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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4
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0answers
9 views

Can we detect smoothness of a norm by its behavior along paths?

We say a norm $\| \cdot \|$ on $\mathbb{R}^n$ is smooth if it is smooth as a function $\mathbb{R}^n\setminus \{0\} \to \mathbb{R}$. (i.e, after restricting the domain). We say a norm is smooth along ...
2
votes
2answers
27 views

Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
0
votes
1answer
20 views

Finding the area between two curves using a set of transforms and their Jacobian

I have the following transforms: $\begin{align} x &= u^2 - v^2 \\ y &= 2uv \end{align}$ and am tasked with finding the area between the following curves: $\begin{align} x &= 4 - ...
6
votes
3answers
47 views

How to show that a continous function $f:\mathbb{R}^m \to \mathbb{R}$ has a maximum?

My task is this: Suppose $f:\mathbb{R}^m \to \mathbb{R}$ is a positive, continous function such that $\lim_{\mid \textbf{x}\mid \to \infty} f(\textbf{x}) = \textbf{0}$. Show that $f$ has a maximum. ...
0
votes
0answers
25 views

Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
0
votes
1answer
18 views

How to set the limit for interated integral of $f(x,y)$ over diagonally partitioned region

I would like to compute $$I = \int_{\mathcal{R}} f(x,y) d\mathcal{R}$$ $$ f(x,y) = \begin{cases} x^2, \quad 0 < x < y < \pi \\ y^2 , \quad 0 < y < x < \pi \end{cases}$$ ...
1
vote
2answers
20 views

Show that Set in $M:=\{x\in \Bbb R^3 : x_1^2\ge2(x_2^3+x_3^3) \}$ is closed

I have to show this regarding the Euclidean metric. I've already shown that it isn't bounded by showing that the $d(x,y)\:\forall x,y \in M$ isn't bounded. I know that in order to show the ...
1
vote
1answer
30 views

Problem with multiple integrals of $\cos(x+y)$

I have a problem with this integral $\int_{0}^{\pi}\int_{0}^{\pi}\mid \cos\left(x+y\right)\mid dxdy$ I work with this problem, but the result of the book does not match with my result Note: The book ...
1
vote
1answer
23 views

Question Regarding Proof of Taylor Remainder Theorem in Tu's “An Introduction to Manifolds”

The statement: Let $f$ be a $C^{\infty}$ function on an open set $U\subseteq \mathbb{R}^n$ which is star shaped with respect to a point $p=(p^1,...,p^n) \in U$. Then there are functions ...
0
votes
1answer
33 views

The region where the two variable function $xy/(x-y)$ is differentiable

I need to found the area where this function is differentiable $$ f(x,y) = \frac{xy}{x-y} $$ How do I need to proceed? For partial derivatives I got: $$ \frac{\partial f}{\partial x} = ...
2
votes
2answers
23 views

Langrange Multiplier, to find maximum volume of a cone

Question: A right-angled triangle is rotated about one of its sides that form the right angle to a cone. Given that the sum of the lengths of two sides of the triangle that form the right angle is ...
0
votes
0answers
6 views

Extremum of Laplacian of a function

Let f(x,y,z) be any arbitrary continuous function. Let's denote Laplacian of f by $\nabla^2 f$. 1) How do we denote the extremum of $\nabla^2 f$ mathematically ? 2) how do we solve for such ...
1
vote
1answer
52 views

Product of limits when one is zero and the other does not exist

We have two functions $f(x)$ and $g(x)$, such that $\lim_{ x\to 0}f(x)=0$ but $\lim_{x\to 0} g(x)$ does not exist. Would that mean that $\lim_{x\to 0}f(x)g(x)= 0$, assuming we don't divide by zero ...
1
vote
0answers
29 views

Uniform current in cylinder and straight wire causing same magnetic field?

