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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0
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1answer
22 views

find the volume of the half-cone $\sqrt{x^{2}+y^{2}}<z<1,\ x>0$

I need to find the volume of the half-cone $$\sqrt{x^{2}+y^{2}}<z<1,\ x>0$$ I have found the range of $x$ and $z$ which are $0<x<1, \sqrt{x^{2}+y^{2}}<z<1$, but I cannot figure ...
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0answers
18 views

Derivative functions and level curves and gradient

Let n(P) = P • P. Find Dn. Let r(P) = ||P|| for P in R^2. Find Dr. What does the graph of r look like in this case? What are the level curves? What is the gradient?
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0answers
23 views

Chain rule for general derivative operator.

We have the derivative operator which is defined as: $$D^{\alpha}f(x) : = \frac{\partial^{|\alpha|}f(x_1,\ldots, x_n)}{\partial^{\alpha_1}x_1\ldots \partial^{\alpha_n}x_n}$$ where $|\alpha | = ...
0
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1answer
62 views

Need help calculating this limit for $\varepsilon \to 0$

I used Gauss' identity to derive $$(\ast) {1\over \varepsilon^{n-1}}\int_{\partial B(a,\varepsilon)} f dS = {1\over r^{n-1}}\int_{\partial B(a,r)} f dS $$ where $0<\varepsilon<r$ and $f$ is ...
0
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1answer
26 views

Multivariable functions limits and paths

In order to approach a point as (0,0) there many directions to do so. A whole 360 degrees actually. So between [0,360) degrees there are actually infinite directions. My question is why does it ...
1
vote
1answer
55 views

What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?

A Polydisc of center $z^o=(z_1^o,\dots,z_n^o)\in\Bbb C^n$ and multiradius $r=(r_1,\dots,r_n)\in(\Bbb R^+)^n$ is defined as $$ P_{z^o,r}:=\prod_{j=1}^n\Delta_{z_j^o,r_j} $$ where ...
2
votes
1answer
34 views

Proving this function is an open map

Prove the function $f(x, y, z) = (x^3, y^2-z^2, yz)$ is an open map from $\mathbb{R^3}$ to $\mathbb{R^3}$ (i.e for every open set $U$ of $\mathbb{R^3}$, $f(U)$ is open). I know, as an application of ...
0
votes
1answer
23 views

Contradiction: finding potential function of $\nabla f$ where $f=e^x y z$ due to $e^0=1$ - violates path independence

I have got the following contradiction: I take the function $$f(x,y,z)=e^x yz$$ and compute its gradient $$\nabla f=\langle e^xyz,e^x z,e^x y \rangle .$$ Now I want to find the potential (i.e. $f$) ...
1
vote
1answer
30 views

Surface integral of $x^4+y^4+z^4$ over the sphere $x^2+y^2+z^2=a^2$

After doing regular methodology have reached upto integral shown in figure , but when i eliminate z from it it becomes very complicated to solve .Is there any other way to solve this .Thanks
5
votes
2answers
203 views

Is this a correct use of the squeeze theorem?

I have to find the following limit: $$ \lim_{x\to0,y\to0} \frac{x^2y^2}{x^2+y^4}=[\frac{0}{0}] $$ I try to reach the origin moving on the y-axis ($x=0$): $$ \lim_{y\to0} \frac{0}{y^4}=0 $$ I get ...
1
vote
1answer
50 views

Mistake in my proof: what is the normalisation factor of the surface integral of a sphere?

