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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2answers
35 views

Limit of multivariable function

I need to find the limit: $ \lim \limits_{P \to P_.0} \frac{x^2 y^2}{x^2 + y^2}$ where $P_0 = {0,0}$ I know that i can find the limit using polar coordinates, BUT how can i find it with $\epsilon - ...
1
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2answers
968 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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2answers
48 views

$f : \mathbb R^n \to \mathbb R$, what is the gradient of $f(tx)$?

Fairly simple question, suppose there is a function $f: \mathbb R^n \to \mathbb R$, and a scalar $t \in \mathbb R$. is it possible to find $D_f(tx)$ using only $t$ and $D_f(x)$? Perhaps using chain ...
0
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1answer
27 views

How does the transformation $u=x+y$, $v=x/y$ transform the first quadrant?

How is the region $(x,y) \in [0,\infty] \times [0,\infty]$ transformed under the change of coordinates given by $$u=x+y$$ $$v=x/y$$ Would appreciate any hints on how to find the image of such ...
1
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4answers
72 views

Center of mass of $x^2+y^2 \leq z \leq h$

I'm trying to find the center of mass of this shape $x^2+y^2 \leq z \leq h$but im having difficulties founding the limits of integration. using cylindrical coordinates, $x=rcos \theta$, $y=r \sin ...
1
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2answers
81 views

solving $ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} $

in solving $ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} $, my textbook says to replace $x^2+y^2$ with $r^2$, thus making the equation $ \lim_{r \rightarrow 0} \frac{\sin(r^2)}{r^2} ...
2
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0answers
34 views

Poisson equation on a square

Studying PDEs from the notes of my professor, and there's a part I don't understand about seeking a solution for the Poisson equation on a square. Let's start from the beginning though. We want to ...
3
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2answers
66 views

Where is my mistake $\iint_{Q} (x+y)^{2013}dxdy$

I'm preparing for a calculus exam, and I tried to solve the following question. $Q$ is square $[-1,1]^2 \subset \mathbb R^2$ We are asked to evaluate $\iint_Q (x+y)^{2013}dxdy$ Here is what I did: ...
2
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1answer
45 views

Calculate $\int_D x^3y\ dx\,dy$

Let $D$ the bounded region by the $y$-axis and the parable $x= -4y^2 + 3$. How can I calculate the integral $$\int_D x^3y\ dx\,dy$$ I am stuck with this problem some help to solve this please.
4
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0answers
71 views

Is this correct and sufficient to show limit does not exist?

Find limit or show that it does not exist: $$\lim_{(x,y) \to (0,0)} \frac{ 2x^{2}y^{3/2} }{y^{2}+x^{8}}$$ using the path $x=m y^{1/4}$: $$\lim_{(my^{1/4},y) \to (0,0)} \frac{ ...
1
vote
1answer
35 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
1
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2answers
162 views

Proving inexistence of limit

Prove that limit does not exist $$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}$$ Obviously, since it is symmetric in $x$ and $y$, classic approach of substituing $x$ as a "simple" function - linear, ...
0
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4answers
47 views

Total differential of a function

Let's consider a function $$f(x,y)=\begin{cases}\dfrac{xy^3}{\sqrt{x^2+y^2}},& (x,y)\neq(0,0)\\ 0,& (x,y)=(0,0)\end{cases}$$ does it have a total differential in point $(0,0)$? I say that it ...
0
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1answer
82 views

The chain rule: differentiation in several variables

Hi! I am trying to study for an upcoming exam by doing online problems, but this one has me completely stumped. If someone can help answer or explain how to do this problem I would really appreciate ...
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0answers
54 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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1answer
45 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
1
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2answers
60 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
0
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0answers
33 views

Prove or disprove the statement related to the definition of multivariable-differentiable function

The question: Let $f,f_1,...,f_n \; (n > 0)$ be functions from $\mathrm{D} \subset\mathbb{R}^n$ to $\mathbb{R}$ satisfying $$\left ( \sqrt{\sum_{i=1}^n x_i^2} \right ) f(\mathrm{x}) = \sum_{i=1}^n ...
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1answer
114 views

Quasi-concavity of a function of two variables such as $z=(x^a + y^b)^2$

If I have a function such as $z=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
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1answer
33 views

Proof of Equality with Mixed Partials

Here is the link: Hessian I understand everything but this line: $$g(x_0 + \Delta x) − g(x_0) = \frac{dg}{dx} (ξ) \Delta x$$ i.e., $$S (X_0, \Delta x, \Delta y) = \frac{∂φ}{∂x} (ξ, y_0 + \Delta y) − ...
2
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1answer
105 views

A set is compact if and only if every continouos function is bounded on the set?? [duplicate]

I was asked to prove the following statement: let $K \subseteq R^n$. show that $K$ is compact (meaning closed and bounded) if and only if every continouos function is bounded on $K$. What I did: ...
2
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1answer
53 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
0
votes
1answer
30 views

Direction for greatest derivative

Suppose I have a function like $f(x,y) = e^x e^y x^2 y^2$, and I want to know in which direction the derivative will grow fastest at a stationary point. $(0,0)$ is a stationary point of the example ...
0
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1answer
68 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.
2
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1answer
58 views

Spherical-Coordinate Reference Frame

my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q ...
1
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0answers
27 views

What is the 1st derivative i.r.t. coordinates for a vector function?

