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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23 views

Using polar co-ordinaries on an integral whose domain is a disk not centred at the origin

I need to find the volume of the region in $z > 0$ that lies within the cylinder $x^2 + y^2 = 2x$ and is bounded by the cone $z^2 = x^2 + y^2$. I have been struggling to set up the integral for ...
3
votes
3answers
108 views

Fundamental limit in two variables

Can I write that $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{u\to0}\frac{\sin(u)}{u}$$ and, hence, that $\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=1$? If so, why can I do it?
2
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0answers
44 views
+250

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
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2answers
60 views

How to parametrise the curve [closed]

C : curve of intersection of sphere centered at (1,1,0) and radius sqrt2 and the plane X+Y=2 direction of curve is taken as such that it begins at (2,0,0) goes below the XY plane and then comes to ...
0
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2answers
18 views

Doubt in proving differentiable when both partial derivatives are equal

I had a problem with a step in this: I have to prove that: $|xy|^{\alpha}$ is differentiable at $(0,0)$ if $\alpha > \frac{1}{2}$. In this case both partial derivatives exist and have the ...
1
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1answer
30 views

what is divergence? [closed]

I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To give you a sense of ...
1
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0answers
14 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
2
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1answer
21 views

calculating partial derivatives at $(0,0)$

Let $f:\mathbb R^2 \to \mathbb R$ given by := $$f(x,y) = \begin{cases} 0 & \text{, if xy=0 } \\ 1 & \text{, if xy $\neq$ 0} \end{cases}$$ I've to show that $\partial_1 ...
0
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2answers
29 views

If a continuous function of two variables has finite zero set, then it does not change sign

If $f$ is a continuous function from $\mathbb R^2 \rightarrow \mathbb R$ such that $f(x)=0$ for only finitely many values of $x\in\mathbb R^2$. Can we conclude that $f(x)\leq 0$ or $f(x)\geq 0$ for ...
1
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0answers
27 views

How to parametrize the volume of the intersection of cube and a right tetrahedron?

This is an extension of my previous question. I am trying to find the volume of the region which is the intersection of a cube given by $\vec r_1 = (x,y,z)$, where $$\begin{cases}0 \le x \le 1 \\ 0 ...
0
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2answers
24 views

The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
4
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1answer
39 views

Why can I not combine integrals this way?

Evaluating the triple integral $\int^1_0 \int^{1-z}_0 \int^{1-y-z}_0 \text{dxdydz}$, I get $\frac 16$. Evaluating the triple integral $\int^1_0 \int^1_0 \int^1_0 \text{dxdydz}$, I get $1$. So I ...
0
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2answers
35 views

Proving Multivairble Limit Exists [duplicate]

How do you deal with multivariable limits? We'll use the example $f: \mathbb R ^2 \rightarrow \mathbb R$ $$\lim _{(x,y) \rightarrow (0,0)}\frac{\sqrt{|xy|}}{\sqrt{x^2 + y^2}}$$ The limit doesn't ...
2
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1answer
31 views

Existence of partial derivative

I know how to compute partial derivatives of functions with more than one variable. But how can i assert that the partial derivatives of a given function exist at a point without computing it? ...
0
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1answer
51 views

I need help finding the mean radius of a cylinder

The question is: What is the mean radius $\overline{r}$ from the midpoint of a cylinder of radius $a$ and height $h$ to its boundary surface? Evaluate $\overline{r}$ for $a = h/2 = 10~\mathrm{cm}$. ...
0
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1answer
34 views

Is this vector identity accurate?

Does this identity hold true for vectors $A$, $B$ and the gradient operator? $(\nabla \cdot A)B = (A\cdot \nabla)B + (B\cdot \nabla)A$
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1answer
11 views

Limit equivalence

"Let $f:A\subset\Bbb{R}^n\to\Bbb{R}$ be a function and denote $\Bbb{x}=(x_1,\dots,x_n)$ and $\Bbb{p}=(p_1,\dots,p_n)$. Show the following equivalence: ...
0
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2answers
19 views

(Inequality) $p \cdot (z-x) \leq \frac{a}{R} | z- x| \Leftrightarrow |p|\leq \frac{a}{R}$

I need to solve this inequality: Let $z \in \mathbb{R}^N$ and $a,R > 0$, prove that $$(\forall x\in B_R(z)) \quad p \cdot (z-x) \leq \frac{a}{R} | z- x| \ \Longleftrightarrow \ |p|\leq ...
0
votes
2answers
36 views

Green's theorem exercise

I am trying to solve the following problem: Show functions $P,Q:\mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$ of class $C^1$ that verify $P_y=Q_x$ but $$\int_\gamma P(x,y)dy+Q(x,y)dy \neq 0$$ where ...
2
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1answer
22 views

Doubt on understanding continuity .

