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).

learn more… | top users | synonyms

0
votes
1answer
44 views

Multivariable Calculate $\int\int_D(x^2 + y^2 )dx dy$

Calculate Double integral $$\iint_D (x^2 + y^2 ) dxdy$$ where: $$D=\{(x,y)\in\mathbb{R}^2 : x\le x^2+y^2\le2x, -x\le y \le x \}$$ What i did? I tried to use polar coordinates and i got this ==> $\...
2
votes
0answers
47 views

Question related to Milnor's proof of the hairyball theorem

I am trying to understand Milnor's proof of the Hairy Ball Theorem (here's the proof I am trying to read: http://people.ucsc.edu/~lewis/Math208/hairyball.pdf). In lemma 1, he first considers the case ...
0
votes
0answers
20 views

Double integral $\sin(x+y)$ between boundaries

$R=[0,{{\pi}\over 2}] \times [0,{{\pi}\over 2}]$ Justify that $0\le\iint_R\sin(x+y)dA\le{\pi^2 \over 8}$ But, $\iint_R\sin(x+y)dA = 2\gt{\pi^2 \over 8}$ It asks the volume of $\sin(x+y)$ over the ...
3
votes
1answer
45 views

How can we evaluate the following limit?

How can this problem be solved? $$ \lim_{(n,r) \rightarrow (\infty, \infty)} \frac{\prod\limits_{k=1}^{r} \left( \sum\limits_{i=1}^{n} i^{2k-1} \right)}{n^{r+1} \prod\limits_{k=1}^{r-1} \left( \sum\...
0
votes
1answer
42 views

Line Integral of an unknown curve?

Solved: This is a case where the Fundamental Theorem of Line Integrals can be applied. I am confused about the wording of this question: Evaluate $\int_C\ (xy^2+1)\,dx\, +\ x^2y\,dy$ where $C$ is an ...
2
votes
1answer
54 views

How does one calculate the boundary of a set?

I am trying to understand more advanced calculus theorems like the Reynold's transport theorem which require one to find the boundary of a set. Is there a systematic way to calculate those boundaries? ...
3
votes
2answers
94 views

Must $\vec{n}$ be a Unit Normal Vector (Stokes' Theorem)?

If $S$ is an oriented, smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation, then for some vector field $\vec{F}$: $$\oint_C \vec{F} \cdot d\vec{r} =...
1
vote
1answer
33 views

Double and Triple Intergral

This may be a silly question but I have trouble grasping this very basic concept. In equations, sometimes we have $$ \int_0^\pi \int_x^\pi \frac{\sin y}{y} dydx$$ function given as f(x,y). And ...
1
vote
1answer
35 views

How do I evaluate the line $\int F \cdot dr$ when $F = (5xy^3)i + (3x^2y^2)j$?

On the curve C consisting of the x-axis from x=0 to x=4, the parabola $y=16−x^2$ up to the y-axis, and the y-axis down to the origin. I can't seem to get the right answer. Since the vector field is ...
1
vote
1answer
45 views

Find mass and moment of inertia using triple integration

A solid lies inside the cylinder $r=2$, within the sphere $$x^{2}+y^{2}+z^{2}=16,$$ and above the xy-plane. The density at a point $P$ is directly proportional to the distance from the xy-plane. Find ...
0
votes
0answers
21 views

Fourier transform of a scaled function.

Related to: Question concerning the differentiation of a scaled function. I am considering the homogeneous Sobolev norm of a function $u_{\lambda} = \lambda^{\alpha} u (\lambda^{\beta}t, \lambda^{\...
0
votes
1answer
36 views

Multivariable Calculus - Chain rule - Partial derivatives computation

Suppose you are given that $f$ is a differentiable function of two variables $u$ and $v$. Let $g(r,s)=f(r^2-s^2, 2rs)$. Compute $\nabla g(r,s)$ in terms of $\partial f\over\partial u$, $\partial f\...
0
votes
1answer
28 views

Partial Derivatives of Compositions of Functions

Suppose the variables $x, y, z$ are related by $z=f(x,y)$ and $y=g(x,z)$. Show that $$\frac{\partial f}{\partial y}\frac{\partial g}{\partial z}=1$$ Intuitively this makes sense. By abusing notation, ...
0
votes
0answers
7 views

Question concerning the differentiation of a scaled function

Consider $u_{\lambda}(t,x) = \lambda^{\alpha} u (\lambda^{\beta}t,\lambda^{\gamma} x)$, for some smooth function $u$, with $0 < \lambda \in \mathbb{R}$. I differentiate this wrt $t$ giving me $$ \...
0
votes
1answer
33 views

Implicit function for a convex gradient.

