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
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
46 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
vote
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
53 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
42 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
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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 $𝑥 + 𝑦 - 𝑧 + 2 = 0$ 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 ...
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
23 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
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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
62 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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0answers
18 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
30 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?
0
votes
1answer
17 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 ...
0
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1answer
29 views

Solve system of variables [closed]

How do I solve the following system of variables: \begin{align} I=\left(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}\right) \end{align} My notes go that is ...
1
vote
1answer
54 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 ...
3
votes
1answer
35 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 $\mathbb{R}^3$ ...
0
votes
1answer
23 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 ...
0
votes
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) \\ ...
0
votes
1answer
20 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 ...
0
votes
1answer
26 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 ...
0
votes
1answer
32 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 ...
4
votes
5answers
72 views

To determine if set is open , closed [closed]

Define a function $f : \mathbb{R}^{2} \rightarrow \mathbb{R} $ by $$f(x, y) =\begin{cases}1 & \text{if $xy=0$} \\ 2& \text{otherwise} \end{cases}$$ If $S = \{(x, y): f \text{ is ...
0
votes
1answer
30 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 ...
-1
votes
1answer
39 views

$(2n-1)$-form is closed [closed]

Consider in $\mathbb{R}^{2n}$ differential forms$$\omega = dx_1 \wedge dx_2 \wedge \dots \wedge dx_n\text{ and }\theta = \sum_{n+1}^{2n} (-1)^{j-1}x_jdx_{n+1} \wedge \dots \wedge dx_{j-1} \wedge ...
3
votes
1answer
67 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 ...
1
vote
1answer
22 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 ...
1
vote
0answers
37 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 ...
0
votes
0answers
14 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 ...
1
vote
1answer
15 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 ...
0
votes
0answers
22 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 ...
-1
votes
1answer
20 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.
2
votes
1answer
57 views

Points at which partial derivatives exist and are continuous

Given $ E = (xy : xy \neq 0 ) $ . Let $ f : \mathbb R^{2}\longrightarrow \mathbb R$ be defined by $$f(x,y)=\cases{0,& if $xy=0$\\y\sin\left(\frac{1}{x}\right)+x\sin\left(\frac{1}{y}\right),& ...
0
votes
1answer
50 views

To check differentiability of Multivariable function defined at rationals [closed]

Given $$f(x,y) = \begin{cases} x^{2} + y^{2}, & \text{ if } x, y\in \mathbb Q\\ 0, & \text{ otherwise} \end{cases}$$ How do i check its differentiability at origin ? I was having problem ...
1
vote
2answers
33 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. ...
0
votes
1answer
36 views

Evaluating an integral by changing the order of integration.

Problem: Evaluate $$ \int_{1/4}^1 \int_{\sqrt{x-x^2}}^{\sqrt{x}}\frac{x^2-y^2}{x^2}\, dy\,dx $$ by changing the order of integration. I have divided the region into the following three segments: ...
0
votes
1answer
27 views

Vector Integral

Let ${\bf F} = \langle y, x+2y\rangle$. Calculate $\int_C {\bf F}\cdot {\rm d}{\bf r},$ where $C$ is the upper semicircle that starts at $(0,1)$ and ends at $(2,1)$. In order to calculate this ...
-2
votes
1answer
74 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
58 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 ...