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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43
votes
2answers
1k views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_0^1 \! \cdots \! \int_0^1 \left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} ...
0
votes
2answers
38 views

Critical points of $(x-y)(1-xy)$

$f(x,y) = (x-y)(1-xy)$ $\frac{df}{dx} = 1-2xy + y^2 = 0$ $2xy + y^2 = -1$ $x^2 - 2xy + y^2 = x^2 - 1$ $(y-x)^2 = x^2-1$ $y-x = \sqrt{x^2+1}$ $y = x + \sqrt{x^2+1}$ $\frac{df}{dy} = -1-x^2 + 2xy ...
1
vote
3answers
116 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
1
vote
2answers
3k views

Open / Closed domain

I was calculating a domain of a function $f(x,y)$ and I need to say if the domain is an open set or closed set, and if it is bounded. At the end of my calculations, I got $xy \geq 1$, which is the ...
3
votes
1answer
115 views

Computing the volume of this weird object,

Let $f: [-1,1] \to \mathbb{R}$ be a continuously differentiable function such that $f(-1) = f(1) = 0$ and $0<f(x)\le 1$ for all $x \in (-1,1)$. Let $S$ be the surface in $\mathbb{R}^3$ obtained by ...
0
votes
1answer
54 views

Simplifying an expression of the second partial derivative w.r.t x

I've been simplifying an expression for a few hours, but I have a little trouble in understanding the last few steps to get to the desired expression. Can anyone help me or point me in the right ...
0
votes
1answer
24 views

Composition of $C^\infty$ maps between Banach spaces is $C^\infty$.

Let $V$, $W$, and $X$ be Banach spaces, and let $A \subset V$ and $B \subset W$ be open. Suppose that $F \in C^\infty(A,W)$, $G \in C^\infty(B,X)$, and $F(A) \subset B$. Is there a non-combinatorial ...
-1
votes
1answer
23 views

Finding domain of vector field $\frac{-yi+xj}{\sqrt{x^2+y^2}}$

Let $\frac{s}{r}$ be a vector field such that: $s=-yi+xj,r=\sqrt{x^2+y^2}$ What is the domain $D$ of $\frac{s}{r}$? My attempt: The domain is $\{x,y\mid x^2+y^2>0\}$ Is it correct?
5
votes
1answer
82 views

What is the set $\mathbb{R}^{\nvDash}$ defined as?

The set $\mathbb{R}^{\nvDash}$ is used in my multivariable-calculus assignment, but I do not know what the superscript '$\nvDash$' means on the set of real numbers. The context in which it used is ...
2
votes
1answer
84 views

Is the function differentiable at $(0,0)$

Given the function $$ f(x,y)= \begin{cases} \frac{\sin(x)^4 \ln(1+x^2)}{(1+\cos(x))^2+y^4}, & \text{if $(x,y)\neq (0,0)$} \\ 0, & \text{if $(x,y)=(0,0)$} \end{cases}$$ I want to check if it ...
2
votes
2answers
77 views

Solution to system of linear ODE's

Let $\Delta_n$ be the closed unit simplex in $\mathbb R^n$. For any $a,b \in \Delta_n$, define the differential equation: $$ a'(u) = b-a(u) \quad\quad\quad a(0) = a $$ How does one go about solving ...
2
votes
1answer
48 views

Compute the derivative $ \frac{d}{dR}\iiint_{\{(x,y,z)\in\textbf{R}^3: \sqrt{x^2+y^2+z^2} \leq R\}}f(x,y,z)\,dx\,dy\,dz. $

Let the function f and its first-order partial derivatives be continuous in $\textbf{R}^3$. Suppose that $$ \iiint_{\textbf{R}^3}|f(x,y,z)|\,dx\,dy\,dz < \infty. $$ Compute the derivative $$ ...
3
votes
1answer
120 views

What is the significance of the integral of the Hessian determinant?

The integral of a function over some region measures the total value of the function in that region: $$T(u)=\int u\thinspace\mathrm{d}V$$ The integral of the squared norm of the gradient of the ...
-1
votes
1answer
105 views

the equivalence of a absolute value function $|D^2 u|$ in problem 10(b) evans pde chapter 5

Can someone tell me whether $|D^2 u|$ is equivalent of writing $\frac{\nabla u}{|\nabla u|}\, D^2 u$? This relates to the post Integrate by parts to prove this inequality I wasn't sure why ...
2
votes
1answer
44 views

Stokes' theorem generalized the FTC part 2. Is there a known generalization for part 1?

