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).
1
vote
1answer
50 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
0
votes
0answers
30 views
Euler's theorem for homogeneous functions
Let $\textbf{R}_{+}$ be the set of positive real numbers.
The following is a well-known theorem due to Euler:
A differentiable function $f:\textbf{R}^n_{+} \rightarrow \textbf{R}_{+}$ is positively ...
9
votes
0answers
67 views
$\int_{\mathbb{R}}|f(t)|^2dt=\int_{\mathbb{R}}|f'(t)|^2dt$ implies $f(t)=\mathbb{x}_{i}|f(t)|$
Let $f \in C^{1}(\mathbb{R},\mathbb{R}^m)$ be such that $f$ and $f'$ are integrable and
$$\{t:f(t)=0\} \subset \{t:f'(t)=0\}$$
$$ |\{t:f(t)=0\}|=n\in \mathbb{N}$$
Prove that if ...
1
vote
2answers
45 views
Being inside or outside of an ellipse
Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside
$E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$
if some line passing trough $A$ ...
1
vote
0answers
27 views
Forward Time Centered Space Scheme on unit square, Stability Analysis
I'm stumped on the following problem:
Show that
$u(x,y,t)=\exp(1.68t)\sin(1.2(x-y))\cosh(x+2y)$ solves
$\frac{\partial u}{\partial t}-2\frac{\partial ^2 u}{\partial x^2}-\frac{\partial ^2 ...
0
votes
0answers
36 views
Consider the function $f(x,y)=2 \cos(x^2 +y^2) +xy \sin (x+y)^2 - x^2y^2 $
Consider the function $f(x,y)=2 \cos(x^2 +y^2) +xy \sin (x+y)^2 - x^2y^2 $
Determine the fourth order Taylor of $f$ in $(0,0)$. And use the inequality: $x^4 + y^4 + x^2y^2 - yx^3 - xy^3>0$ when ...
1
vote
1answer
42 views
Can Green's theorem be used in a plane other than the xy-plane?
In the following 2D case, Green's theorem solves the following problem:
$$\vec{F}=\langle{xy+\ln{(\sin{e^{x})},x^2+e^{y^2}}}\rangle$$
$$\oint_C\vec{F}\cdot{d\vec{r}}=\iint_Dx\space{dA}$$
where C is ...
0
votes
2answers
38 views
Vector Line Integral Question
I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.
Once again, I'm ...
0
votes
1answer
32 views
Work to provide explanation on the definition of the area of a Jordan-measurable set
The problem is as follows:
Given this theorem:
Let $D$ be bounded & Jordan-measurable set
Let $f$ be a bounded function on $D$
And $f$ is continuous except for a set of zero ...
1
vote
2answers
58 views
Volume of a set in phase space. How many dimensions?
Suppose I have a $6N$ dimensional space with points looking like this:
$$(r_x^{(1)},r_y^{(1)},r_z^{(1)}, p_x^{(1)}, p_y^{(1)}, p_z^{(1)},...,r_x^{(N)},r_y^{(N)},r_z^{(N)}, p_x^{(N)}, p_y^{(N)}, ...
2
votes
3answers
96 views
Vector calculus for ellipse in polar coordinates
I'm having trouble with this question, can somebody please help me with it! I'll thanks/like your comment if help me =)
I know that for a ellipse the parametric is $x=a\sin t$ , $b= b \cos t$, ...
3
votes
1answer
24 views
Any arbitrary closed smooth curve bounds a orientable surface?
I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ?
I have no ...
1
vote
1answer
33 views
Integration with change of variables (multivariable).
The following are the problems that I have been working on. It involves change in variables with 2,3 variables respectively.
(1)Let $R$ be the trapezoid with vertices at $(0,1),(1,0),(0,2)$ and ...
3
votes
1answer
57 views
Is the function identically zero?
Let $f(x, y)$ be a continuous, real-valued function on $\mathbb{R}^2$.
Suppose that, for every rectangular region $R$ of area 1, the double integral of $f(x, y)$
over $R$ equals 0. Must $f(x, y)$ be ...
0
votes
0answers
16 views
Finding rate of maximum temperature increase along surface
So I know that the rate of maximum increase of some function (say, $f(x,y)$) is given by the gradient ($\nabla f$), where the direction is the direction of maximum increase of the function, and the ...
3
votes
1answer
62 views
Linearization of an implicitly defined function.
