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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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 ...
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1answer
28 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 ...
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1answer
18 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 = ...
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2answers
125 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
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1answer
28 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) = ...
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1answer
30 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 ...
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0answers
24 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} = ...
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1answer
68 views

If every composition of a differentiable path and a function is differentiable at 0, means the function is differentiable at 0

I'll write the question more formaly: Let $f :\mathbb{R^n} \rightarrow \mathbb{R}$ a certain function. Assume that for every differentiable path $p: [-1,1] \rightarrow \mathbb{R^n}$ so that $p(0) = 0 ...
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2answers
101 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 ...
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1answer
18 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), ...
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2answers
72 views

Improper integral (is it convergent?)

I would like to either prove or disapprove the following: Let $\alpha\in (-1/2,0)$ be given. Then we can find $\gamma \in (1,2)$ such that $$\int_0^1 \int_0^{u} ...
2
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1answer
46 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 ...
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0answers
47 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 ...
2
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1answer
122 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 ...
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4answers
2k views

Why does dust gather in corners?

I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is? Edit: No, really, this is a ...
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1answer
10 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 ...
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0answers
42 views

Point of intersection closest to the origin

How do I find the point of intersection of $𝑥 + 𝑦 - 𝑧 + 2 = 0$ and $𝑧^2 = 𝑥^2 + 𝑦^2$ that is closest to the origin? I know I have to use the LaGrange multiplier in order to minimize the ...
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2answers
55 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 ...
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1answer
66 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 $h(x, y, z) = 0$. ...
3
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1answer
54 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 ...
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1answer
41 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 ...
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2answers
70 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 ...
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1answer
54 views

vector component form from symmetric equation

I'm working through some practice problems in one of my math textbooks, and I'm told to find both the parametric and symmetric equations of the line through $(1,-1,1)$ and parallel to the line $ x + 2 ...
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2answers
24 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 ...
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0answers
32 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 ...
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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?
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2answers
75 views

Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
3
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1answer
53 views

Proving that something is a manifold from the definition

Consider a set $$M = \{ (s\cos t, s\sin t, t) \colon s,t\in \mathbb{R}\}\subset \mathbb{R}^3.$$ I am asked to show from the definition that $M$ is a 2-dimensional submanifold of $\mathbb{R}^3$ ...
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1answer
24 views

Find a polynomial $Q$ of degree $k$ and a remainder function $E$ for $f(x)=\frac{1}{1-x}$.

There is a theorem in our textbook saying that rather than calculating all the derivatives needed to compute the taylor polynomial, if one can find, by any means, a polynomial $Q$ of degree $k$ such ...
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2answers
217 views

do Carmo Differential Geometry Exercise 4.4.20 [closed]

Let $T$ be a torus of revolution which we shall assume to be parametrized by$${\bf x}(u, v) = ((r\cos u + a)\cos v, (r\cos u + a)\sin v, r \sin u).$$Prove that If a geodesic is tangent to the ...
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1answer
34 views

Minimum value of $F(a,b)$.

Let $$F(a,b) = \sum_{i=1}^n \left[ y_i - (ax_i+b) \right]^2$$ Find the minimum of $F$. Evaluating the dirctional derivatives: $$\frac{dF}{da} = \sum_{n=1}^n 2\cdot (y_i - (ax_i+b))(-x_i) \\ ...
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1answer
56 views

Using Stoke's theorem evaluate the line integral $\int_L (y i + zj + xk) \cdot dr$ where $L$ is the intersection of the unit sphere and x+y = 0

Evaluate $$\int_L (y i + zj + xk) \cdot dr$$ where $L$ is the intersection of the unit sphere and $x+y = 0 $ traversed in the clockwise direction when viewed from $(1,1,0)$. My attempt: $∇ \times A ...
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1answer
21 views

Finding the general solution of a 2nd order ODE?

