivative fail to be linear?

Is it possible for the directional derivative for a function $f$ in the direction of a vector $v$, $D_vf(x) = \lim_{h \to 0} \frac{f(x + hv) - f(x)}h$ to exist for every vector $v$, and yet $v \mapsto ...
0
votes
1answer
36 views

How to use Implicit Function Theorem for this function?

Let $x = (x_{1}, x_{2}, x_{3})^{T} \in \mathbb{R}^{3}$, and take a sufficiently differentiable function $f: \mathbb{R}^{3} \mapsto \mathbb{R}$. Suppose that I am searching for zeroes of $f$ such ...
2
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1answer
40 views

Can we find a regular ($C^k$) parametrization for this surface?

I have here a surface whose curvature properties I want to study, represented in cylindrical coordinates: $$f(r,\theta) = r^2\cos4\theta$$ The problem, however, is that the parametrization is not ...
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3answers
28 views

Show that $1/|x|$ is not Lipschitz continuous on $|x|<1$.

$x$ is a $3$d vector. This is what I have so far, don't know if it is enough to prove , that $f(x) = 1/|x|$ is not Lipschitz-continuous on $|x|<1$: First we have to show, that for all $L>0$ ...
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1answer
38 views

Triple integral vs double integral to find volume of an object

Is it possible to find the volume of an object bounded by two surfaces in both of these two ways?: -a triple integral of 1 dV (I know this works) -a double integral of the top surface - bottom ...
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0answers
15 views

Gaussian region in $\mathbb R^n$

My question pertains to the paper "A Simplified Proof of the Divergence Theorem" by Djairo Guedes de Figueiredo. The paper says: A Gaussian region is an open connected bounded set $V$ in $\mathbb ...
6
votes
3answers
531 views

Why is this set not a manifold?

Set $M = \{ \, (x, y) : x^2 = y^2 \, \}$. If for every point $(a, c)$ in $M$, there exists a neighborhood $U$ containing $(a, c)$ and function $\phi(x, y)$ such that: $\phi(x, y) = 0$ on $M \cap ...
0
votes
1answer
15 views

Estimating the curvature of a discretized curve in 3d with cubic splines

I have a computer simulation in which I'm modeling a physical curve by discretizing it and updating the locations of these points. I want to find/estimate the location of the maximum curvature of the ...
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2answers
23 views

Show that $f(x) = x \cdot |x|^2$ with $|x|<1$ is Lipshitz continuous.

I am reading Body & Soul, Part $3$ and got stuck with this exercise: Show that $f(x) = x \lvert x\rvert^2$ with $\lvert x\rvert<1$ is Lipschitz continuous, where $x$ is a $3$d vector. I ...
2
votes
1answer
20 views

Total Derivative at a point

Let $f(x,y)=x^{3}+y^{3}$ . How do I find the total derivative at (0,0) by using the definition? I am confused of what to take as the Error function.
4
votes
1answer
33 views

Integrating both sides of an equation with respect to different variables [duplicate]

So im reading a book called "Ordinary Differential Equations" (Tenenbaum & Pollard) and in the introduction(ish) they are doing an example using a carbon dating problem, represented as: ...
0
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0answers
39 views

The most general way to prove differentiability over an interval

Preface: I'm going to try and make this question as general as I possibly can, as there are many different extensions of Calculus (Single-Variable, Multi-Variable, Vector, Tensor etc.), in which ...
0
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2answers
40 views

Prove that $\lim \limits_{(x,y)\to (c,0)} \frac{\sin(x^2y)}{x^2-y^2}=0$, $c\ne0$ with (ε, δ)-definition

So the problem is to work with $\left|\frac{\sin(x^2y)}{x^2-y^2}\right|$ and show that that is less than a formula involving δ, let´s call it g(δ), formula which I will later equal to ε in order to ...
1
vote
1answer
27 views

Solving a multivariate polynomial system involving the power sums

I would like to know if there is a way to solve or simplify the system of equations given by: $$ x_1^1+x_2^1+\cdots x_n^1 = c_1\\ x_1^2+x_2^2+\cdots x_n^2 = c_2\\ \vdots\\ x_1^n+x_2^n+\cdots x_n^n = ...
1
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1answer
28 views

Can a two-variables function be an odd function in one variable?

Like in single variable, we use $f(-x)=-f(x)$ to show that a function is odd. Similarly, for two variables, we can use $f(-x,-y)=-f(x,y)$. If we have a two variable function like this ...
0
votes
0answers
28 views

How to calculate partial derivative of many values?

I have a function for example : $$f(x,y) = x^5 + 3xy + \cos(xy)$$ It's easy to calculate the partial derivative of $x$ or $y$. But how to calculate the partial derivative of $x$ AND $y$, $[ f'x,y ]$