Problem:
Given the equation: $xz^{2}+y^{2}z^{5}=19$
Also given: (3,4,1) is a solution to the equation. This point is not the only solution.
1) Find dz/dx and dz/dy (through implicit ...
-1
votes
0answers
62 views
What is $\lim\limits_{|a| \to \infty} \int_0^1 (G(x) - a_0 - a_1x)^2\,dx$?
Assume the integral of g from 0 to 1 is a finite #.
$$\lim_{|a| \to \infty} f(a) = \lim_{|a| \to \infty}\int_0^1 (G(x) - a_0 - a_1x)^2\,dx$$
$a= [a_0, a_1]$, as $|a| \to \infty$, we have $a_0^2 + ...
0
votes
0answers
18 views
Prolate spheroidal coordinates
If $\alpha \in (0,\infty)$, $\beta \in (0,\pi)$ and $\theta \in (0,2\pi)$.
$$\varphi (\alpha,\beta,\theta) =(
\sinh(\alpha)\sin(\beta)\cos(\theta),
\sinh(\alpha)\sin(\beta)\sin(\theta),
...
0
votes
0answers
12 views
A general formula for the partial derivatives of $\sigma(\xi_1,\ldots,\xi_j,-(\xi_1+\cdots+\xi_m),\xi_{j+1},\ldots,\xi_{m})$.
Let $\sigma$ be defined on $(\mathbf{R^n})^m\backslash \{0\}$ and suppose it is adequately differentiable (that is, we can take as many derivatives as required to show this next statement). If ...
0
votes
1answer
17 views
prove that a function is a diffeomorphism
anyone can help me to prove that $f(\alpha,x,y,z) = (\sinh(\alpha) x, \sinh(\alpha)y,\cosh(\alpha)z)$ is a diffeomorphism? In fact, i'm not sure if it's a diffeomorphism.
2
votes
2answers
54 views
Finding the range of a vector valued function
For a single valued function, I can infer if the function is monotone from its derivative.
For a vector valued function, is it possible to infer monotonicity from the directional derivative?
For ...
1
vote
1answer
32 views
If $f:U\to \mathbb{R}$ is continuous and $(x^2+y^4)f(x,y) + (f(x,y))^3=1$, then $f$ is $C^\infty$
Let $f:U\to \mathbb{R}$ be continuous in $U \subset\mathbb{R}^2$, such that
$$(x^2+y^4)f(x,y) + (f(x,y))^3=1$$
for all $(x,y) \in U$. Prove that $f\in C^{\infty}$.
I'm learning the implicit ...
4
votes
1answer
69 views
Find the volume of the region bounded by $z = x^2 + y^2$ and $z = 10 - x^2 - 2y^2$
So these are two paraboloids. My guess is I would want to find the intersection of these two which would be $2x^2 + 3y^2 = 10$ and construct a triple integral based on its projection. No idea how to ...
1
vote
0answers
59 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
1
vote
2answers
36 views
Vector Field Generating Variation Along Curve
I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following.
Suppose ...
0
votes
1answer
52 views
Liouville's formula
I have some questions concerning a proof of Liouville's formula:
$$W'(t)=\text{tr}(A) W(t)$$ where $W$ is the Wronskian of the homogenous ODE.
If the vectors in the columns of the fundamental matrix ...
0
votes
0answers
41 views
Let $f,g :\mathbb{R}^n \to \mathbb{R}$, such that $g(x) = f(x) + (f(x))^5$. If $g \in C^k$ then $f \in C^k$.
Someone can help me on this question ?
Section on the implicit function theorem.
1
vote
1answer
13 views
Neumann problem, stuck on a boundary condition.
I am stuck on a problem that I am trying for exam practice and I would very much appreciate a hint to help me out, here is the section where I am stuck:
A solution is sought to the Neumann problem ...
2
votes
0answers
17 views
Area of $ M=\{[x,y] \in R^2; x (x^2+y^2) < x^2-y^2; x>0 \} $
I started out with expressing $y$ in terms of $x$:
$$
\begin{equation}
\sqrt{\frac{x^2-x^3}{x+1}} <y
\end{equation}
$$
Now I integrate over $x \in (0,1)$ since I've graphed the above expression.
...
1
vote
1answer
51 views
Curvature and Torsion problem
Calculate the curvature and torsion of
$$x= e^t\sin(t),\quad y= e^t\cos(t),\quad z= e^t$$
I'm not sure if I am doing this correctly since I am getting quite complicated results.