SO here's a problem that I'm not having much progress with: Using substitution $u=cosx$, how can I find the general solution of $sinx(d^2y/dx^2)-cosx(dy/dx)+2ysin^3x=0$ Thank you so much for ...
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1answer
33 views

Simple question about $\nabla f(\mathbf x).(\mathbf y - \mathbf x)$

For the function $f:\mathbb R^n\rightarrow\mathbb R$, why if $\nabla f(\mathbf x)\cdot (\mathbf y - \mathbf x)\le 0$ for all $\mathbf x$, then $\mathbf y$ maximizes $f(\cdot)$? I know $\nabla ...
3
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1answer
101 views

Exterior derivative of a 2-form

I want to prove that the exterior derivative of a 2-form in $\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial ...
0
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1answer
32 views

If $\|Df\|<M$ in a disk, then $M$ is a Lipschitz constant

Some notes on multi-variable calculus I was reading, they quote a "standard result": Suppose $f:\mathbb{R}^n \to \mathbb{R}^n$ is such that $\| f'(x)\| \leq M$ for $x\in D$, where $D$ is a ...
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1answer
110 views

Proof of Multivariable Implicit Differentiation Formula

If the equation $F(x,y,z)=0$ defines $z$ implicitly as a differentiable function of x and y, then by taking a partial derivative with respect to one of the independent variables (in this case x), you ...
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1answer
23 views

Show the $i$-th row of $D_f$ is $\nabla f_i$

Let $f:\mathbb{R}^m\to\mathbb{R}^n$. Show that the $i$-th row of the differential, $D_f$ is the gradient of $i$-th function, $\nabla f_i$ I understand it intuitively, because I know that ...
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0answers
44 views

Continuity of the function $f$ defined by $f(x,y)=1$ if $xy=0$ and $f(x,y)=2$ otherwise

Define $f:\mathbb R^{2}\to \mathbb R$ by $$f(x,y)=\begin{cases} 1 &\text{ if } xy=0\\2 &\text{ otherwise } \end{cases}$$ Let $S$ be the set of all those points of $\mathbb R^{2}$ at which ...
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6answers
7k views

Why do we need vectors and who invented it?

It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what? EDIT: As George Lowther pointed out, the problem is too broad; I added the ...
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0answers
20 views

Interpretation of Partitions of Unity

this is my first post. I have been working through Spivak's Calculus on Manifolds and have finally arrived at a section devoted to partitions of unity. Up until now, I have not had very much trouble ...
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1answer
54 views

Vector flux through a segment of a sphere

Given the vector field $\vec A(\vec r) = \vec r$, I have to calculate the vector flux through a sphere whose center is located in the origin. I want to apply Gauß-Theorem and use spherical ...
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0answers
24 views

Area of the surface $S=\{(x,y,z)\mid z^2=x^2+y^2,0\leq z\leq\sqrt{x}\}$

I want to compute the area of the surface $S=\{(x,y,z)\mid z^2=x^2+y^2,0\leq z\leq\sqrt{x}\}$. Is the following attempt correct? I think a parametrization of the surface $S$ can be as follows: If ...
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1answer
40 views

multivariable calculus charge density question

The sphere given by $x^{2} + y^{2} + z^{2} = 4$ is submerged in an electric field with charge density given by $f(x, y, z) = x^{2} + y^{2}$. Find the total amount of electric charge on this surface. ...
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0answers
62 views

Where is the curvature minimal?

Where is the curvature minimal? Now I have found where the curvature is maximum. How ever I don't know how to solve for the minimum. $$\vec{r}(t)=2\cos(t)\vec{i}+3\sin(t)\vec{j}$$ I used the ...
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1answer
27 views

Taylor expansion for two-variable function.

Expand the function $ f (x, y) = e ^ {x-2y} $ in a Taylor series at the point $ (- 1,2) $. Please help me with it. I don't know how to do it although I did try to do it.
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1answer
93 views

Basic Initial Value Problem

Given the initial value problem $$x''+4x=0, \qquad x(0)=1, x'(0)=4$$ (a) Find the matrix $A$ for which $\begin{bmatrix}x'\\x''\end{bmatrix} = A \begin{bmatrix}x\\x'\end{bmatrix}$. (b) Find ...
3
votes
2answers
79 views

Green's Theorem; computing a double integral

This is the last part of an exercise in Apostol Vol. II. (p.385, 1 (e), to be precise.) No doubt there's a trick I'm missing, because evaluating the double integral over the region involved seems ...
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2answers
37 views

To determine whether range of f is closed , connected etc

Let $E= \{ (x,y) : |x| + |y| \leq 1 \}$ . Define $f : E \to \mathbb R$ by $f(x, y) = x + y / 1 + x^{2} + y^{2} $ Then range of $f$ is A . Connected open set B . Connected closed set C. ...