I was trying to prove $$ {1\over \varepsilon} \int_{\partial B(a,\varepsilon)} f dS = {1\over r} \int_{\partial B(a,r)} f dS$$ where $0<\varepsilon < r$ and $f$ is harmonic on $\mathbb R^2$ ...
1
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2answers
39 views

Prove that a limit in two variables does not exists

I am trying to solve the following limit and Wolfram Alpha says that it does not exist, but I am not able to prove it. $$ \lim_{(x, y) \to (0,0)} \frac{\sin(xy)}{x^2+y^2} $$ I have tried to move ...
1
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1answer
36 views

Fubini theorem related proof

Let $C={x,y; x>0,y>0}$ and $f(x,y)=1/(x^2+\sqrt x)(y^2+\sqrt y)$. I need to show that $f$ is integrable over $C$. My idea was to set up a rectifiable set $C_N=(1,N)$ and then use Fubini theorem, ...
1
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0answers
20 views

Potential of central force - Does it matter that $n$ is an integer in $\rho^n$?

In problem 6G-2 of the following link to a pdf-file from ocw.mit.edu (Link) it is pointed out that $n$ is an integer. Problem 6G-2: Show that the fields $\mathbf{F} = \rho^n(x\hat\imath + ...
1
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0answers
59 views

Mistake in Spivak's *Calculus on Manifolds*?

Didn't see this in any of the errata: Problem 1-8(b): Suppose $T$ is a linear transformation. If there is a basis $x_1, \dotsc, x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \dotsc, \lambda_n$ ...
0
votes
2answers
53 views

Analysis questions [closed]

What condition must the constants a,b and c satisfy to guarantee that $$\lim_{(x,y) \to (0,0)} \frac{xy}{(ax^2 + bxy + cy^2)}$$ exists? Explain your answer and find the value of the limit.
3
votes
1answer
38 views

Derivative with Respect to a Ratio of Variables

We have two strictly positive real-valued variables $x$ and $y$ and a third one defined as $z=\frac{x}{y}$. Question 1: How do I compute the derivative $\frac{\partial x}{\partial z}$? Is it ...
0
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1answer
29 views

Is there any known method to fit plane onto sampling data?

For example I have the variables x, y (or higher dimensional data in general) and a probability density distribution p(x,y). I want to approximate p(x,y) as a linear function, a plane in this case, at ...
0
votes
1answer
23 views

To find Area bounded by curve and my attempt

I amnot getting correct answer , i am getting 7/2 but textbook states 3/2 .Can someone throw light on my path ? THANKS
1
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1answer
37 views

Real/Complex differentials forms

Given $f:\Bbb C^{n}\to\Bbb C$ identified with $f:\Bbb R^{2n}\to\Bbb C$, in a book I read that $$ \partial_x f\,dx+\partial_yf\,dy=\partial_zf\,dz+\partial_{\bar z}f\,d\bar z $$ and that this could be ...
1
vote
1answer
83 views

Is changing variables the same as substitutions?

I have asked several on a similar matter. This time the question is tad different. $$\int_{\mathbb{R}} e^{-x^2} dx$$ We let $x=y \implies dx=dy$ $$\int_{\mathbb{R}} e^{-(x^2 + y^2)} dxdy$$ But ...
2
votes
1answer
54 views

Double integral: How to switch to polar coordinates with a difficult domain

i have this double integral: $$ I=\int \int_{R} (x+y),\;\; R=\left \{ (x,y):\frac{x^{2}}{3} \leq y\leq 3,\; -1\leq x\leq 3\right \} $$ and this is the domain of integration NOT in polar coordinates: ...
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0answers
33 views

Best practice to sketch difficult functions.

what's the best practice to sketch a difficult function such as: $$ y=x^3+3cos(x)+x $$ or $$ y=e^{x}-\frac{4}{9}e^{3x}+\frac{1}{3}x+\frac{4}{9} $$ when i'm on a test, how can i sketch them in a small ...
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votes
0answers
25 views

Try the correctness of one component of the composite

I want to help me with this doubt: If $f,g:\mathbb{R}²\rightarrow \mathbb{R}³$ are functions of class $C1$ ang $h$ homeomorphism in $\mathbb{R}²$ such that $$f=g\circ h$$ Then $h$ is differentiable ...
2
votes
1answer
31 views

