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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1answer
72 views

Calculate $\int_D x^2 dxdydz$ for $D$ an ellipsoid

Let $D$ be the ellipsoid $$x^2/a^2 + y^2/b^2 + z^2/c^2 \leq 1.$$ Compute $\int_D x^2 dx dy dz$. Map to the unit ball by $\varphi: (x,y,z) \mapsto (x/a, y/b, z/c)$. Now $$\int_D \frac{1}{abc}x^2 ...
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1answer
87 views

Implicit Function Theorem -Confusion-

I am trying to figure out what the implicit function theorem is. Can anyone explain it to me? I was reading the wiki: http://en.wikipedia.org/wiki/Implicit_function_theorem but still don't understand ...
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2answers
212 views

Evaluate $\displaystyle \int_A x^{-1}$

We are asked to evaluate $\displaystyle \int_A x^{-1}dV(A)$, with $A=\{ (x,y):2<x+y<4,y>0,x-y>0\}$. From the solutions we know that$\displaystyle \int_A x^{-1}dV(A)=2Log(2)$. The point is ...
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3answers
66 views

Dimension of the boundary of a polytope for applying divergence theorem in $\Bbb{R}^n$

Let's use the unit hypercube in $\Bbb{R}^n$ as an example. The unit $n$-cube is formed by the intersection of $2n$ half-spaces, $n$ of them being defined by $x_i \ge 0$ and the other $n$ of them being ...
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1answer
80 views

Scalar surface integral help.

$\iint xy \,\mathrm dS$ where $S$ is the surface of the tetrahedron with sides $z=0$, $y=0$, $x + z = 1$ and $x=y$. The answer is given as: $$\dfrac{3\sqrt{2}+5}{24}$$ ∫∫ xy dS = ∫∫ xy √(1 + ...
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1answer
48 views

Differentiability of two variable function

Is $f(x,y)=\begin{cases} \cfrac{x^3+x\sqrt{y^4+y^5}}{x^2+y^2}, &(x,y)\neq (0,0)\\ 0, &(x,y)=(0,0) \end{cases}$ differentiable at $(x,y)=(0,0)$?
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1answer
62 views

In the change-of-variables theorem, must $ϕ$ be globally injective?

In the above theorem, doesn't $\phi$ need to be injective too? The inverse function theorem merely implies that $\phi$ is locally injective -- is this sufficient? I ask because Marsden, in his ...
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1answer
82 views

Integration problem on multiple variables

I would like to ask help on this integration problem: $$\int_0^1\int_0^1\sqrt{x^2+y^2}\;\left[1-\alpha(1-2x)(1-2y)\right]dxdy$$ I was wondering if polar substitution is possible. I have done the ...
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1answer
53 views

In the change-of-variables theorem, must $\phi (A)$ be open?

In the above theorem, is $\phi (A)$ necessarily an open set in $\mathbb R^n$ ? The author (subsequently) suggests so, but I can't see why. p.s. In the above, "a set has volume" is equivalent to ...
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2answers
90 views

How to derive these calculus identities?

The following are calculus identities, but I never memorized them and don't know how to derive them: $$ \cos(\mathbf{n},x_1) = \frac{\frac{\partial f}{\partial x_1}}{\pm \sqrt{1 + \left( ...
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1answer
56 views

Differentiation of $f(f(x,y),f(x,f(x,y)))$

Let $f(x,y)$ be a continuously differentiable function of two variables. How do I calculate the derivative with respect to $x$ of $f(f(x,y),f(x,f(x,y)))$. Thanks.
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3answers
477 views

Property of critical point when the Hessian is degenerate

Let $f:{\Bbb R^2}\to{\Bbb R}$ be a function such that $$ f(x,y)=5x^2+xy^3-3x^2y. $$ Is $(0,0)$ a local maximum, local minimum or a saddle point? Calculation shows that $(0,0)$ is a degenerate ...
1
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1answer
310 views

Does integration by parts work for partial derivatives?

Does integration by parts works for partial derivatives? Can we write $$\int_a^b \frac{\partial f(x,y)}{\partial x}g(x,y) dx = f(x,y)g(x,y)|_a^b - \int f(x,y)\frac{\partial g(x,y)}{\partial x}dx$$
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3answers
225 views

Finding global minumum maximum

How can I find the global min/max in this problem? Find the critical points of the function : f(x,y)=$2x^3-3x^2y-12x^2-3y^2$ and determine their type. Are there any global min/max? The critical ...
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2answers
111 views

Frechet derivative and linear transformation question

It's been a while now I am studying multivariable calculus and the concept of differentiation in space (or higher dimension). I saw relative posts but one question remains. I can't understand the ...
2
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2answers
68 views

Continuity in multivariable calculus

I want to find out the points, where the function $f(x,y)=\dfrac{xy}{x-y}$ if $x\neq y$ and $f(x,y)=0$ otherwise, is continuous. I have shown that at all the points $(x,y)$, where $x\neq y$, $f$ is ...
4
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1answer
57 views

Surface integrals and parametrisation and limits.

