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
15 views

Tangent plane and normals in $\mathbb{R}^2$

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function given by $$z = f(x, y) = x^4 + y^4.$$ Find the point on the surface $z = f(x, y)$, where the normal to the surface is perpendicular to the chord ...
0
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1answer
12 views

Inconclusive second derivative test at (0,0) for $x^{4} + y^{4} - 2x^{2} - 2y^{2} +4xy $

Second derivative test is inconclusive here , given f( x, y) is $x^{4} + y^{4} - 2x^{2} - 2y^{2} +4xy $ At (0,0) how do i check nature ? Also i would like to know general tactics when things like ...
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0answers
39 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
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0answers
26 views

Prove that exists $\delta>0$ such that, if $(x,y)\in S$ satisfies $\lVert(x,y) \rVert < \delta$, then $f(x,y) \leq f(0,0)$.

This exercise appeared on my Calculus II exam, and I didn't know even how to start doing it. Any hint is appreciated. Let $\ f, \ g : \mathbb{R^2}\to \mathbb{R}$ two $C^2$functions over the plane. ...
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1answer
22 views

Is every point of rational number boundary point?

While studying first chapter of multivariable calculus, I am wondering if every point of the rational number is boundary point. It is obvious that $\Bbb{R}^n$ is the union of interior, exterior, ...
1
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1answer
15 views

Expression defined by exponential random variables, probability of being nonnegative

Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle ...
2
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1answer
25 views

Volume of a sphere with two cylindrical holes.

Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the ...
4
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2answers
38 views

Find that the limit is $0$

I have to prove that the following limit is $0$: \begin{equation} \lim_{(x,y)\to (0,0)}\frac{\lvert x\rvert^2y^2}{x^2+y^4}=0. \end{equation} This is a part of an exercise where I have to study the ...
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0answers
34 views

Integration: Step in paper unclear

I've seen in a paper the following step: $$2\operatorname{Re}\int_{\mathbb{R}^n} r \partial_r \bar u \Delta u \, dx=(n-2)\int_{\mathbb R^n} |\nabla u|^2$$ This is not clear to me as I calculated: ...
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1answer
24 views

If a continuous function is nonzero at a point $a$, there is a ball around $a$ in which it has the same sign as $f(a)$

Let $f$ be a scalar field continuous at an interior point a of a set $S\in \mathbb{R}$. If $f(a)\ne 0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The ...
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1answer
23 views

An unusual “multivariate Gaussian integral” that comes up when trying to translate results about a standard Gaussian to the general case

I am trying to solve this question and it leads me to a strange looking integral that I do not know how to solve. Let $\Sigma$ be positive semidefinite, and $1>\lambda>0$. I am not certain I am ...
1
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1answer
18 views

Trick to finding length of parametric curve

I was giving the parameters of the curve: $x = 2cos(2t)$ $y = 2sin(2t)$ and $z = 1$, where $ 0 \leq t \leq 10 \pi$ This curve describes a cylinder in the $z$ direction, and seems very straight ...
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0answers
21 views

Understanding the Jacobian past calculus

What's taught in calculus: In the calculus of multiple variables I learned that the Jacobian $$\textbf ...
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0answers
9 views

Continuous scalar field at an interior point of S and same sign proof.

Let $f$ be a scalar field continuous at an interior point $a$ of a set $S \in R$. If $f(a)$ is not $0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The above ...
1
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1answer
26 views

Double integral of $e^{3+y^2}$ over a triangle

Evaluate $\iint_{A}^{} e^{3+y^2}dxdy$ where $A$ is a triangle with vertices $(0,0)$, $(0,-1)$ and $(1,-1)$. I don't know how to bite that. I tried multiplying it by $e^{x^2}$ and then changing the ...
2
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1answer
45 views

Problem with Lagrange multipliers

I am asked to find local extrema of $f(x,y,z)=ax+by$ ($a,b$ non-zero and fixed) defined on $\{(x,y,z)\colon (x,y)\neq 0\}$ subject to $$\left (R-\sqrt{x^2+y^2}\right)^2 + z^2 - r^2 = 0.$$ (here ...
1
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1answer
55 views

Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make ...
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0answers
11 views

Finding 1st,2nd and 3rd derivative for funtion of 2 variable

$E=g(p,v)$ $\frac{dp}{dv}=F$ $\frac{dE}{dv}$=$g_pF+g_v$ $\begin{align}\frac{d^2E}{dv^2}&=(g_pF+g_v)_pF+(g_pF+g_v)_v \\ &=g_{pp}FF+g_pF_pF+g_{vp}F+g_{pv}F+F_vg_p+g_{vv} \end{align}$ ...
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2answers
15 views

Why does the slope of a smooth simple closed curve have winding number one?