For a vector function $f(x,y,z)$, we have the divergence $$\nabla \cdot f(x,y,z) = \frac{\partial{f}_{x}}{\partial x}+\frac{\partial{f}_{y}}{\partial y}+\frac{\partial{f}_{z}}{\partial z}$$ , the ...
0
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1answer
61 views

Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Prove without using the divergence theorem. The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem. Thanks in advance.
1
vote
1answer
55 views

Find the partial derivatives of second order of $f(x,y)=\varphi(xy,\frac{x}{y})$

Ok guys, I'm given this smooth function $\varphi(u,v)$ defined in $R^2$. So that $f(x,y)=\varphi(xy,\frac{x}{y})$. I have to find all partial derivatives of second order of $f$ using the partial ...
3
votes
2answers
84 views

How to find the following limit? $\lim\limits_{t \to {\pi}/{2}}\frac{ \int_{\sin t}^{1}e^{x^2\sin t}dx}{\int_{\cos t}^{0}e^{x^2 \cos t}dx}.$

I am trying to solve the limit $$\lim\limits_{t \to \pi/2}\frac{ \int_{\sin t}^{1}e^{x^2\sin t}dx}{\int_{\cos t}^{0}e^{x^2 \cos t}dx}$$ My first method was to try with L'Hopital, i derived using ...
0
votes
1answer
30 views

Tranforming to polar co-ordinates

$$I = \int_0^1\int_0^{\sqrt{1-x^2}} xy \, dy\, dx$$ By transforming to circular polar co-ordinates, evaluate I. How do I do this? Is there a formula/strategy for doing this that works with ...
1
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2answers
48 views

“Orthonormal” parameterization of solid sphere?

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is ...
3
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2answers
90 views

very strange phenomenon $f(x,y)=x^4-6x^2y^2+y^4$ integral goes wild

I am going over my lecture's notes in preparation for exam and I saw something a bit strange I would like someone to explain how it is possible. Look at the function $f(x,y) = x^4-6x^2y^2+y^4$ if we ...
4
votes
1answer
147 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
1
vote
1answer
52 views

Unable to solve system of equations in Lagrange multiplier problem.

The problem: Find the right triangular prism of given volume and least area if the base is required to be a right triangle. As for parameters of the right triangular prism, $V$ is volume, $A$ is ...
1
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0answers
41 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
4
votes
4answers
101 views

$\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or converges?

Question I'm trying to solve to prepare for an exam. I need to find out if $\displaystyle\iint_{\mathbb R^2} \frac{dx\,dy}{1+x^{10}y^{10}}$ diverges or converges. What I did: I switched to polar ...
3
votes
1answer
72 views

Prove that a function is differentiable if…

I'm trying to prove that given a differentiable function $f: \mathbb{R}^2 \to \mathbb{R}^m$ in $p =(p_1, p_2) \in \mathbb{R}^2$, the function $$ g(x, y) = f(x, y) - \frac{\partial f}{\partial x}(p)(x ...
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votes
2answers
62 views

Find partial derivatives of $f(x, y)=\sqrt[3]{xy}$

Let $f(x, y)=\sqrt[3]{xy}$. Find $f_x(0,0)$ and $f_y(0,0)$. Is $f$ differential at (0,0)? How can I do?
2
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1answer
41 views

What is a Hyper-Sphere?

I am interesting about the geometric properties of 3-D spheres and I know nothing about hyper-spheres. Please can you describe me, what is a hyper-sphere?
1
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2answers
75 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
1
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3answers
36 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 ...
2
votes
1answer
85 views

A confusing vector field differential

In my notes on theoretical mechanics, I wrote that my professor stated this vector identity: $$\mathrm{d}\mathbf{P}(\mathbf{r})=[\nabla\cdot\mathbf{P}(\mathbf{r})] \mathbf{dr} + ...
3
votes
1answer
84 views

Check my proof $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sin(x^2+y^2) \, dx \, dy$ diverges

I am trying to prove that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sin(x^2+y^2)\, dx\,dy$ diverges and I did it like this: $x=r\cos \theta$, $y=r\sin\theta$, $\theta \in [0,2\pi]$, $r\in ...
1
vote
1answer
38 views

Applying Green's theorem for a line integral of a vector field

Integrate the vector field $F(x,y)=(e^y+\frac{1}{y+3},xe^y-\frac{x+1}{(y+3)^2})$ over a curve that goes from $(-1,0)$ to $(-1,2)$ to $(0,1)$ to $(1,2)$ (in a linear fashion). Now, I'm almost certain ...
0
votes
0answers
16 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
3
votes
3answers
102 views

calculate $\int_{0}^{\pi} \int_{0}^{x}\log(\sin(x-y))dydx$

I was asked to find the integral $\iint_A \log(\sin(x-y))dxdy$ where $A$ is the triangle $y=0, x=\pi, y=x$ in the first quadrant. I was given a hint: evaluate $\int_{0}^{\pi}\log(\sin(t))dt$ using ...
1
vote
0answers
23 views

Calculate integral over $S = \{(x,y,z) \in \mathbb{R}^3 | 2z=x^2+y^2<\sqrt{z} \}$

Calculate $$\int\limits_{S} \sqrt{\frac{z}{1+2z}} \, d \sigma_S,$$ where $S = \{(x,y,z) \in \mathbb{R}^3 \; | \; 2z=x^2+y^2<\sqrt{z} \}$.
0
votes
2answers
83 views

Boundaries for area with double integration.

I have to find the area using double integral for the domain bounded by $$y^2=x$$ and $$y=x-2$$ Now, I want to find my integral boundaries: I did $y=x^2, y=x+2$, solved this system and get ...
1
vote
0answers
18 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
0
votes
1answer
50 views

Implicit function derivation

I have function $h(x,y)=e^{xy^2-1}+\log{\frac{x}{y}}-1$ and I have to find if a function $y=f(x)$ around $[1,1]$ exists. I have to check some conditions in order to find out if $y=f(x)$, ...