Tagged Questions

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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-3
votes
2answers
63 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
votes
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
vote
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
votes
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
votes
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
vote
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
votes
2answers
29 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
votes
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
votes
2answers
37 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
votes
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
votes
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
votes
1answer
34 views

Is this vector identity accurate?

Does this identity hold true for vectors $A$, $B$ and the gradient operator? $(\nabla \cdot A)B = (A\cdot \nabla)B + (B\cdot \nabla)A$
0
votes
1answer
12 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
votes
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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
156 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
40 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
votes
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 ...
0
votes
0answers
34 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
votes
1answer
34 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
votes
1answer
29 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
votes
1answer
21 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
votes
1answer
36 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 ...
1
vote
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
vote
0answers
24 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
70 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 ...
1
vote
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
votes
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 ...
1
vote
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 ...
1
vote
1answer
34 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
37 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
36 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
25 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
38 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 ...
1
vote
1answer
24 views

Gradient vs Conservative vector field: What's the difference?

From the definitions I'm reading between the two: The gradient vector field is defined by its construction: gradient of a scalar (or real) function generally over two or more variables. The ...
0
votes
0answers
19 views

Can I use Fubini's theorem to show that the order of this double integral can be changed?

$\int_0^1\int_{\frac{\pi}{2}}^\frac{3\pi}{2}\frac{1}{1 - ysin^2x}dxdy$ My notes say that I can use Fubini's theorem to interchange the order of the double integral if it is defined over a rectangle ...
1
vote
0answers
32 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
1
vote
0answers
32 views

multivarible calculus-directional derivaties

Let $f$ and $g$ be functions from $\mathbb{R}^n \to \mathbb{R}^m$. Assume that $f$ is differentiable at $c$, that $f(c)=0$, and that $g$ is continuous at $c$. Let $h(x)=g(x)f(x)$. Prove that $h$ ...
3
votes
0answers
12 views

Multiple Integral Substitution Error

I just started learning about the substitution rule for multiple integrals and I decided to give myself an example problem: Calculate $\iint_R{(x^2 + y^2)dA}$ with $R = \{(x, y) \in \Bbb{R} \ |\ 0 ...