Let $Q \colon \mathbb{R}^n \to \mathbb{R}$ denote a convex function with $g(x) = \nabla Q(x)$ well-defined. I am interested in defining the following variables \begin{eqnarray} c & = & x - g(x)...
0
votes
0answers
29 views

Evaluate the line integral $\int_C (x^2-y+3z)\;ds$

Evaluate the line integral $\int_C (x^2-y+3z)\;ds$ where C is the line segment from $(0,0,0)$ to $(1,2,1)$ So I did the following: $$r(t)=(1-t)\langle 0,0,0 \rangle+t\langle1,2,1\rangle$$ $$r(t)=\...
0
votes
1answer
24 views

Convert $\int^{2}_{-2}\int^{\sqrt{4-x^2}}_{-\sqrt{4-x^2}}\int^{4}_{x^2+y^2}\;x\;dz\;dy\;dx$ to cylindrical or spherical coordinates and evaluate.

Convert $\int^{2}_{-2}\int^{\sqrt{4-x^2}}_{-\sqrt{4-x^2}}\int^{4}_{x^2+y^2}\;x\;dz\;dy\;dx$ to cylindrical or spherical coordinates and evaluate. To do this I first evaluated the innermost integral ...
0
votes
2answers
43 views

Finding extrema with Lagrange multipliers

I'm trying to find the extrema of $f(x,y)= \cos(x^2-y^2)$ constrained to $x^2+y^2=1.$ Using Lagrange Multipliers I get this far: $-x(\sin(2x^2-1)=\lambda x$ $-y(\sin(-2y^2+1)=\lambda y$ But I don't ...
6
votes
1answer
74 views

If a separately continuous function $f : [0,1]^2 \to \mathbb{R}$ vanishes on a dense set, must it vanish on the whole set?

Assume $f(x,y)$ is defined on $D=[0,1]\times[0,1]$, and $f(x,y)$ is continuous of each separate variables(i.e. if we fix $y$ to $y_0$, then $f(x,y_0)$ is continuous and vice versa). If $f(x,y)$ ...
0
votes
1answer
15 views

How would you use cylindrical polar coordinates to find the area of a cone (and why does my method not work?

The following question was recently asked in a lecture: Using cylindrical polar coordinates find the area of the curved surface of a cone of height $h$ and radius $a$. My attempt to do this was ...
0
votes
1answer
21 views

Showing that $\varphi_{i_1}\otimes \dots \otimes \varphi_{i_k}$ is a basis for $J^k(V)$(Calculus on manifolds)

Let $V$ be a vector space and $J^k(V)$ be the set of $k-\text{linear}$ maps from $V^k$ to $\mathbb{R}$. Also, let $\{v_1,\dots, v_n\}$ be a basis for $V$ and let $\varphi_1,\dots,\varphi_n$ be the ...
0
votes
0answers
25 views

A $C^1$ function from $\mathbb{R}^n$ to $\mathbb{R}$ is not injective [duplicate]

How to prove that a function $C^1(\mathbb{R}^n,\mathbb{R})$ cannot be injective? I know that $D_1f(x)$ is not $0$ for all $x$ in some open set $A$ of $\mathbb{R}^n$. Can I consider a function $g$ ...
0
votes
0answers
59 views

Parameterization of Ellipsoid

I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed ...
1
vote
1answer
91 views

How to design a closed rectangular box of minimum cost without Lagrange Multipliers

Suppose the box is to be of volume $V_0$ cubic cm; and the cost of material for the front and back sides is $b$ dollars per square cm, $c$ dollars per square cm for the left and right two sides, and $...
1
vote
1answer
34 views