Just preparing for my multivariable-calculus exam and wanted to clear these things: I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the ...
0
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0answers
11 views

Unit tangent vector

Let $f:I\to \mathbb R^3$ a vector valued function. When we define the unit tangent vector: $T(t)=f´(t)/||f´(t)||$ , $||f´(t)||\neq 0$ is it neccesary that $f$ is a $C^1$ function? or just ...
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0answers
11 views

Regarding functions from R² to R: continuity and differentiability

Let $f : U \rightarrow \mathbb{R}$ where $U \subseteq \mathbb{R}^2$ is an open set and $P \in U$. I am almost sure the following statements are correct, but please confirm: The only requirement for ...
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2answers
28 views

Showing a function is convex on $x^2+y^2\leq a^2$

This is a question from my assignment about which I have no idea: Let $f(x,y)=\phi(x^2+y^2)$,where $\phi$ is of class $C^2$ ,increasing and concave. Show that $f$ is convex on disk $x^2+y^2\leq a^2$ ...
0
votes
1answer
21 views

Show that $\sum_{i=1}^{n} \Big(\frac{\partial u}{\partial x_i}\Big)^2=|f'(r)|^2$

Can anyone help with this: Let $x\in \mathbb R^n$ and $u=f(r)$,where $r=\|x\|$ and f is differentiable . Show that $\sum_{i=1}^{n} \Big(\frac{\partial u}{\partial x_i}\Big)^2=|f'(r)|^2$ . I can't ...
3
votes
2answers
151 views

Finding the limits of a multivariable function

Given the following function, determine whether the following function is continuous at $(0,0)$ $$f(x,y)=\begin{cases}\frac{x^2y^2}{x^4+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ ...
2
votes
0answers
39 views

Normalize gradient

I want to minimize a function $f \, : \, \mathbb{R}^{N} \, \longrightarrow \, \mathbb{R}$ (with $N \in \mathbb{N}^{\ast}$. In my problem, $N = 315$). I know that $f$ is differentiable on ...
0
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0answers
15 views

Question involving Stokes' theorem

Given a cector field F= (x^2-y^2 , -x^2 + y^2 , z ) S: portion of surface x^2+y^2 -2by + bz =0 whose boundry lies in xy plane . here im to evaluate doule integral of curlF.n dsigma ....from stokes ...
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0answers
32 views

A PDE problem related to the ratio of populations

Let $m>0$ in $\bar{\Omega}$ be a given nonconstant function, where $\Omega\subset \mathbb{R}^n$ is a bounded smooth domain. Then consider the following elliptic modeling problem: $$ \Delta ...
0
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1answer
32 views

Why i got negative value for volume?

I want to find the indicated volumes under the surface $z=\frac{1}{y+2}$ and over the area bonded by $y=x$ and $y^2+x=2$. After sketching the graph for $x=2-y^2$, and $x=y$ i found that $y=0$ and ...
0
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1answer
28 views

Find the point on a parameterized line closest to another line

Let $x_1 = (1, 2, 3)$ and $x_2 = (3, 2, 1)$. Consider the two lines $x_1(s) = x_1 + su_1$ and $x_2(t) = x_2 + tu_2$. $u_1 = (\frac{2}{\sqrt{5}}, \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}})$, $u_2 = ...
0
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1answer
20 views

Parameterizing $y = 2 -\sin \frac{\pi x}{2}$

I am trying to parametrize the part of the curve $$ y = 2 -\sin \frac{\pi x}{2} $$ from (0, 2) to (1, 1). I tried the difficult paramaterization $x=t$ and obtained $$ y=2-\sin \frac{\pi t}{2} $$ ...
0
votes
1answer
42 views

Evaluate $ \lim_{ x \to 0, y \to 0}\frac{x^2+y^2+x+y}{x+y}$

How do you find $$\lim \limits_{x \to 0 , y\to 0}\frac{x^2+y^2+x+y}{x+y}$$or prove that it doesn't exist? I've tried every method I know, but I can't find anything conclusive.
0
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1answer
34 views