But I understand ...
0
votes
1answer
46 views
Level Sets Questions
1) In the following link, question 1:
http://mathquest.carroll.edu/libraries/MVC.student.14.01.pdf
Is it true that both partial derivatives are negative ? If so, can someone help me find an example ...
2
votes
2answers
30 views
Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
2
votes
0answers
19 views
Given $\Sigma$ a surface parameterized by $\Phi : D \to \Sigma$, prove a certain formula for $area(\Sigma).$
Let $\Sigma$ be a surface parameterized by $\Phi : D \to \Sigma$, and let $$A=\Phi_u \cdot \Phi_u~,~B=\Phi_u \cdot \Phi_v,~ C=\Phi_v \cdot \Phi_v.$$ Prove $$area(\Sigma)=\int\int_D \sqrt{AC-B^2} ...
2
votes
1answer
28 views
Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
The question is as follows:
Given:
(1) function $f: U \subset \mathbb R^n ==> \mathbb R$
(2) $U$ is open and convex set
(3) $f \in C^1$ in $U$
Goal: Show that $f$ is ...
0
votes
2answers
38 views
Integral of a vector field
I'm trying to evaluate the following integral:
$ \int_C(y+\sin x) dx +(z^2+\cos y)dy+(x^3)dz$
Where $C$ is the curve: $c(t) = (\sin t, \cos t, \sin 2t) $. Note that
$C$ lies on the surface ...
0
votes
0answers
18 views
Double solid angle integration with integrand only dependent on relative angle
Suppose one has an integral of the following form,
$$
\int \text{d} \Omega_{1} \text{d} \Omega_{2} f(\gamma).
$$
Where gamma is the relative angle between $(\theta_1, \phi_1)$ and $(\theta_2, ...
2
votes
1answer
49 views
Is the inverse function smooth?
Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often ...
4
votes
0answers
44 views
Evaluate $\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} dx$
I found two different approaches, both is giving the same answer.
Fubini:
$$
\begin{align}
\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\
...
1
vote
0answers
17 views
Finding gradient of a norm
How to calculate the gradient of a function $f(\mathbb x) = \| \mathbb x \|$ where $\mathbb x$ is a $n$ dimensional vector, $\|\cdot\|$ could be either a $L_1$ norm or a $L_2$ norm or a ...
1
vote
1answer
66 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
0
votes
1answer
52 views
Prove if $\nabla f(x_0) = 0$ and $\nabla^2 f(x_0)$ is positive definite, then x$_0$ is a point of local minimum
Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function.
If $f$ is twice differentiable, and there exists a point $x_0\in\mathbb R^n$ such that $\nabla f(x_0) = 0$ and $\nabla^2f(x_0)$ is ...
2
votes
1answer
88 views
How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$
Could you help me solve this?
$$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$
$M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$
I know that the region would look like this and I need to solve it as ...
4
votes
2answers
74 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
1
vote
1answer
46 views
Line Integral of Every Positively Oriented Simple Closed Path - Green's Theorem
This question is from Example #5, Section 16.4 on P1059 of Calculus, 6th Ed, by James Stewart.
Given Question: If $\mathbf{F}(x,y) = \left(\dfrac{-y}{x^2 + y^2}, \dfrac{x}{x^2 + y^2}\right)$, show ...
16
votes
6answers
252 views
Why is boundary information so significant? — Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
1
vote
1answer
29 views
How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\frac{dV}{dt} = \frac{dV}{dx_1}\frac{dx_1}{dt} + ...
1
vote
1answer
53 views
How to compute the second derivatives?
Motivation:
In isogeometric analysis, state variables(e.g. displacement) are defined in the parametric domain, which can be mapped to the physical domain by $\boldsymbol{\xi}\mapsto \boldsymbol{x}$ ...
0
votes
1answer
35 views
Vector valued Mean value theorem: Norm for the gradient
The wikipedia article on the vector valued Mean value theorem, says
For $f:\mathbb R^n \to \mathbb R^n$, if the gradient is bounded,
$$
\| \nabla f \| \le M,
$$
then
$$
\|f(x)-f(y) \| \le M ...
1
vote
0answers
114 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
0
votes
2answers
43 views
The closes point to a curve in space.
I am working on the following problem.
Find the point closest to the origin, of the curve of intersection of the plane $2y+4z =5$ and the cone $z^2 = 4(x^2+y^2)$
I was able to see that the ...