$\def\RR{\mathbb{R}}$Let $S^1$ be the circle and let $\gamma : S^1 \to \RR^2$ be a smooth injective map with $\gamma'(t)$ everywhere nonzero. What is the easiest way to show that $t \mapsto ...
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3answers
41 views

integrals calculation got wrong with the extra 2

Given $$ f(x, y) = \begin{cases} 2e^{-(x+2y)}, & x>0, y>0 \\ 0, &otherwise \end{cases}$$ For $ D: 0 <x \le 1, 0 <y \le2$, I'm trying to calculate this $$ \iint_D f(x,y) \, dxdy ...
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0answers
14 views

Simplification of integral region (no integration skills needed)

We have the following "formula" or simplification for integrals: Let $f_i:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for $i=1,\dots,n$ and $g_j:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for ...
2
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3answers
53 views

How to draw a contour map of $ f(x,y)=x^2+y^2+xy$

I have used a program to see that it is an ellipse but I want to know the process of thinking to actually draw the contour map myself. $x^2+y^2+xy=C$ for $C=0,1,2,3,...$ I can't seem to get it into ...
1
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1answer
29 views

Is $f(x,y)=-xy$ neither concave nor convex?

Is $f(x,y)=-xy$ neither concave nor convex? I used the definition for first differentiable functions and determined it depends on the choice of points, hence it is neither.
2
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1answer
35 views

Change of variables in multi-variable calculus?

About the last equality, I know it is change of variables. Let $\xi=x+t,\eta=-x+t$, but I don't know how to get the integration domain? I have been thinking for an hour and I can't get the ...
2
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1answer
32 views

Extending Taylor's theorem from one to several variables

In my calculus class we are dealing with Taylor´s theorem in several variables. When we were looking at the function $f(x,y)=\sin(xy)$ my teacher said that instead of applying the theorem in several ...
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1answer
29 views

three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
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2answers
27 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
1
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1answer
41 views

Let $f$ and $g$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $|f(1)-f(0)| \le g(1)-g(0)$

Let $f:[0,1] \rightarrow \mathbb{R}^m $ and $g:[0,1] \rightarrow \mathbb{R}$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $$|f(1)-f(0)| \le g(1)-g(0)$$ Comments ...
1
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1answer
32 views

To check if (0,0) is local minima for$F ( x, y) = x (x - 2y^{2}) $

Hello Thanks for your time $F ( x, y) = x (x - 2y^{2}) $ . I have applied second derivative test which does not give any result . By looking at function i see that when x is greater than $2y^{2} ...
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0answers
16 views

second derivative fail, classify the nature of the critical point

f(x,y,z)= $\frac{(x+y)^2}{2}+z^3$ the critical point I calculated is span{(1,-1,0)} the eigenvalue of the Hessian of point (1,-1,0) are 0,0,2, which means that this point is degenerate and the ...
0
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1answer
22 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
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0answers
14 views

Restricted Boltzmann Machine Derivation

From book chapter, RBM probability is shown as $$ P(x,h;W) = \frac{1}{Z(W)} \exp \bigg[ \frac{1}{2} y^T W y \bigg] $$ wnere $y \equiv (x,h)$ The book mentions after maximum log likelihood, he ...
1
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1answer
31 views

How to guess that $f(x,y)$ has no limit?

I need to determine if the limit as $\mathbf{x}\rightarrow \mathbf{0}$ exists for the following functions: ($f:\mathbb{R}^2 - \{(0,0^T)\} \rightarrow \mathbb{R}$) $f(x_1,x_2) = ...
4
votes
4answers
449 views

Find the volume of the set.