The tridimensional version of the Biot-Savart law says that the magnetic field generated at the point $\boldsymbol{r}\in\mathbb{R}^3$ by a tridimensional distribution of current defined by the current ...
1
vote
1answer
12 views

Surface integral of function over intersection between plane and unit sphere

I've been asked to compute the integral of $f(x, y, z)= 1 - x^2 - y^2 - z^2$ over the surface of the plane $x + y + z = t$ cut off by the sphere $x^2 + y^2 + z^2 = 1$ for $t \leq \sqrt3$ and prove it ...
0
votes
1answer
317 views

norm form of remainder of Taylor's expansion

Let $\mathbf x=(x_1,x_2,\cdots,x_n), \mathbf a=(a_1,a_2,\cdots,a_n)$, Can we write Taylor's expansion of $f(\mathbf x)$ at $\mathbf a$ as $f(\mathbf a)+(f_1(\mathbf a),f_2(\mathbf ...
3
votes
3answers
559 views

interval of convergence of $\sum n \exp (-x \sqrt n)$

$$\sum^{\infty}_{n=1} n \exp (-x \sqrt n)$$ How to find the interval of convergence? Obviously, 0 is not in the interval because the series becomes divergent. could you help me?
0
votes
1answer
19 views

Volume 4-dimensional sphere

I'm studying Fubini's Theorem and Change of Variables Theore in class, and one of the exercises from last year exam was calculate the volumen of the 4D sphere. I searched on Internet how can I do that ...
4
votes
1answer
311 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
0
votes
0answers
19 views

How to use the extreme value theorem on a vector valued function?

My task is this: Suppose that $A \subset\mathbb{R}^m$ is closed, bounded and that $\textbf{F}:A\to \mathbb{R}^k$ is continious. Show that $\exists ! \: K\in\mathbb{R}:\: ...
0
votes
0answers
11 views

Use of covariance matrix in noise creation

I read it on Wiki Multivariate Gaussian noise is generalization of one-D univariate normal distribution to higher dimensions. In univariate normal distribution we use standard deviation to specify ...
0
votes
0answers
13 views

All possible paths to evaluate a multi variable limit

Most of the books that I have (H.K Dass) say that (or at least that's what I have understood) for the limit of a multivariable function (say f(x,y) ) to exist the limit along every possible path ...
2
votes
1answer
382 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
0
votes
1answer
10 views

finding the expectation of the MLE for $\mathbf{\Sigma}$ in a multivariate Gaussian

I am trying to find the expectation of the MLE for $\mathbf{\Sigma}$ for the multivariate gaussian. $E(\mathbf{\Sigma}_{ML}) = E\left (\dfrac{1}{N} \sum (\mathbf{x}_n - \mathbf{\mu})(\mathbf{x}_n - ...
0
votes
0answers
38 views

Global extremum when the constraint is not compact?

When the constraint is compact, the function must have both a global maximum and a global minimum somewhere in the constraint. However, if the constraint is not compact, the global extremum may not ...
0
votes
1answer
6 views

Multivariate Normal cdf differentiation respect to dispersion

I am interesting in how to differentiate multivariate normal cdf respect to diagonal elements of covariance matrix (that is, I am interested only in variances). Problem similar to mine has been ...
0
votes
1answer
14 views

Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
0
votes
1answer
27 views

How to integrate an equation with multiple non-independent variables

I'm a little lost with this particular equation, I have three variables which need to be integrated and can't quite wrap my mind to get the correct result. I have this: $$ \frac{dH}{dt}=8\pi ...
1
vote
0answers
9 views

Problem about find the extreme of a function (Multipliers of Lagrange)

Good morning, i have a problem with this: Find the maximum and minimum distances from the origin to the curve $g\left(x,y\right)=5x^{2}+6xy+5y^{2}$ I make this: Function to optimize: ...
3
votes
2answers
730 views

A function whose partial derivatives exist everywhere, but is nowhere continuous?

Consider $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. Unlike functions of one variable, the partial derivatives may exist at a point even though $f$ is not continuous there. I have seen examples where ...
1
vote
0answers
13 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...
1
vote
1answer
39 views

How can I use inverse/implicit function theorem to find a function and its inverse?

My task is this: Let $f:\mathbb{R}^3 \to \mathbb{R}$ be the function $f(x,y,z) = xy^2e^z + z$. Show that there exists a function $g(x,y)$ defined around $\textbf{x} =(-1, 2)$ s.t. $g(\textbf{x}) = ...
0
votes
1answer
14 views

How can I find the limits of this iterated polar integration?

