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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2
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1answer
33 views

Extending Taylor's theorem from one to several variables

In my calculus class we are dealing with TaylorΒ΄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 ...
0
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1answer
30 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 ...
1
vote
2answers
31 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
vote
1answer
42 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 ...
2
votes
1answer
35 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} ...
-1
votes
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 ...
1
vote
1answer
26 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} ...
1
vote
1answer
33 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
460 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.
0
votes
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 ...
0
votes
2answers
29 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
votes
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: ...
1
vote
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 ...
2
votes
0answers
26 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
votes
2answers
51 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
votes
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 ...
0
votes
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 $F \colon \Omega \to \mathbb R^n$ ($\mathcal C^1$ ...
3
votes
2answers
29 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, ...
0
votes
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: $(dl)^2 = (dx)^2 + (dy)^2$ ...
1
vote
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(π‘₯, 𝑦) = \frac{1}{2} + \frac{π‘₯}{4} + \frac{π‘₯^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) β†’ ...
0
votes
1answer
38 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
votes
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
votes
0answers
56 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 ...
0
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0answers
22 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) ...
1
vote
0answers
44 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 ...
0
votes
1answer
17 views

Show that $\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}$

As the title states, I'm trying to show that $$\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}, $$ where $\hat{v}$ is the unit velocity vector of a particle $a = ...
0
votes
1answer
27 views

Integral on part of n-sphere

Let $S^n$ be the $n$-sphere and $0<c<1$. Show that $$ \int_{\{x \in S^n | c\le x^2_1+x^2_2\}} \ln \left (\frac{1}{\sqrt{1-x_1^2 -x^2_2}}-1\right ) dx < \infty$$ Since we are integrating ...
0
votes
2answers
45 views

Is my proof that empty set is open and R is open correct?

Claim: The empty set is open. Proof. Assume that the empty set is closed. Then, there must be one point such that any point in its ball is not inside of the empty set. However, the empty set has no ...
2
votes
1answer
17 views

Nature of stationary points

I have $$f(x_1,x_2) = 2x^4_1 + 2x_1x_2 + 2x_1 + (1+x_2)^2$$ How can I determine the nature of the stationary points? I know; $$f_{x_1,x_1}(x) = 24x_1^2$$ $$f_{x_2,x_2}(x) = 2$$ $$f_{x_1,x_2}(x) = ...
0
votes
1answer
25 views

linear function which does not change the geometry of R^{n}

The linear function is given by $T(\mathbf{x}) = P\mathbf{x}$, where the transpose of $P$ is equal to the inverse of $P$. For any two vectors $x$ and $y$ of $R^{n}$, how can I show that ...
1
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0answers
19 views

Function represented as composition

Question:Prove that if $\vec{g} : \mathbb{R}^n \rightarrow \mathbb{R}^n $ and $ \det(\vec{g}') \neq 0$, then in some open set $V \subset \mathbb{R}^n $ such that $\vec{x} \in V$ we have: $\vec{g} = ...
-1
votes
1answer
11 views

Finding the stationary point of a multivariate function

For $f (x) := 2x^4_1 + 2x_1x_2+ 2x_1 + (1 + x_2)^2$ what are the stationary points? $\nabla f(x,y) = \langle f_{x_1}(x_1x_2), f_{x_2}(x_1x_2) \rangle $ $\nabla f(x,y) = \langle 2(4x_1^3 + x_2 + 1), ...
1
vote
2answers
52 views

Integral over the unit ball

This question has been asked before, but I did not understand it, so I worked on it on my own and got stuck. Any help would be appreciated. Let $A$ be the region in $\Bbb R^2$ bounded by the curve ...
2
votes
1answer
39 views

Show that $F(x)=(x,f(x))$ is differentiable where $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ is.

Show that $F(x)=(x,f(x))$ is differentiable where $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ is and show the matrix $dF(x)$. This is an exercise of my homework but I'm insecurity with this. So a ...
2
votes
1answer
45 views

$y'' - y' = e^x$ (Variation of Parameters)

I've solved multiple differential equations in this practice set, and even a few with variation of parameters, but no matter how many times I restart this problem I can't get it. I must be doing ...
1
vote
2answers
26 views

Using eigenvalues of a hessian matrix vs D operation to classify critical points.