Let $$S=\{x=(x_1,x_2,\cdots,x_n)\in \Bbb{R}^n:0\le x_1\le x_2\le \cdots \le x_n \le 1\}$$ Find the volume of the set $S$. I tried writing it as a multiple integral but it got complicated.
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1answer
26 views

'Meaning' of a triple integral with $f(x,y,z)\neq 1$

I'm studying for my Calculus II exam, and this question came to my mind while I was practising integals with spherical coordinates. Probably this question doesn't have sense at all, but there's a ...
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2answers
28 views

Gradient of dot product of two vectors

I am taking a class in which knowledge of gradients is a prerequisite. I am familiar with gradients but don't have too much experience, so I am having trouble understanding the following example. ...
0
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1answer
16 views

The Point of Tangency Between a Sphere and a Tangent Plane

Find the equation of the sphere centered at (2,0,-3) that is tangent to the plane x=y. What is the point of tangency? Describe the interior of the sphere with an inequality. What I have thus far: ...
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0answers
35 views

What is the intersection between $x + y - z = -2$ and $z^2 = x^2 + y^2$

I got the answer as $4x + 4y + 2xy + 4 = 0$ by substituting $z = x + y + 2$ into the second equation, but I feel as this is wrong since I am missing $z$ in the function. How do I approach this ...
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0answers
18 views

Lagrange multiplier over two constraints

I'm having two constraints $g_{1}$=x+y-z+2=0 and $g_{2}$=$z^{2}$-$x^{2}$-$y^{2}$=0 and I want to determine the point on the intersection which is closest to the origin. The question asks us to use ...
0
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2answers
49 views

How to find the limit $ \lim_{(x,y)\to (0,0) }\frac{\sin(x^2+9y^2+|x|+|3y|)}{|x|+|3y|} $?

How to find this limit? $$ \lim_{(x,y)\to (0,0)}\frac{\sin(x^2+9y^2+|x|+|3y|)}{|x|+|3y|} $$ I considered approaching along the sequence $\{(1/n,0)\}$; then the limit is $1$.
2
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0answers
24 views

Differentiating Under the Integral

This problem has been giving me trouble-perhaps you can help. I met this problem in the context of an exam, so it may be that I am on the right track and what I have done so far is as far as a grader ...
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0answers
7 views

Divergence theorem and almost everywhere smooth boundary

Let $\Omega \subset \mathbb R^n$ be an open set whose boundary is almost everywhere regular and oriented ($\mathcal C^2$ class). For each vector field $F \colon \Omega \to \mathbb R^n$ ($\mathcal C^1$ ...
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2answers
25 views

If $x$ is a boundary of $S$ in $\mathbb{R}$, then $x$ must contain both interior points and exterior points of $S$

Above is the statement that I am given to prove or disprove. I think it is false. For $Q$ a rational number, there is no interior point nor exterior point. so every point in $Q$ is boundary point, ...
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0answers
14 views

How can the surface integral (for a surface defined with parameters) be derived without using vectors?

It is possible to derive the arclength analytically by using the Pythagorean theorem: given a curve y(x), infinitestimal length dl along the curve can be given as: $(dl)^2 = (dx)^2 + (dy)^2$ ...
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1answer
27 views

Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → ...
0
votes
1answer
37 views

Can interior set or exterior set be empty?

I am trying to prove or disprove that if x is a boundary of S in R, then every ball B(x) contains both interior point of S and exterior point of S. I am trying to think of counter example, and one ...
0
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0answers
13 views

Differentiation of an inclination function

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. Define $F \colon \mathbb{R}^2 \to \mathbb{R}$ by $$F(x,y) = \begin{cases} \frac{f(x)-f(y)}{x-y} & x\neq y \\ f'(x) & ...
2
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0answers
53 views

Multivariable Calculus or Differential Geometry (Analysis on Manifolds) after single variable calculus

Background: Applied Mathematics program, finished with single variable calculus, and in parallel with basic analysis. (Not knowledge of multivariable calculus yet) Please feel free to recommend ...
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0answers
21 views

Use Frenet Frame and Pythagorean Theorem.

Suppose we have a curve $c(t)$ where $t$ goes from $a$ to $b$. $c$ has positive curvature and a frenet frame(TNB). Choose $\rho > 0$ and small. and define: $f(t) = c(t) +\rho B(t)$ $g(t) = c(t) ...
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0answers
42 views

Basic Differential geometry: Shortest path between two points in R^3 is straight.

Given two points P and Q in $\mathbb{R^3}$, we want to show that the shortest distance between them is through a straight line. let $c(a) = P$ and $c(b) = Q$ and $c(t)\neq P$ for $t>a$(One ...