Stokes' theorem generalizes the fundamental theorem of calculus (part 2) using differential forms. Is there a known generalization of part 1? edit In case anyone is unaware, The fundamental theorem ...
2
votes
1answer
56 views

Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
2
votes
1answer
46 views

Proving linearity of derivative

The derivative for a function $f : \mathbb{R}^n \to \mathbb{R}^m$ defined in some open set containing $x$ at $x$ is defined (at least in Rudin and other references), if it exists, to be the linear ...
2
votes
2answers
154 views

Evaluate the integral by type1 or type 2

Evaluate $\displaystyle\int_{0}^{2} \int_{0}^{\log(x)}(x-1)\sqrt{1+e^y}\,dy\,dx$. I have tried integration by substitution but can't connect to type 1 or type 2. Any help.
1
vote
0answers
29 views

Product of Integral over General Region

Given $f(x,y)=g(x)g(y)$ and $R=\{(x,y):a\le{x}\le{b}, c\le{y}\le{d}\}$, where $a,b,c,d$ are constants, prove the following result: $$\iint\limits_R{f(x,y)dA=\int_a^b{g(x)dx}\int_c^d{g(y)dy}}$$ So far ...
1
vote
1answer
43 views

Confidence ellipse for a 2D gaussian

For a 1D gaussian, the interval +/- 1SD about the mean will comprise ~68% of the area under the curve. Consider a 2D gaussian with a mean of zero and a diagonal covariance matrix (i.e., it is not ...
5
votes
2answers
68 views

Calculating an area between circles with Double Integral

Hey I need to answer the following question: find the area of $D=[(x,y):(x-1)^2+y^2\leq 1, x^2+y^2\geq 1,0\leq y\leq x] $ I know how to solve this kind of problems with normal integrals but how ...
2
votes
1answer
64 views

An optimization problem, in the form of a word problem,

The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the ticket price is $50$ dollars. A market survey indicates that $10$ additional seats will remain ...
3
votes
1answer
151 views

For which values does this series converge?

p and k are real numbers. For which values of p and k does the following double series converge $$\sum_{n,m=1}^\infty \frac{1}{n^p + m^k}$$ I am trying to find a better (and quicker) way to solve ...
5
votes
3answers
382 views

a function with differentiable partial derivatives but unequal mixed derivatives

I am looking for an example of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are both differentiable at some point, say ...
0
votes
1answer
38 views

Integrating $\prod_{i=k+1}^{kN} \left(\int_{-\infty}^{\infty}dp_i\right)\times$ with conditions on $p_i$

I am trying to integrate this expression which came up in a derivation of the momentum distribution function for an ideal gas. $\Theta(x)$ is the Heaviside step function which is $1$ when $x$ is ...
1
vote
2answers
64 views

Multivariable Functions - Second Derivative Problem

So here is the problem: Calculate the second class derivative on $(1,1)$ of the equation $x^4+y^4=2$ I found this problem on my proffesor's notes. However it doesn't state whether a partial or a ...
1
vote
1answer
59 views

What does $d \| \vec{x} \|$ represent?

I am doing some review problems for my upcoming exam and something has come up that I don't understand. $$\int_{\gamma} f(x,y) \space d \|\vec{x} \|$$ $\gamma(t)=(\cos(t),\sin(t))$ ...
0
votes
4answers
77 views

Calculate the volume bounded by 2 balls.

I need to calculate the volume bounded by: $$x^2+y^2+z^2\:=\:1,\:x^2+\left(y-1\right)^2+z^2=1$$ My solution: Because the volume I want to calculate is symetric, Ill calculate only one half of it and ...
0
votes
2answers
39 views

Chain Rule and Multivariable calculus

I would be very grateful if anyone could assist with the following: Given that $$ z = yg(x^2-y^2) $$ I'm trying to show that: $$\frac{1}{x}\frac{\partial z }{\partial x} + \frac{1}{y}\frac{\partial ...
0
votes
1answer
81 views

Sum of Two Continuous Random Variables

Consider two independent random variables $X$ and $Y$. Let $$f_X(x) = \begin{cases} 1 − x/2, & \text{if $0\le x\le 2$} \\ 0, & \text{otherwise} \end{cases}$$.Let $$f_Y(y) = \begin{cases} ...
2
votes
0answers
27 views

Definition of Integration of a Differential From in Lee's Introduction to Smooth Manifolds.

On pg. 402 of Lee's Introduction to Smooth Manifolds (Second Edition), the following is said to define the integral of a differential form on $\mathbf R^n$: Let $D$ be an open domain of integration ...
0
votes
3answers
77 views

Calculate the divergence of the polar coordinate vector field $\partial_\phi$ [closed]

I have to solve this problem: $v=\partial_\phi$ on $M=\mathbb{R}^2\backslash{0}$ where the components of $v$ are in polar coordinates. Calculate the divergence of $v$. Even with the help of ...
0
votes
2answers
63 views

Directional Derivative with Unit Vector

Suppose that $f(x,y)=x^2+xy-y^2$. How can I find the largest and the smallest possible values of the directional derivative $D_{u}f(1,2)$ in which $u$ is a unit vector?
1
vote
2answers
85 views

