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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2
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1answer
66 views

Double Integral with abstract functions of 2 variables

I am required to prove something, and so far I have come to set up an integral $$\int_0^l{\int_0^T{u(x,t)\, \frac{d}{dt}u(x,t) dt }dx}.$$ I was just wondering how to think about these ...
0
votes
1answer
47 views

Can all curves be parametrised?

In my current multivariable calculus course, the definition of a line integral over a vector field has been stated as: $$ \int_C {\bf F} \cdot d{\bf r} = \int_0^1 {\bf F}({\bf r}(t)) \cdot {\bf ...
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0answers
46 views

Path test versus iterated limits for proving existence of a limit

Which is the strongest method for proving existence of limits (or the non-existence of a limit), the path test or using iterated limits? Indeed, there are cases where the iterated limit shows that ...
1
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2answers
110 views

Real analysis homework problem need help

The open cube in $\mathbb{R}^{n}$ with center $a$ and radius $r$ is the set $C_{r}(a):=\big\{x\in \mathbb{R}^{n} : |x_{i}-a_{i}|<r_{i},\text{ for }i=1,2,...,n \big\}$. Prove that every open subset ...
1
vote
0answers
93 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
1
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2answers
50 views

Evaluating $\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA$

I'm trying to evaluate the double integral $$ \frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA. $$ Any ideas?
1
vote
1answer
927 views

calculate Jacobian matrix without closed form or analytical form

The question is probably clear in the title. In many of my applications mostly computer vision, I might not have the closed-form or analytical form of f (a multivariable function). It's calculated ...
0
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1answer
37 views

spherical coordinates to find the triple integral

Use spherical coordinates to evaluate the triple integral where E is the region bounded by the spheres and
1
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1answer
44 views

Volume within the sphere

Find the volume of the solid that lies within the sphere , above the xy plane, and outside the cone My problem is finding the integral function and the limits
3
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1answer
57 views

please help me to do Triple integral

Integrate over the region in the first octant above the parabolic cylinder and below the paraboloid I could not get the limits right even that I tried many one but I still could not get it
1
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1answer
74 views

Triple Integral

Use cylindrical coordinates to evaluate the triple integral $$\iiint_{\mathrm{E}}\sqrt{x^{2}+y^{2}}\, dV,$$ where $\mathrm{E}$ is the solid bounded by the circular paraboloid $z=1-9(x^{2}+y^{2})$ and ...
2
votes
2answers
326 views

Is my understanding of space correct?

I'm learning more about dimensions in multivariable calc, and have been able to make connections by studying level curves and level surfaces. I've learned that a function of 2 variables is really a 2 ...
2
votes
1answer
55 views

Two Multivariable Limits help!

Will someone please help me understand the following? $\lim \limits_{(x,y) \to (0,0)} \dfrac{\tan y \cdot \sin^2(x-7y)}{x^2+y^2} $ which is the expression I get after doing the substitution $x-7=x , ...
0
votes
1answer
35 views

Calculus-based proof that $ x_1^{p_1}\cdots x_n^{p_n}\le p_1x_1+\dots+p_nx_n$ when $\sum p_i=1$

Let $$g(x_1...x_n)=x_1^{p_1}\cdot...x_n^{p_n}$$ $$u(x_1...x_n)=p_1x_1+...p_nx_n$$ Where $\sum p_i = 1$. I have to show that $f(x)=g(x)-u(x)$ is always negative or $0$ over $\Bbb R_+^n$. I've ...
2
votes
3answers
69 views

Differentiation of functions w.r.t. a composed argument

I need help with the following derivative involving inner products: $$\frac{d\, \log(x)^T\,y}{d\,x^T\,y}$$ Here $x$ and $y$ are $n$-dimensional vectors, $T$ indicates transpose, and the logarithm of ...
1
vote
3answers
265 views

Find three positive numbers $x$, $y$, $z$ whose sum is $10$ such that $x^2y^2z$ is a maximum

I'm self learning from the Vector Calculus book available online in PDF form (page 88). The question is: Find three positive numbers $x$, $y$, $z$ whose sum is $10$ such that $x^2y^2z$ is a maximum. ...
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4answers
51 views

Direction of gradient vector

How do we know that gradient always points in direction of greatest increase of function and not the greatest decrease?
3
votes
4answers
122 views

Optimization with a constrained function

Okay so I understand how to find points of extrema when for example, We have $3x^2 + 2y^2 + 6z^2$ subject to the constaint $x+y+z=1$. I followed the method of the Lagrange multiplier and resulted in ...
1
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0answers
45 views

How to use 2nd Derivatives test to determine critical point at (0,0)?