How to calculate the angle between 2 vectors in 3D space given a preset function

In my application, I am attempting to connect 2 points in 3d space with a cylinder via a function taking in 2 vectors. I understand that I need the angle to apply to the cylinder. As I understand, I ...
0
votes
1answer
22 views

Finding the curl of a cross product

Let $\mathbf{x}$ be the position vector, $\mathbf{a}$ be a constant vector. I need to show that: $$\text{curl}(\mathbf{a}\times\mathbf{x})=2\,\mathbf{a}$$ The problem is, I keep getting ...
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0answers
23 views

Notation for gradients analogous to partial derivatives

Is there an equivalent of partial differentiation for functions taking multiple vectors as input? I mean the following. If we have a function $f(x,y)$, then a partial derivative is denoted as ...
0
votes
1answer
18 views

Prove Tetrahedron Opposite Vectors add to $0$

I really need help on this problem, I'm in Multivariable Calculus (Calc III) and I just can't solve this. Let $v_1$, $v_2$, $v_3$, and $v_4$ be vectors whose lengths are equal to the areas of the ...
1
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0answers
20 views

Multivariate calculus (Lagrange multiplier)

If we need to use the method of Lagrange multipliers to find extreme values of a function $f(x, y)$ on a triangle-shaped region in $R ^2$ , how many times would we have to run the method? How many ...
0
votes
3answers
68 views

construct a path between (-1,0) and (0,2)

So we are given a region S which is above the x-axis and between the semicircle of radius 1 and 2 centred at the origin. we are asked to construct a path that connect the point (-1,0) and (0,2)..and ...
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1answer
21 views

Verify the divergence theorem for a sphere

Question i cannot work out. I assume you need to get both sides in terms of u and v (parameterized), but im getting pretty confused after completing the first few steps.
0
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0answers
29 views

Directional derivatives in two directions

How can I take a directional derivative in two directions? I mean,$$D_{xy}f(0,0)$$ Because when I have something like $$D_{x}f(0,0),$$ I just use that my direction is in the x axis, $ \vec ...
1
vote
2answers
40 views

How do I find this partial derivative

I have the following function u(x,y) defined as: $$u(x,y) = \frac {xy(x^2-y^2)}{(x^2+y^2)}$$ when x and y are both non zero, and $u(0,0)=0$ I want to compute its partial derivative $u_{xy}$ at ...
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0answers
30 views

Computing volume with triple integrals

I'm confused with this problem. Determine the volume of the solid limited for $x = 1-y$, $x = 3-y$, $y = 0$, $z = 0$ and $z = 1-y^2$. What I tried to do: well, first I suppose that the function I ...
0
votes
0answers
17 views

Composition of functions in vector form

Is H equal to the matrix multiplication of G*F How do I use the chain rule to calculate H'? G'(F(x))*F'(x)? How does this work in practice for matrices? For (b) I will compute the matrix of ...
0
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0answers
147 views

Find a scalar potential of $v(x, y)= g(y/x)(-1/x, 1/y)$

Let $v(x, y)= g(y/x)(-1/x, 1/y)$ be a vector field on $Ω$ where $Ω := [(x, y) : x > 0, y > 0]$. $g:\mathbb{R}→\mathbb{R}$ is continuous. Find a scalar potential of $v$ in terms of an ...
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1answer
33 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
1
vote
2answers
47 views

How to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis?

Given $\int \int dxdy$, I want to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis. I think the limits of integral in $y$ axis are from $y=\ln\left(x\right)$ to ...
0
votes
1answer
36 views

Determining the Euler-Lagrange equations for a minimizataion problem

I'm working on a problem in computer vision and I've ended up trying to minimize the functional $$\int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$ where $\lambda$ is ...
2
votes
1answer
35 views

What is an exact differential?

My book says "A differential expression $M(x, y)dx+N(x, y)dy$ is an exact differential in a region $R$ of the $xy$-plane if it corresponds to the differential of some function $f(x, y)$ defined on ...
1
vote
1answer
24 views

Deriving a high ordered Euler-Lagrange equation.

I've been able to derive the Euler-Lagrange equation for $$\int_a^b F(x,y,y')dx$$ relatively easily by using the total derivative and integration by parts. However, I was unable to apply the same ...
4
votes
2answers
36 views

Show Laplace operator is rotationally invariant

I'm trying to show the Laplace operator is rotationally invariant. Essentially this boils down to showing $$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 ...