How can compute the area of the triangle whose corners are at the origin, (1,0) and (1,1). I solved this with r integral first but I could not find the correct limits for theta integral first order. ...
1
vote
1answer
60 views

Solving $\int_0^2\int_{y/2}^1 ye^{-x^3}\,dx\,dy$ [on hold]

So I have to solve the integral above but I wasn't really sure how to start? I know it is integrable since it is an integral over a nice boundary and that I can solve it using iterated integrals but ...
0
votes
1answer
27 views

If $h:\mathbb{R}^2 \to \mathbb{R}$ is some function, and $g(r,\theta)=(r\cos{\theta},r\sin{\theta})$, compute the matrix $Dh_{g(r,\theta)}$.

The question says to express $Dh_{g(r,\theta)}$ only in terms of $r$, $\theta$, and the two entries of $D(h\circ G)_{(r,\theta)}$, but I'm not really sure how. Could someone point me in the right ...
1
vote
1answer
39 views

Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...
1
vote
1answer
23 views

find value or prove limit doesn't exist.

Given: Find or prove it doesn't exist: .... My attempts thus far include: I can show that doesn't exist using y=kx and showing path dependancy, but dunno if it's enough to prove that ...
0
votes
1answer
434 views

Solid Tetrahedron with vertices

Use only double integrals to find the volume of the given solid. The solid Tetrahedron with vertices (0,0,0), (0,0,1) (0,2,0) and (2,2,0) I know you plot the points on xyz-plane, but how do you get ...
0
votes
2answers
39 views

Polar equation of the curve y = sinx

I am looking for the polar equation of the following curve given in Cartesian Coordinates. y = sinx Any kind of hint or help is appreciated.
0
votes
1answer
9 views

How do I convert a integral over a region R into a double integral using Green's Theorem?

I have been given an integral ∫(xdx+2xdy) over the region R defined as the region bounded above by y=ex-x+1 and below by y=e^x, and I have been asked to convert it to a double integral using Green's ...
2
votes
3answers
3k views

A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be ...
0
votes
0answers
16 views

Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
0
votes
0answers
19 views

How do i show an iteration using Newtons method in $\mathbb{R}^m$?

My task is this: Let $\textbf{F}:\mathbb{R}^m \to \mathbb{R}^m$. A fixpoint for $\textbf{F}$ is the same as a zero for $\textbf{G}(\textbf{x}) = \textbf{F}(\textbf{x}) - \textbf{x}$. Show that when ...
-1
votes
0answers
19 views

integral of partial derivative for continuity equation

Task in question is deriving velocity parallel to z component in continuity equation. $$ \frac{\partial n}{\partial t} + \left(\nabla \cdot n \textbf{u}\right) = 0 $$ $$ \frac{\partial n}{\partial t} ...
0
votes
0answers
8 views

Concave or convex function defined on convex set.

I have a question regarding the definition of concave and convex functions for many variables. Both are defined for some convex set. I am wondering what happens for nonconvex sets. If anyone can help ...
0
votes
3answers
39 views

Implicit Differentiation - What am I doing wrong?

I need to find $y'$for the following equation: $$ e^{\frac{x}{y}} = x-y $$ Before differentiating I decided to perform a quick rewrite: $$ \begin{align*} e^{\frac{x}{y}} &= x-y \newline ...
2
votes
0answers
19 views

Maximum of a Line Integral of the Vector Field

The problem: Let $S$ be the graph of $z=\sqrt{1-x^2-y^2/2}$. Let $F=<x+y,xy,sin(e^x)>$ be a vector field. To which level curve does the line integral of the vector field attains maximum? How do ...
1
vote
2answers
19 views

Find all Points on the Surface at which the Tangent is Parallel to the Plane

The problem: Find all points on the surface $z=x^3+xy^2$ at which the tangent plane is parallel to the plane $2x+2y+z=0$ So I established $f(x,y,z)=x^3+xy^2-z$ and the normal vector determined from ...
1
vote
1answer
24 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. ...
0
votes
1answer
64 views

Find the mass of the half circle

Find the mass of the half circle that is defined by $x^{2} + y^{2} \le 4$ $(y \le 0)$ if the density at point $(x,y)$ is proportional to its squared distance from the point $(0, -2)$ and the density ...