I just have question about which method I should be using to do the second derivative test: For the function: $f(x,y)=x^3y+xy^3-xy+1$ at the point (0,0). I did this by solving: det$[H_{f}(0,0) - ...
5
votes
5answers
430 views

Indeterminate two-dimensional limit

I'm pretty sure that \begin{equation} \lim_{(x,y) \rightarrow (0,0)} \frac{x^4y}{x^2 + y^2} = 0, \end{equation} but I'm having some trouble proving it. The only technique I'm aware of that can be ...
3
votes
2answers
75 views

Parametrizing a 3D surface

Find a parametrization of the surface $x^3 + 3xy + z^2 = 2$, $z > 0$, and use it to find the tangent plane at $x = 1$, $y = \dfrac{1}{3}$, $z = 0$. I know how to find the tangent plane once I have ...
0
votes
1answer
53 views

3D Fourier transforms of $e^{-\beta r} $ and $re^{-\beta r} $

I am trying to find the integrals $$\large\int\limits_{\mathbb{R}^3} e^{-\beta \left|\vec{r}\,\right|}e^{i \vec{q} \cdot\,\vec{r}} \mathop{d^3r}$$ $$\large\int\limits_{\mathbb{R}^3} ...
1
vote
1answer
272 views

Find the Global maximum and minimum values on the closed disk of this function

just wondering if I am doing this correctly: For $f(x,y) = x^3y + xy^3 - xy + 1 $ Find the Global maximum and minimum values on: $ D = {(x,y) \in R | x^2 + y^2 \le 4} $ I found that the critical ...
2
votes
0answers
47 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
1
vote
1answer
42 views

Help with multivar. chain rule

I am having trouble with the following problem. I feel that I do understand the multivariable chain rule in general, but applying it here is more difficult. I am lost on where to start. Any help would ...
0
votes
1answer
34 views

Surface integral question help.

If $S$ is the surface of the sphere $x^2 + y^2 + z^2 = a^2$, compute the value of the surface integral $$\iint_S xz\,{\rm d}y\,{\rm d}z + yz\,{\rm d}z\,{\rm d}x + x^2\,{\rm d}x\,{\rm d}y$$ ...
4
votes
5answers
116 views

Multivariable limit $\lim\limits_{(x,y,z)\to(0,0,0)}\frac{3xyz}{x^2+y^2+z^2}$

Can anybody give me a hint on how to bound $$\frac{3xyz}{x^2+y^2+z^2}$$ I'm trying to prove that it converges to zero. Thanks!
0
votes
1answer
46 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
0
votes
1answer
29 views

An integral inequality with little information

$u,v$ are scalar fields on $V\subset\mathbb{R}^3$ such that $\nabla^2 u=0$ on $V$ and $u=v$ on $\partial V$. Prove that: $$\int_V|\boldsymbol{\nabla} ...
1
vote
3answers
76 views

Multivariable limit $\lim\limits_{(x,y)\to(0,0)}\frac{x^2y^2}{|x|^3+|y|^3}=0$

I need to prove $$\lim\limits_{(x,y)\to(0,0)}\frac{x^2y^2}{|x|^3+|y|^3}=0$$ I sort of know how to do it using polar coordinates, but I was trying to find an upper bound. Any ideas? I also wouldn't ...
1
vote
1answer
53 views

$\int_{-c}\mathbf{F}.\mathbf{dl}=-\int_{c}\mathbf{F}.\mathbf{dl}$-what is wrong here?

We know about line integral that $\int_{-c}\mathbf{F}.\mathbf{dl}=-\int_{c}\mathbf{F}.\mathbf{dl}$. Suppose my $\mathbf{F}$ is $\frac{\mathbf{r}}{r^3}$ and path is radial path from $r=a$ to $r=b$. so ...
1
vote
1answer
62 views

Multivariable limit $\lim \limits_{(x,y) \to (x_0, 1)}\frac{x}{y^2-1}$

I have to prove that this limit does not exist. I´ve already tried with polar coordinates, and by approximation with curves, and I'm running out of methods to do it. Any ideas? $$\lim \limits_{(x,y) ...
0
votes
3answers
81 views

Finding a limit of a two-variable function

I want to find this limit: $$\lim\limits_{(x,y)\to(0,0)} \frac{x^4-y^4}{x^2+y^2}$$ Approaching the point $(0,0)$ from the x-axis, I set $y=0$ then evaluate the limit of the resulting one-variable ...
1
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2answers
82 views

