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).
0
votes
0answers
14 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
0answers
29 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
39 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:
$$
\begin{align}
\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\
...
1
vote
0answers
15 views
Finding gradient of a norm
How to calculate the gradient of a function $f(\mathbb x) = \| \mathbb x \|$ where $\mathbb x$ is a $n$ dimensional vector, $\|\cdot\|$ could be either a $L_1$ norm or a $L_2$ norm or a ...
1
vote
0answers
31 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
0
votes
1answer
42 views
Prove if $\nabla f(x_0) = 0$ and $\nabla^2 f(x_0)$ is positive definite, then x$_0$ is a point of local minimum
Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function.
If $f$ is twice differentiable, and there exists a point $x_0\in\mathbb R^n$ such that $\nabla f(x_0) = 0$ and $\nabla^2f(x_0)$ is ...
2
votes
1answer
79 views
How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$
Could you help me solve this?
$$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$
$M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$
I know that the region would look like this and I need to solve it as ...
2
votes
1answer
28 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
1
vote
1answer
24 views
Line Integral of Every Positively Oriented Simple Closed Path - Green's Theorem
This question is from Example #5, Section 16.4 on P1059 of Calculus, 6th Ed, by James Stewart.
Given Question: If $\mathbf{F}(x,y) = \left(\dfrac{-y}{x^2 + y^2}, \dfrac{x}{x^2 + y^2}\right)$, show ...
16
votes
6answers
233 views
Why is boundary information so significant? — Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
1
vote
1answer
26 views
How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\frac{dV}{dt} = \frac{dV}{dx_1}\frac{dx_1}{dt} + ...
1
vote
1answer
52 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
$$
\|f(x)-f(y) \| \le M ...
1
vote
0answers
80 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
0
votes
2answers
38 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 ...
0
votes
1answer
54 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
30 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
21 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
60 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
104 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
36 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 $\alpha$?
$$\large{\int_{0}^{a} \int_{0}^{2\pi} ...
0
votes
2answers
17 views
Finding the limit for functions with two variables
I know that when we have a limit of a function with $2$ variables, the limit must be the same, regardless of the path we take. So this is useful for proving that a limit doesn not exist. But when ...
2
votes
2answers
84 views
What is the (parametric) intersection of a plane and a sphere?
Can someone please show me how to prove that the intersection of the plane
$$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$$
That's simple enough. But when I see a $z$ in the equation, I get stumped. I know $z=f(x,y)$. I ...
2
votes
0answers
37 views
Multivariate Differentiation on Composite Functions
Given $F(x,y,z) = (f(x,y,z),g(x,y,z),f(x,y,z) + g(x,y,z))$ I'm a little confused about what the derivative matrix would be.
Is it the 3 by 3 matrix $(DF_1, DF_2, DF_3)$ where $DF_1 = (\frac{\partial ...
2
votes
1answer
29 views
Find expression for $dy/dx $ + state where it is valid
hopefully you guys can shed some insight into this question I'm working on.
Given
$xy+y^{2}-e^{x^{2}} = 6$
find an expression for $dy/dx$ and state where it is valid.
So, what I did was ...
2
votes
2answers
63 views
Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function.
If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
Where can I find a proof for this theorem? ...
1
vote
2answers
46 views
Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\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
20 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 $$f(x,y)=\left\{\begin{array}{lr}\frac{xy}{x^2+y^2} ...
0
votes
1answer
45 views
Determine what kind of stationary point you have in $(0,0)$ where $f(x,y)=(x^2+y^2+1)^2 - 2(x^2+y^2) +4\cos(xy)$
Given the function
$$f(x,y)=(x^2+y^2+1)^2 - 2(x^2+y^2) +4\cos(xy)$$
I know that the taylor polynomial of degree $4$ of $f$ is equal to $5+x^4+y^4$. And I know that $x^4+y^4 \geq ...
2
votes
3answers
89 views
$x^4+y^4 \geq \frac{(x^2+y^2)^2}{2}$
I'm doing some exercise to prepare for my multivariable analysis exam. I didn't understand the second part of this question.