Parametrization of a cylinder that is parallel to x axis

The answer is no it does not matter. The surface is $y^2+z^2=4$, I parametrized it so: $\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$ But Pauls Outline works through the ...
1
vote
1answer
58 views

Mechanics derivation that I don't understand

I am reading the section of method of calculus of variations from Goldstein, where he tries to find a curve for which given line integral has a stationary value. After some steps into the derivation ...
0
votes
0answers
7 views

Moments and Centers of Mass

The moment of inertia is given by equation: $I_L = \lim_{n\to \infty}\sum_{k=1}^n\Delta I_k = lim_{n\to \infty}\sum_{k=1}^nr^2(x_k,y_k,z_k)\delta(x_k,y_k,z_k)\Delta V_k = \iiint_{D}r^2\delta dV.$ ...
0
votes
1answer
33 views

Compute $\int_C xe^y\;dx+x^2y\;dy,\; y=3,\ 0\le x \le 2$

Compute $\int_C xe^y\;dx+x^2y\;dy,\; y=3,\ 0\le x \le 2$ In most of the problems I've done just far they have defined points which helped me to build a position vector. How do I get started with ...
0
votes
0answers
70 views

Triple Integral: Volume between a cylinder and 2 planes

My solution is correct. Thank you. This is a question that I had on one of my midterms. I got this wrong originally but may have found a solution. Can Someone please check if I did it right? $\int_0^...
0
votes
1answer
98 views

find an appropriate parametrization for the given piecewise-smooth curve in $\mathbb{R^2}$

...for the curve $C$, which goes along the circle of radius $3$, from the point $(3,0)$ to the point $(-3,0)$, and then in a straight line along the $x$-axis back to $(3,0$). So I set the half-circle ...
0
votes
0answers
41 views

Rigorous Definition On composition of Multivariable Functions

Suppose you have $f(x_1,.....x_n)\colon R^n \to R$ and $g_1,...g_n\colon R^m \to R$. Then $f(g_1,....g_n)$=$f\circ(g_1,...g_n)$ :$R^m \to R$, right? Any clear definition on composition of ...
2
votes
2answers
45 views

How do I prove that $\lim_{(x,y) \to (0,0)} \frac{2y^2}{\sqrt{x^2+xy}}$ exists?

Now I have learnt that to prove a function of 2 variables exists we must have the both the repeated limits as equal, which is $\lim_{(x=0,y\to0)}f(x,y) = \lim_{(x\to0,y=0)}f(x,y)$ , now in this case $...
0
votes
0answers
25 views

Difficulty with Theorem 13.6 of Munkres Analysis on Manifolds

Theorem 13.6. Let $S$ be a bounded subset of $\mathbb{R}^n$; let $f:S\rightarrow \mathbb{R}$ be a bounded continuous function; let $A=$Int $S$. If $f$ is integrable over $S$, then $f$ is integrable ...
3
votes
1answer
118 views

How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?

Let $U\subset\mathbb{R}^n$ be a open set and $f:U\to\mathbb{R}$ a function in $C^\infty_c(U)$. Evans PDE book uses the following result $$\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)\...
3
votes
0answers
19 views

Integral exists over the closure but not the set itself

Is there an open set $S$ with a bounded function $f$ such that $\int_\overline{S}f$ exists but $\int_Sf$ does not?
0
votes
2answers
68 views

Let $Q:=[0,2] \times [0,2]$ ; then how to evaluate ${\int\int}_Q\lfloor x+y\rfloor dxdy$ ?

Let $Q:=[0,2] \times [0,2]$ ; then how to evaluate ${\int\int}_Q\lfloor x+y\rfloor dxdy$ ? where $\lfloor . \rfloor$ denotes the greatest integer function . Please help . Thanks in advance
0
votes
1answer
25 views

Having problem with dtermining the path over which a line integral is to be evaluated

Let $C$ be the curve of intersection of $z=xy ; x^2+y^2=1$ traversed once in a direction that appears counterclockwise when viewed from High above the $xy$-plane ; then how to evaluate $\int_Cydx+zdy+...
2
votes
1answer
40 views