This is a question in preparation for an exam I am going to sit. It is from a previous years exam and no solution has been provided. Let S be the surface in R3 given by $$z = (x^2 + y^2)^{1/2},x^2 + ...
2
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2answers
67 views

Local estimate for a $C^1$ map with negative definite Jacobian

Let $U$ be an open set in ${\Bbb R^n}$ and $f=(f^1,\cdots,f^n):U\to{\Bbb R^n}$ be $C^1$. Suppose $f'(x_0)$ is negative definite for some $x_0\in U$. Show that there exists $\epsilon>0$ and a ...
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1answer
28 views

Extending the derivative

A linear functional $\mu: C^1(\mathbb{R^n})\rightarrow \mathbb{R}$ which follows the Liebniz rule $\mu(fg)=\mu(f)g(0)+\mu(g)f(0)$ is called a derivative at $0$. It is easy to show that such a ...
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3answers
58 views

Finding a limit

I want to check whether the limit $\lim\limits_{(x,y)\to (0,0)}\dfrac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}$ exists? I proceed as follows: $\vert \dfrac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}\vert \leq ...
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1answer
56 views

Parametrize $|x|+|y|+|z|=1$

How can we parametrize the surface $|x|+|y|+|z|=1$? Here I mean differentiable parametrize. I think we may need to divide it into 8 pieces and consider them respectively.
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1answer
51 views

$f(x ,y)$ differentiable all over the plain. $g(u, v) = f(u^2 - v^2, u^2v).$ if $\nabla f (-3, 2) = 2 \vec i + \vec j$ , calculate $ \nabla g(1,2)$

unfortunately, I had to miss the lecture that gradients leant and I don't know how to solve this question. Let $f(x ,y)$ differentiable all over the plain. Let $g(u, v) = f(u^2 - v^2, u^2v).$ ...
2
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2answers
40 views

What information can one get from $f(x,y)\geq -3x+4y$ provided that $f$ is continuously differentiable near $(0,0)$?

Let $V$ be a neighborhood of the origin in ${\Bbb R}^2$ and $f:V\to{\Bbb R}$ be continuously differentiable. Assume that $f(0,0)=0$ and $f(x,y)\geq -3x+4y$ for $(x,y)\in V$. Prove that there is a ...
1
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1answer
154 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
1
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1answer
30 views

One-Sheeted- Hyperboloid

I am given the following surface: $z^2-xy=1$ . How can I show it is of the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}=1$ ? I can't find an appropriate coordinate change... Will you ...
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3answers
141 views

Finding critical points of multivariable function

Find the critical points of $f(x,y)=x^y+4xy-y^2-8x-6y$ I found the derivative of the function and got $$f^\prime_x=yx^{y-1}+4y-8 \\ f^\prime_y=\ln x\, x^y+4x-2y-6 $$. I want to find point ...
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2answers
259 views

What is the interpretation of the eigenvectors of the jacobian matrix?

I'm trying to think about the jacobian matrix as a abstract linear map. What is the interpretation of the eigenvalues and eigenvectors of the jacobian?
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1answer
38 views

Normal to a parameterized surface- Can't prove

Given a surface $ F: D \in R^2\longrightarrow S \in R^3 $ with smooth parameteric representation: $F(u,v) = (x(u,v),y(u,v), z(u,v)) $ . Denote by $N = F_u \times F_v $ , how can one prove that $N$ ...
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1answer
57 views

Solid bounded by regions

Find the volume of the region bounded by $x^2+2y^2=2,\space\space z=0,\space\space x+y+2z=2$ From Triple Integrals Multivariable Calculus - Marsden, Tromba, Weinstein (Springer) Page 316, ...
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1answer
47 views

If $r = \langle x,y\rangle$, $r_1 = \langle x_1, y_1\rangle$, and $r_2 = \langle x_2,y_2\rangle$, describe the set of all points $(x,y)$ …

[2D - vector application] If $r = \langle x,y\rangle$, $r_1 = \langle x_1, y_1\rangle$, and $r_2 = \langle x_2,y_2\rangle$, describe the set of all points $(x,y)$ such that $|r - r_1| + |r - r_2| = ...
0
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1answer
93 views

Finding the determinant of an $n\times n$ matrix… and the inverse

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. ...
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1answer
47 views

Given $f(t)$ differentiable. let $u(x,y) = yf(\cos(x-y))$. find $u'_x, u'_y$.