Calculating the limit $\lim \limits_{(x,y) \rightarrow (0,0)} \frac{f(x,y)}{g(x,y)}$

I want to prove that $\displaystyle\lim \limits_{(x,y) \rightarrow (0,0)} \frac{f(x,y)}{g(x,y)}$ is $0$ if the degree of $f(x,y)$ is greater than the degree of $g(x,y)$. Here $f(x,y)$ and $g(x,y)$ are ...
1
vote
1answer
58 views

Calculate the surface integral $\iint_S (\nabla \times F)\cdot dS$ over a part of a sphere

How can I calculate the integral $$\iint_S (\nabla \times F)\cdot dS$$ where $S$ is the part of the surface of the sphere $x^2+y^2+z^2=1$ and $x+y+z\ge 1$, $F=(y-z, z-x, x-y)$. I calculated that ...
0
votes
1answer
107 views

An Interesting Resource Allocation Problem

Here is the problem: \begin{array}{ll} \text{minimize} & \sum_{i=1}^N \frac{1}{1 + \textrm{exp}(C_i + x_i)}\\ \text{subject to} & \sum_{i=1}^N x_i \le R \\ & x_i \ge 0, ~ i = 1,2,...,N ...
3
votes
2answers
149 views

Zero Partials imply Constant Function Theorem or Proof

Let $f(x,y)$ be a function of two variables. Given that $$\frac{\partial f}{\partial x}=0$$ for all $y$, and $$\frac{\partial f}{\partial y}=0$$ for all $x$, is there a theorem that states $f(x,y)=$ ...
-1
votes
1answer
21 views

Implicit function theorem and Taylor polynomial

Prove that equation $$x\ln w+w\ln y=0$$ sits in a neighborhood of $(x_0,y_0)=(1,1)$, variable $w$ as function of $x$ and $y:$ $w=g(x,y)$. Prove that $g$ is of class $C^\infty$. Write second Taylor ...
0
votes
2answers
43 views

To check continuity of Multivariable functions

To check continuity of function at origin given by $$F (x, y) = \begin{cases}\dfrac{xy^{2}}{x^{2} + y ^{4}}&;& \mbox{otherwise},\\ 0&;&\mbox{ at origin}. \end{cases}$$
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2answers
74 views

Replacing Variables in Integration [duplicate]

I have posted questions about this, but they werent clear, here is my actual misunderstanding. $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I dont understand, we say: $$I = \int_{-\infty}^{\infty} ...
1
vote
0answers
41 views

L'hopital's rule in multivariate calculus

Suppose $f ,g : \mathbb{R}^{n} \to \mathbb{R}^n$ are two differentiable functions with an isolated fixed point at the origin. In general the limit of the real valued function $$ \phi(x) := \frac{ | ...
2
votes
1answer
89 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
1
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0answers
30 views

saddle point versus local extermum

Suppose a function $f$ from $\mathbb{R}^n \to \mathbb{R}$, is differentiable. We know that $c$ is a critical point of $f$, i.e. $\nabla f(c) = 0$. Our goal is to find out if $c$ is a local extremum, ...
3
votes
0answers
61 views

Differentation uder the integral sign

Let $F(x)=\int_{\sin x}^{\cos x} e^{x\sqrt{1-y^2}} \, dy $. My task is to calculate $F'(x)$. My idea is to use http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign and I get: ...
0
votes
1answer
37 views

Boundaries of the triple integral

I need to calculate the triple integral $\iiint_V y\,dx\,dy\,dz$ where $V$ is deterimned by $y\geq x^2, z\leq 4-y, 0\leq z \leq3$. I get $\int_{-1}^1 dx \int_{x^2}^1 y \, dy \int_0^3 dz + 2\int_1^2 dx ...
4
votes
3answers
137 views

Dirichlet's integral $\int_{V}\ x^{p}\,y^{q}\,z^{r}\ \left(\, 1 - x - y - z\,\right)^{\,s}\,{\rm d}x\,{\rm d}y\,{\rm d}z$