Having recently covered using the discriminant, $D(x_0,y_0)$, for classifying critical points of equations of two variables. For example: $$R(x,y)=-x^2+4x+2xy+8y-2y^2$$ to find that $(6,8)$ is the ...
0
votes
1answer
4 views

Bounds of a Bivariate Function

I am given that $h(x, y) = \frac{x}{(x+y)}$ , $x > 0$ , and $y > 0$. I am supposed to deduce that the bounds for $h(x, y)$ are $0 < h(x, y) < 1$, but I do not understand how to arrive at ...
0
votes
0answers
33 views

Point of intersection closest to the origin

How do I find the point of intersection of $π‘₯ + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = π‘₯^2 + 𝑦^2$ that is closest to the origin? I know I have to use the LaGrange multiplier in order to minimize the ...
1
vote
2answers
48 views

Why S=B(0;1) is a open set?

If I have a $S=B(0,1)$ usual notation for Ball with center at $0$ and with radius $=1$, then it is an open set in $\mathbb{R^2}$. My book explains that every point of $S$ is the center of a circle ...
3
votes
1answer
53 views

Approaching $\infty$ in $\mathbb R^n ; n=2$ or higher.

say I have a double limit in the sense of having a function from $\mathbb R^2 \rightarrow \mathbb R$ in which there are two variables approaching infinity:. $$\lim_{n,m \to \infty} f(m,n) $$ I am ...
1
vote
1answer
12 views

Finding the conic section given equations of double cone and plane

Given the function of a double cone and a plane, how do we find the intersection between them? Suppose the equation of the cone is $f(x, y, z) = 0$ and the equation of the plane is $h(x, y, z) = 0$. ...
0
votes
1answer
19 views

Linear transformation of variable under the integral sign. Easy change of variables question

I realize this might be a basic question, but I need a sanity check. Let $f(\vec{x})$ be a function that takes $n$-dimensional vectors and returns a real number. Suppose the goal is to compute ...
0
votes
0answers
41 views

Finding $\lim\limits_{(x,y,z) \to (1,2,-3)} \arctan\left(\frac{x+z}{y}\right)$

this is a homework problem, so I am just looking for a hint to get me going in the right direction. I am asked to find the following limit and prove my result, or to show that the limit does not ...
1
vote
2answers
50 views

Triple Integral of $1/\sqrt{2 + x^2 + y^2 + z^2}$ over unit sphere

I'm studying triple integrals (physics major), and I'm having trouble solving this little beast: $$ \iiint_V \frac{1}{\sqrt{2 + x^2 +y^2 +z^2}} \,dx\,dy\,dz$$ where V is $$x^2+y^2+z^2=1$$ Of ...
0
votes
2answers
57 views

How do I calculate $ \iiint_D|z|\,dx\,dy\,dz$ without using spherical coordinates?

I have the following integral: $$ \iiint_D|z|\,dx\,dy\,dz $$ which I need to integrate over the set: $$ D = \{x,y,z \in \mathbb{R}: x^2 + z^2 \leq y^2, y^2 \leq 4 \} $$ I have a problem ...
1
vote
0answers
17 views

Tricky vector derivatives

If $n_i=n_i(x_1,x_2)$ are the components of a unit vector ($\sum_i n_in_i =0$), and $i=1,2$, I know that $$\sum_in_i\nabla_jn_i=\sum_i \frac{1}{2}\nabla_j(n_in_i)=0$$ If $\nabla_i := ...
0
votes
2answers
19 views

Lagrange method over two constraints

plane $x+y-z=-2$ intersects $z^2=x^2+y^2$ I need to use Lagrange multipliers to determine the point of intersection which is the closest to the origin. As far as I understand, to use Lagrange I need ...
1
vote
0answers
30 views

Prove that $f$ is differentiable at $\underline{0}$.

Let $f:\mathbb{R}^n\to\mathbb{R}$. Lets assume that for every differetiable curve $\gamma:[-1,1]\to\mathbb{R}^n$ where $\gamma(0)=\underline{0}$, $f\circ\gamma[-1,1]\to\mathbb{R}$ differentiable at ...
-1
votes
0answers
29 views

Surface and Volume of revolution of a cardioid about axes

please find the surface and volume of revolution of the cardioid $r=1-\cos(\theta)$ about the axes (polar axis & vertical axis). Regards Yegan
1
vote
1answer
42 views

Calculate the area enclosed in $(x^2+y^2)^5=x^2y^2$

Calculate the area of the plane contained within the curve $$(x^2+y^2)^5=x^2y^2$$ Any suggepstion please?