I've never seen a question like this and I'll be happy if you can help me solve it. This is the WHOLE question: (in case it looks wierd) Given function $f(t)$ differentiable in every point in the ...
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1answer
38 views

What is an integral with a higher order differential? d^3 v

Pauli's lectures on statistical mechanics start out with: Let $ f(v) d^3 v $ be the number of molecules with velocity contained in $ d^3 v $, and let $n=\int f(v)d^3 v$ be the number of molecules ...
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1answer
40 views

Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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1answer
41 views

Integral converges in $E^n$

How do I prove that $\displaystyle \int_{E^n}|\mathbf{x}|^{-|\mathbf{x}|}d\lambda(\mathbf{x})$, where $\lambda$ is lebesgue measure, converges? I was thinking of finding an upper bound function for ...
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3answers
74 views

Local graph for $F(x_1,x_2,x_3,x_4)=\left(x_1^2+x_2^2+x_3^2-x_4^2,\sum x_i\right)=(0,0)$ when implicit function theorem fails?

Let $X=\{(x_1,x_2,x_3,x_4)\in{\Bbb R^4}:x_1^2+x_2^2+x_3^2-x_4^2=2,\ \sum x_i=2\}$ and let $p=(1,1,1,-1)$. Then $p\in X$. Is it possible to find a product of open set $V$ containing $p$ such that ...
2
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1answer
44 views

multi-variate normal distribution distance from vector sub-space

let $X\sim {\cal N}(\mu,C)$ be a random variable obeying multi-variate normal distribution in $\mathbb{R}^n$ and $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=n-1$. What is the probability ...
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2answers
46 views

how to find out the region between two surfaces

Describe the region cut out of the ball $x^2+y^2+z^2\le4$ by the elliptic cylinder $2x^2+z^2=1$ i.e the region inside the cylinder and ball I equated $4-x^2-y^2=1-2x^2$ getting $y^2-x^2=3$. I guessed ...
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0answers
72 views

The inverse matrix |${\delta}$| does it have an application

The jacobian is the determinant of |${\delta}$| this means that |${\delta}$| is invertible. Does this inverse have any use in the real world? Maybe I am not clear in my question: Does the inverse of ...
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3answers
164 views

Derivative of function with 2 variables

I've leart in Calculus 1 that the derivetive of $f(x)$ is: $$\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$. suppose $f(x,y)$ is a function with 2 variables, does $$f'(x,y) = \lim_{h\to0} \frac{f(x+h, y+h) ...
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2answers
73 views

Existence of $G$ for $\nabla \times G=F$?

What are the necessary and sufficient conditions on functions $h,k:{\Bbb R}^2\to{\Bbb R}$ such that given any smooth $F:{\Bbb R^3}\to{\Bbb R}^3$ of the form $F=(F_1(y,z),F_2(x,z),0)$ and whose ...
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2answers
145 views

Del operator ($\nabla$) in spherical co-ordinate system

I am teaching myself about vector fields and came across the following question: Is the following force field $\vec{F}$ conservative, where $\vec{F}(r,\theta,\varphi)$ is defined by: ...
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0answers
79 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
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2answers
939 views

volume of a largest rectangular parallelepiped inscribed in an ellipsoid

Show that the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ is $\frac{8abc}{3\sqrt3}$. I proceeded by ...
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3answers
145 views

Details of $z=u \cos(v) \sin(u), u=e^{xy^2}, v=x^2+y \quad \frac {\partial z} {\partial x}=?,\frac {\partial z} {\partial y}=?$ [closed]

Given $$z=u \cos(v) \sin(u), \quad u=e^{xy^2}, \quad v=x^2+y$$ How to show details of finding $\dfrac {\partial z} {\partial x}$ and $\dfrac {\partial z} {\partial y}$?
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1answer
36 views

How can I show that $ \int_{\Gamma_1}F\cdot dr-\int_{\Gamma_2}F\cdot dr=2k\pi $?

Let $F:{\Bbb R}^2\to {\Bbb R}^2$ be such that $$ F(x,y)=\left(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}\right). $$ Suppose we have to one-to-one $C^1$ curves: $\gamma_j:[0,1]\to{\Bbb R}^2$, such ...
2
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2answers
55 views
1
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3answers
172 views

a function whose every point is a saddle point

i came across this particular problem which says Suppose that $z=f(x,y)$ is defined, has continuous second partial derivatives and satisfies the Laplace equation $\frac{\partial^2 z}{\partial y^2} + ...
1
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1answer
159 views

Calculate normal vector to $2$-face of polytope in $\Bbb R^n$

I am trying to work through a divergence theorem application for a function integrated over an $n$-dimensional convex polytope, but I can't seem to figure out how to properly calculate the normal ...
1
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1answer
96 views

Why is the Jacobian ${\rho}$

Just a little confused. When I find the volume of a cone (or a sphere) for that matter I multiply the partial derivatives by the Jacobian. ${\rho}$ for a cone. and ${\rho^2 \sin \phi}$ for a sphere. ...