Complex value for volume, using triple integrals

I'm trying to calculate the volume a hyperboloid, within $$z=0$$ and $$z+\frac 12 x-3=0.$$ The hyperboloid: $$x^2+\left(\frac y2\right)^2-z^2=5.$$ I calculated the projections on $xz$, $yz$, to use ...
1
vote
0answers
83 views

Calculation of double integral: $\iint_D e^\frac{x-y}{x+y}\,dx\,dy$

I'm trying to solve the following double integral: $$I=\iint_D e^\frac{x-y}{x+y}\,dx\,dy$$ where $$D=\{(x,y)\in\mathbb R^2: 0\le x\le 2,\,1-x\le y\le 2-x\}$$ Let $$\left\{ \begin{array}{c} u=x+y \\ ...
0
votes
2answers
71 views

Gradient steepest direction and normal to surface?

From this Maths SE question I now understand the gradient to be the directional derivative that returns the steepest slope at a point. However reading my textbooks they all say that the gradient is ...
1
vote
1answer
38 views

A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
3
votes
1answer
123 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 ...
1
vote
1answer
32 views

Extrema using Lagrange Multiplier

Let $f(\textbf{x})=\sum_{j=1}^n\frac{a_j}{x_j}$, determine its extrema on the set $$S=\{x\in \mathbb{R}^n|\prod_{j=1}^n x_j=C\}$$ The answer is here ...
1
vote
1answer
32 views

Calculation of triple integral over an elliptic cone

The task at hand is to calculate the following integral: $$I=\iiint_V z^2\,dx\,dy\,dz$$where$$V=\{(x,y,z)\in\mathbb R^3:\,\frac{x^2}{a^2}+\frac{y^2}{b^2}\le\frac{z^2}{c^2},\, 0\le z\le h\}$$ Using the ...
1
vote
0answers
50 views

Given an embedded $C^k$ submanifold of $R^n$, is the distance function $C^k$?

I am asking because I am wondering if studying smooth manifolds is the same as studying zero loci of smooth functions with non-vanishing gradient. Let me be a little formal for clarity: Let $M$ be a ...
0
votes
1answer
18 views

Directional deriviation given two points as the direction

Given $$z=x^2y+y^2$$. How can I calculate the directional deriviation at $M(1,2)$ in the direction of $MM_1$ where $M_1(4,0)$? I know how to do this when given a vector and not the two ends of it. ...
0
votes
2answers
29 views

Maximising directional derivative of a polynomial in 3 variables

I am a beginner in multivariable calculus and had started reading Apostol. I have solved all the exercises in the portions I have covered, except one problem. Find values of the constants $a,b,c$ ...
1
vote
2answers
32 views

Continuous of function in a point

Given the function $$f(x,y)=\frac{1-\cos(2xy)}{x^2y^2}$$ I want the function to be continuous in $(0,0)$. If I assume that the limit when $x\rightarrow0$ equals to the limit when $y\rightarrow0$, I ...
0
votes
0answers
35 views

Is this multivariable function continuous?

My function is: $$f\left(x,\:y\right)\:=\:y\left(sin\left(\frac{1}{x-1}\right)\right)\::\:x\:\ne 1$$ $$f\left(x,\:y\right)\:=\:0\::\:x\:=1$$ The question sounds like: "Are this functions continuous?" ...
1
vote
0answers
45 views

Line Integrals, dx, dy and parameters

So I posted a similar question the other day, which I answered myself since I was under the impression that I figured this stuff out. But today I found a question where my technique broke down and I ...
0
votes
0answers
25 views

Nonlinear-Variation of Helmholtz Equation

I was wondering on the solution of the equation $$\nabla^2P(\vec r)=v(\vec r)P(\vec r)^2\phantom{.......}(1)$$ Or more simply, if there exists a coordinate system where: $$\nabla^2P(\vec r)=P^2(\vec ...
0
votes
1answer
69 views

Proof for non-existence of multi variable limit of $2x\cos\left(\frac{1}{x^2+2y^2}\right)/(x^2+2y^2)$ at $(0,0)$

How do I prove that the below limit doesn't exist: $$\lim _{\left(x,\:\:y\right)\to \left(0,0\right)}\left(\frac{2\cos\left(\frac{1}{x^2+2y^2}\right)}{x^2+2y^2}\right)\cdot x$$ Its easy to see that: ...
1
vote
1answer
28 views

Implicit function theorem conclusion notation?

I am working through implicit function theorem for the first time, and I have the following understanding. Given a system of $n$ equations, \begin{equation} f_i(x_1,\dots ,x_m,y_1,\dots , y_n)=0,\ \ \ ...
9
votes
1answer
137 views

What is the Exterior Derivative Trying to Do?

$\newcommand{\R}{\mathbf R}$ Consider a smooth function $f:\R^n\to\R$ and let $Df:\R^n\to \R^{n*}$ be the map which takes a point $\mathbf a\in R^n$ to the linear map $Df_{\mathbf a}:\R^n\to \R$. ...