I found such an exercise: Calculate the Dirichlet's integral: $$ \int_{V}\ x^{p}\,y^{q}\,z^{r}\ \left(\, 1 - x - y - z\,\right)^{\,s}\,{\rm d}x\,{\rm d}y\,{\rm d}z \quad\mbox{where}\quad p, q, r, s ...
0
votes
3answers
29 views

Point closest to curve in $R^m$

I've been working on a problem from a foundation exam which seems totally straightforward but for some reason I've become stuck: Let $f: \mathbb{ R } \rightarrow \mathbb{ R } ^n$ be a differentiable ...
0
votes
1answer
65 views

How does an Iterated Integral Work?

I am simply confused because of planes now. Consider: $$J = \int_{R}\int_{R} e^{-(x^2 + y^2)} dxdy$$ What is the geometrical aspect of this integral? This represents the volume under $h(x,y) = ...
0
votes
2answers
34 views

What is the difference between an undulation point and other critical values?

A necessary but not sufficient condition for a point of inflection is that $$f''(x)=0$$ If the second derivative is 0 and the point is not a point of inflection, Wikipedia tells me that is called an ...
0
votes
1answer
44 views

Change of freebound variables in Integration

Okay, recently I have been looking at: $$\int_{R} e^{-x^2} dx$$ It has been bothering me that, we can say that: $$I = \int_{R} e^{-x^2} dx$$ But also that: $$I = \int_{R} e^{-y^2} dx$$ then ...
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0answers
29 views

Maximization question [duplicate]

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
1
vote
2answers
56 views

Is it differentiable at $x=(0, 0)$?

Let $ \displaystyle f(x, y)=\frac{x^3-y^3}{x^2+y^2} $ be a multivariable function. Examine if it is differentiable at $x=(0,0)$. I proved that the limit of the partial derivatives at $x=(0, 0)$ are ...
1
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0answers
44 views

Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
0
votes
3answers
72 views

Find the maximum value for $x+y+z-xy-yz-zx$

If $x,y,z$ are real numbers for which holds $0\le x,y,z \le 1$, then find the maximum value of $x+y+z-xy-yz-zx$ and find $(x,y,z)$ for which you get the maximum value. This is how did it and I would ...
4
votes
0answers
53 views

Uniqueness of solution to $u_{t} - \Delta u + |\nabla u|^{2} = 0$

The problem I am working on is as follows: Let $\Omega$ be a connected bounded domain in $\mathbb{R}^{n}$ with smooth boundary and let $f, g: \mathbb{R}^{n} \rightarrow R$ be smooth. Show that there ...
1
vote
2answers
56 views

Orientation of multiplying integrals

Consider, $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ The trick is to multiply by $I$ again to get $I^2$ But they often write: $$I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2 - ...
0
votes
1answer
33 views

Gauss-Bonnet Theorem - Notation

Why is the equality in red true? $\bf{e}^{'}$ and $\bf{e}^{''}$ form an orthonormal basis of the tangentplane w.r.t $\gamma(s)$, which is unit speed.
0
votes
0answers
13 views

Volume of region by revolving a curve around $z$-axis

Let $S$ be the surface in $\mathbb{R}^3$ obtained by revolving the curve $$\begin{cases} x&=\cos u \\ y&=\sin 2u \end {cases}, -\frac{\pi}2\leq u \leq \frac{\pi}2$$ in the $xz$-plane ...
4
votes
1answer
98 views

Find $\mathbf{F}$ such that $\nabla \times \mathbf{F} = (-3xz^2, 0, z^3)$

Let $S$ be the surface defined by $z = x^{2} + y^{2}$ for $z \leq 4$, oriented with upward-pointing normal. Use Stokes' theorem to evaluate $\iint_{S}\left(\, -3xz^{2}\ ,\ 0\ ,\ ...