Given the function
$$f(x,y)=(x^2+y^2+1)^2 - 2(x^2+y^2) ...
1
vote
0answers
24 views
Gradient in Cylindrical ccordinates
How can I express grad of $\Phi$ in cylindrical coordinates?. In fact I wanted to check the vector $\nabla\Phi$ is perpendicular to a surface $\Phi=c$ where $c$ is a constant. Where to start? Thank ...
1
vote
2answers
34 views
Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.
This is no homework, but it is in my book and I find it hard to solve:
Determine the points where $f$ is has a local minimum/maximum.
Determine if it strong/weak and absolute/relative and ...
0
votes
0answers
15 views
How do you compute integral over C of F * dr? Need help
Let $C$ be the boundary curve of the surface $z=\sqrt{1-x^2 - y^2}$, and let
$$F(x,y,z) =\langle x^2 + \cos(x^2 + 1), y^3 + \sin(y^2 + 1) + x, \tan^{-1}z \rangle$$
Compute integral $$\oint_C ...
1
vote
0answers
34 views
Questions on Green's Theorem
I have a few questions on the Green's theorem.
It says it needs to be piecewise smooth. It means pieces are differentiable, but how do I check? Can't everything be differentiable? In other words, I ...
0
votes
2answers
34 views
Evaluating $\oint_C (\arctan{y} - 5y)dx + \left(\frac{x}{1+y^2} + 2x\right)dy$
Evaluate integral of
$$\oint_C (\arctan{y} - 5y)dx + \left(\frac{x}{1+y^2} + 2x\right)dy$$
where $C$ is the ellipse $x^2/16 + y^2/25 = 1$.
So I used the Green's Theorem and got double integral of ...
1
vote
3answers
52 views
Using Spherical coordinates find the volume:
Inside the surfaces $z=x^2+y^2$ and $z=\sqrt{2-x^2-y^2}$
I integrated over the ranges:
$0 \leq \theta \leq 2\pi$
$ 0 \leq \phi \leq \frac{\pi}{2}$
$0 \leq r \leq \sqrt{2}$
I get ...
0
votes
1answer
54 views
Theorem or just a change of varibles?
I have a formula in my text:
$$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$
I am really lazy and hate remembering formulas to me this looks like a ...
0
votes
2answers
52 views
How do you use the gradient to find $n$?
$$z = \sqrt{1 - x^2 - y^2}$$
$$W = x^2 + y^2 + z^2 - 1$$$$\overrightarrow{N} = \nabla W = \langle 2x,2y,2z \rangle$$
$$\overrightarrow{n} = \frac{\overrightarrow{N}} {\|\overrightarrow{N}\|} = ...
4
votes
1answer
18 views
Integrate over the region bounded by two regions. Using Polar coordinates.
Using Polar Coordinates integrate over the region bounded by the two circles:
$$x^2+y^2=4$$
$$x^2+y^2=1$$
Evaluate the integral of $\int\int3x+8y^2 dx$
So what I did was said that as $x^2+y^2=4$ ...
4
votes
3answers
122 views
How do you graph $x + y + z = 1$ without using graphing devices?
How can I graph $x + y + z = 1$ without using graphing devices?
I equal $z = 0$ to find the graph on the xy plane. So I got a line, $y = 1-x$
But when I equal 0 for either the $x$ or the $y,$ I get ...
1
vote
1answer
28 views
What is the area of the part of the plane $2x + 2y - z = 10$ that lies in the cylinder $x^2 + y^2 = 4$?
Calculate the area of the part of the plane $2x + 2y - z = 10$ that lies in the cylinder $x^2 + y^2 = 4$.
What can I use to solve this? Can I use the Green's theorem?
1
vote
1answer
42 views
Find local maxima of this quadratic function
How can I find local maxima of this quadratic function?
$$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$
which ...
0
votes
0answers
19 views
Multivariate Normal Product Distribution
I am looking for multivariate case of a distribution of a product of two normally distributed variables X and Y. The variables are independent. Something similar to this:
...
