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).
3
votes
1answer
42 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
29 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
23 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
38 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
29 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
16 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
47 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
33 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
28 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:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
2
votes
0answers
16 views
Given $\Sigma$ a surface parameterized by $\Phi : D \to \Sigma$, prove a certain formula for $area(\Sigma).$
Let $\Sigma$ be a surface parameterized by $\Phi : D \to \Sigma$, and let $$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
21 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
37 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
16 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
45 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
42 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:
$$
\begin{align}
\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\
...
1
vote
0answers
17 views
Finding gradient of a norm
How to calculate the gradient of a function $f(\mathbb x) = \| \mathbb x \|$ where $\mathbb x$ is a $n$ dimensional vector, $\|\cdot\|$ could be either a $L_1$ norm or a $L_2$ norm or a ...
1
vote
0answers
50 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
0
votes
1answer
52 views
Prove if $\nabla f(x_0) = 0$ and $\nabla^2 f(x_0)$ is positive definite, then x$_0$ is a point of local minimum
Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function.
If $f$ is twice differentiable, and there exists a point $x_0\in\mathbb R^n$ such that $\nabla f(x_0) = 0$ and $\nabla^2f(x_0)$ is ...
2
votes
1answer
84 views
How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$
Could you help me solve this?
$$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$
$M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$
I know that the region would look like this and I need to solve it as ...
2
votes
1answer
29 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
1
vote
1answer
27 views
Line Integral of Every Positively Oriented Simple Closed Path - Green's Theorem
This question is from Example #5, Section 16.4 on P1059 of Calculus, 6th Ed, by James Stewart.
Given Question: If $\mathbf{F}(x,y) = \left(\dfrac{-y}{x^2 + y^2}, \dfrac{x}{x^2 + y^2}\right)$, show ...
16
votes
6answers
244 views
Why is boundary information so significant? — Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
1
vote
1answer
26 views
How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\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
31 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
$$
\|f(x)-f(y) \| \le M ...
1
vote
0answers
97 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
0
votes
2answers
41 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 ...
1
vote
2answers
75 views
Find the area of $A = \{ \langle x,y\rangle \in \mathbb{R}^2 \mathrel| (x+y)^4<a x^2 y,\ x>0 \}$?
I can't really think of how to set the limits
1
vote
1answer
37 views
Circulation and line integrals.
The following is the problem I'm working on.
If $\overrightarrow {F} = <x,y^2z,-xy^2>$, calculate the circulation of $\overrightarrow {F}$ over the surface $z=x^2+y^2$ bounded by $C$ using a ...
2
votes
2answers
56 views
Help with Taylor polynomial
I need to find the Taylor polynomial of order $2n$ of the function
$$f(x,y)=\frac{1}{1+xy}$$ on $(x_0,y_0)=(0,0)$.
Can anyone give me a hand please?
Sorry, i make a mistake, their should say ...
1
vote
1answer
18 views
Critical points in multivariable calc
Find the critical points of
$z = x^{3} + 3xy^{2} - 3x^{2} - 3y^{2} + 7$
I understand if it was $f(x,y)$ but this z is really throwing me off..
I could take the partial derivs of x and y, but if I ...
2
votes
0answers
32 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
0
votes
0answers
22 views
How to show that the partial derivatives exist
In general , how to show that the partial derivatives of a multivariable function exists without comupting it .
1
vote
0answers
31 views
Can the mean value of a function be guaranteed to have some degree of regularity?
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuously differentiable function. The mean value theorem tells us that for any $x,y \in \mathbb{R}$ there exists $c=c(x,y) \in [0,1]$ such that:
...
2
votes
1answer
68 views
Evalute this integral using Green's Thereom
Let C be the boundary of the half-annulus
$$1\leq(x^2+y^2)\leq4$$ where $$x\le0$$
in the xy plane, traversed in the positive direction.
Evaluate : $ \displaystyle \int_{c}(7\cosh^3(7x)-2y^3) ...
1
vote
1answer
27 views
I can't see a reason to reject one solution. Parametized surface
Ok so the question is to consider the parametrized surface
x = $u^2 - v^2$, y = u + v, z= $u^2 + 4v$
It asks for me to find a vector normal to the surface in terms of u and v which is no problem.
...
2
votes
3answers
126 views
How to solve this integral for a hyperbolic bowl?
$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
0
votes
0answers
38 views
separating a variable from integral
In the following integral, I would like to separate $\alpha$ from rest of the equation. Can we solve the following equation for $\alpha$?
$$\large{\int_{0}^{a} \int_{0}^{2\pi} ...
0
votes
2answers
17 views
Finding the limit for functions with two variables
I know that when we have a limit of a function with $2$ variables, the limit must be the same, regardless of the path we take. So this is useful for proving that a limit doesn not exist. But when ...
2
votes
2answers
90 views
What is the (parametric) intersection of a plane and a sphere?
Can someone please show me how to prove that the intersection of the plane
$$x+y+z=0$$ and the sphere
$$x^2+y^2+z^2=1$$
can be expressed as
$$x(t)=\frac{\cos t-\sqrt3 \cdot\sin t}{\sqrt6}$$
...
2
votes
1answer
16 views
Finding partial derivatives for equations expressed in terms of $z$ where $z=f(x,y)$ to find tangent plane
I am having troubles finding partial derivatives.
If $f(x,y)=2x^2+y^2$ then,
$$f_x=4x$$
$$f_y=2y$$
That's simple enough. But when I see a $z$ in the equation, I get stumped. I know $z=f(x,y)$. I ...
2
votes
0answers
37 views
Multivariate Differentiation on Composite Functions
Given $F(x,y,z) = (f(x,y,z),g(x,y,z),f(x,y,z) + g(x,y,z))$ I'm a little confused about what the derivative matrix would be.
Is it the 3 by 3 matrix $(DF_1, DF_2, DF_3)$ where $DF_1 = (\frac{\partial ...
2
votes
1answer
29 views
Find expression for $dy/dx $ + state where it is valid
hopefully you guys can shed some insight into this question I'm working on.
Given
$xy+y^{2}-e^{x^{2}} = 6$
find an expression for $dy/dx$ and state where it is valid.
So, what I did was ...
2
votes
2answers
64 views
Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function.
If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
Where can I find a proof for this theorem? ...
1
vote
2answers
47 views
Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$
How would you compute it?
I already solved this problem this way:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( ...
1
vote
1answer
24 views
Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1
The task is as follows:
Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial
z}{\partial y}$ (evaluated at $x$) * $\frac{\partial y}{\partial x}$
(evaluated at $z$) * $\frac{\partial ...
4
votes
1answer
30 views
Can I prove continuity of a function of two variables in this way?
Common approach in handling functions of two variables is to express this function in polar coordinate system. For example, in the classic example $$f(x,y)=\left\{\begin{array}{lr}\frac{xy}{x^2+y^2} ...
0
votes
1answer
46 views
Determine what kind of stationary point you have in $(0,0)$ where $f(x,y)=(x^2+y^2+1)^2 - 2(x^2+y^2) +4\cos(xy)$
Given the function
$$f(x,y)=(x^2+y^2+1)^2 - 2(x^2+y^2) +4\cos(xy)$$
I know that the taylor polynomial of degree $4$ of $f$ is equal to $5+x^4+y^4$. And I know that $x^4+y^4 \geq ...
2
votes
3answers
90 views
$x^4+y^4 \geq \frac{(x^2+y^2)^2}{2}$
I'm doing some exercise to prepare for my multivariable analysis exam. I didn't understand the second part of this question.
Given the function
$$f(x,y)=(x^2+y^2+1)^2 - 2(x^2+y^2) ...