Multivariable limit: $\lim\limits_{(x,y) \to (0,0)}\dfrac{(y^2-x)^3}{x^2+y^4}$

I need to solve the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{(y^2-x)^3}{x^2+y^4}$$ and I can't think on a possible upper bound for it. Any ideas? Thanks
0
votes
1answer
59 views

Problem with path method for limits

I have to prove that this limit does not converge: $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ I've already tried to approximate the limit with $y=mx$, and the result is that it should ...
2
votes
2answers
46 views

How to intuitivly think of graphing a function in $\Bbb{R}^3$

How to intuitivly think of graphing a function in $\mathbb R^3$? Let there be $f: \mathbb R \rightarrow \mathbb R^3$ $$f(t)=\begin{bmatrix} \cos(t) \\ \sin(t) \\ t \\ ...
0
votes
2answers
30 views

Writing out chain rule for the following function

$\frac{dh}{dx}$, where $h(x) = f(x, u(x), v(x))$. First of all, this function doesn't even make sense to me. It's a function of one variable, with domain $\mathbb{R}$ and range $\mathbb{R}$. How can ...
0
votes
2answers
43 views

Show that $y+sin(y) = x$ in a neighborhood of $(0,0)$ can be written as function of $x$.

Show that $y+sin(y) = x$ in a neighborhood of $(0,0)$ can be written as function of $x$. I'm not sure I understand the question. In class we learned the inverse function theorem and the implicit ...
0
votes
2answers
53 views

Is there an easier way to prove a multivariate function is differentiable?

$f\colon U \rightarrow \mathbb{R}, (x,y) \mapsto \sqrt{1 - x^2 - y^2}$ where $U = \{(x,y) \mid x^2 + y^2 < 1\}$. So the definition of differentiability I have is: $$\lim \limits_{(x,y) ...
1
vote
1answer
321 views

Show a function is not continuous at a point

$$ f(x,y) = \begin{cases} \dfrac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y)\neq(0,0) \\ 0, & \text{if }(x,y)=(0,0) \end{cases} $$ For the definition of differentiability, I have: $$\lim_{h ...
0
votes
1answer
17 views

Where a particular equation meets a particular axis

The question asks where the tangent plane to $z = e^{x - y}$ at $(1,1,1)$ meets the $z$-axis. Without performing any computations or even looking at the given function, based solely on the question ...
0
votes
3answers
107 views

How to find normal vector of a function?

The normal vector of $z = x^2 + y^3$ at $(3,1,10)$. I know $\frac{\partial{z}}{\partial{x}}(3,1,10) = 2x = 6$ and $\frac{\partial{z}}{\partial{y}}(3,1,10) = 3y^2 = 1$, but how do I get ...
0
votes
1answer
31 views

Properties of a function $f=f(x,y)$ with maximum in $(x_{0},y_{0})$

I wanted to ask about the following statement. Let $f=f(x,y)$, $f$ is maximum at ($x_{0},y_{0}$). Show that: 1)$\frac{\partial f}{\partial x}(x_{0},y_{0})=0$ y $\frac{\partial f}{\partial ...
0
votes
1answer
40 views

How to find a parametrization of the set $\left\{(x,y,z): e^x+e^{-x}=z-\sqrt3y, 0<y<x<1\right\}$?

I have to find surface area of set $M=\left\{(x,y,z): e^x+e^{-x}=z-\sqrt3y, 0<y<x<1\right\}$ and my problem is to parametrize it, may you help me?
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2answers
44 views

proving that $\max Q(x)=\lambda_\max$

Let $Q(x)$ be quadratic form. Prove that $\max_{\|x\|=1}Q(x)=\lambda_\max$. $Q$ is symmetric so it can be presented as $$\langle Ax,x\rangle$$ where $A$ is matrix which on its diagonal appears ...
1
vote
2answers
35 views

line integral along a curve

Let $ \varphi(x,y) = x^3y+xy^3 ((x,y) \in R^")$, and let $C$ the curve given by $\varphi(x,y)=5$ The question is, how can I calculate the line integral of $\bigtriangledown\varphi$ along the curve ...
3
votes
1answer
104 views

Is it generally difficult to memorize 'multivariable calculus' theorems?

There are many weak forms of "Fubini's theorem" with strong hypotheses in elementary calculus texts. However, these strong hypotheses are very unnatural and are thus hard to memorize. Compared to ...
0
votes
1answer
45 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
1
vote
1answer
73 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
1
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1answer
191 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...