Composition of multivariate functions is Riemann integrable

Let $A\subset \Bbb{R}^n$ be bounded, $f:A\to\Bbb{R}, f$ bounded and integrable on A. Suppose $\phi:\Bbb{R}\to\Bbb{R}, \phi\in C^1(\Bbb{R}), \phi(0)=0$, show that $\phi\circ f$ is bounded and ...
0
votes
1answer
32 views

A function that is integrable over the closure of an open set but not over the set itself

I'm trying to find a bounded function $f$ and an open set $A$ such that $f$ is integrable over $\overline{A}$ but not over $A$. Is there such a function and set?
1
vote
0answers
32 views

Integration Over Spherical Triangle and Change of Variable

Let $T$ denote a spherical triangle on the unit sphere, defined by vertices $u,v,$ and $w$. Let $\triangle$ denote a triangle. This could be the triangle defined by $u,v,w$ in $\mathbb{R}^3$ or the ...
3
votes
3answers
292 views

Calculating volume with a triple integral

I have the following problem. I need to find the volume of the shape that is bounded up by $$u(x,y)=13+x^2$$ and down by $$v(x,y)=7x+4y$$ and from its sides by the cylinder $$x^2+y^2=1.$$ Now I know I ...
4
votes
3answers
274 views

Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
0
votes
0answers
22 views

Show that, for $f \in \mathcal{S}(\mathbb{R}^d) , \Vert f \Vert_{2}^2 = - \frac{1}{d} \int_{\mathbb{R}^d} x_i\partial_{i} (|f|^2) \ dx$.

As the title states, I wish to establish the identity $\Vert f \Vert_{2}^2 = - \frac{1}{d} \int_{\mathbb{R}^d} x_i\partial_{i} (|f|^2) \ dx$. I have attempted integrating by parts which gives me $$\...
0
votes
2answers
20 views

Finding a potential

$F(x,y)= \left(\displaystyle\frac{1-y^2}{(1+xy)^2},\displaystyle\frac{1-x^2}{(1+xy)^2}\right)$. I've been having some troubles to find the potential of $F$. To find it the idea was finding $\int F ...
-2
votes
1answer
67 views

calculate $\lim\limits_{(x,y) \to(0,0)} \frac{x^4y^2}{ (x^4+y)^5}$

I need to calculate $\lim\limits_{(x,y) \to(0,0)} \frac{x^4y^2}{ (x^4+y)^5}$ I get $[0/0]$. i think it doesn't have a limit but i don't know how to prove it. Thank you.
2
votes
0answers
30 views

Convergence of certain integral over torus

Define $F(t_1,s_1,t_2,s_2)=$ $$\big((2+\cos t_1)\cos s_1 - (2+\cos t_2)\cos s_2\big)^2 + \big((2+\cos t_1)\sin s_1-(2+\cos t_2)\sin s_2\big)^2 + (\sin t_1 -\sin t_2 )^2,$$ on $[0,2\pi]\times[0,2\pi]\...
0
votes
1answer
34 views

Consistency Error for Runge-Kutta-Methods

I'm thinking about an old exercise concerning the Friedrich Scheme $$u_{j}^{n+1} = \frac{1}{2}\left(u_{j+1}^n + u_{j-1}^n\right) - \frac{r}{2}\left(u_{j+1}^n - u_{j-1}^n\right)$$ where $u_j^n \...
1
vote
1answer
35 views

Rate of change and gradient

I'm trying to solve the follow problem: Suppose that we are on the point $P=(1/\sqrt{2},1/2,1/2)$ over $z=\sqrt{1-x^2-y^2},$ $z\geq 0, x^2+y^2<1.$ In which direction we have to move over the ...
11
votes
1answer
319 views

Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$

How to find all possible functions $f(x,y)$ such as: $$ \frac{\sqrt{3}}{2}f_x+\frac{1}{2}f_y=0$$ (with $f_x = \frac{\partial{f}}{\partial{x}}$ ) Here's everything I tried: 1) I can guess the ...
0
votes
1answer
26 views

Proving that the vector field $F$ is conservative.

Given : $F$ is a continuous vector field on the open connected set $D$ . We need to prove : If $\int_C F.dR $ is independent of the path within $D$ , then $F$ is conservative. Proving the converse ...