Skip to main content

New answers tagged

1 vote

How do I calculate the parametrization of a 3D surface, given its support function?

If $\vec u$ is a unit vector on the sphere and $h(\vec u)$ is the support function in that direction, then the expression $\vec p(u) = h(\vec u) \vec u - \nabla _S h(u)$ parametrizes your surface. ...
MathFont's user avatar
  • 4,847
2 votes
Accepted

Strange usage of chain rule. Can anyone explain why this derivation was done this way?

(1.35) seems primarily motivated by geometry; if you follow their logic of $\delta x$ and look at the attached picture, it makes sense why they're doing it the way they did. If you want to explicitly ...
cambridgecircus's user avatar
2 votes

Finding the area of an ellipse which is the intersection of $z=x^2+y^2$ and $z=1+2x$.

The ellipse can be written in parametric form as: $$x = 1 + \sqrt{2}\cos t$$ $$y = \sqrt{2} \sin t$$ $$z = 3 + 2\sqrt{2}\cos t$$ Where $x$ and $y$ form the $xy$-plane circle given by $(x-1)^2+y^2=2$, ...
Dan's user avatar
  • 15.2k
0 votes

What coordinate substitution should I perform to evaluate this triple integral?

You are trying to calculate the integral over the region enclosed by the cone $y^2 = 4x^2 + z^2$ and the plane $y = 2$ as is show in the image: Notice that if we view the cone in the $xz$ plane at $y ...
Student2271's user avatar
1 vote

Rotation of a function on the unit sphere

Let $v(x)=u(-R_\theta x)$. The chain rule tells us that $$Dv(x) = -Du(-R_\theta x)R_\theta, \quad\text{and so}\quad \nabla v(x) = -R_\theta^\top\nabla u(-R_\theta x).$$ Note that $-R_\theta^\top = -R_{...
Ted Shifrin's user avatar
0 votes

How do I calculate the parametrization of a 3D surface, given its support function?

Here is one way to do this: The support function $h$ of a convex body $K \subset \mathbb{R}^n$ is a function of $u \in S^{n-1}$. You can extend $h$ to be a homogeneous function $H: \mathbb{R}^n \...
Deane's user avatar
  • 7,627
0 votes

Multiple Integral Problem with Dirac Delta Constraint: Seeking Guidance

Too long for a comment. Using the integral representation of the Dirac delta, $$ \delta\left(\sum_{j=1}^N(x_j^2+y_j^2)-1\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp\left\{ik\left(\sum_{j=1}^N(x_j^...
Gonçalo's user avatar
  • 9,676
1 vote

Multiple Integral Problem with Dirac Delta Constraint: Seeking Guidance

Rather long comment The apparent circular symmetry of the problem inspires the following replacement of variables (change of cartesian to polar coordinates): $$ x_i=r_i\cos\phi_i\quad,\quad y_i=r_i\...
Mostafa Ayaz's user avatar
  • 32.3k
2 votes
Accepted

How to differentiate this functions?

The partial derivatives w.r.t. the $x_i$ are easy because $f(xg(t))$ depends on $x_i$ only through the $i$-th slot of $f(y_1,\dots y_n)\,.$ Therefore, it is very similar to a single variable problem: $...
Kurt G.'s user avatar
  • 14.5k
0 votes

Average distance from a point on a circle to the y-axis.

The average distance is what is made to the center of gravity of the disk from x-axis. $$ x^2 +y^2=a^2$$ So we calculate $$ \bar y= \frac{\int y^2 dx}{\int y~ dx}= \frac{Area First Moment }{Area}. $$
Narasimham's user avatar
  • 40.5k
1 vote
Accepted

Average distance from a point on a circle to the y-axis.

On the circle $x^2+y^2=9$ we have identically $r\equiv 3$, so in the integral we should consider the length of the circle (which is $2\pi r=6\pi$) and integrate only on $\theta$. Thus, the integral is ...
Julio Puerta's user avatar
  • 5,366
3 votes

Finding the area of an ellipse which is the intersection of $z=x^2+y^2$ and $z=1+2x$.

Another solution comes from thinking of the projection of the ellipse onto the $xy$-plane. I realize that this is not the approach you were expected to take, but this might be useful to you in the ...
Ted Shifrin's user avatar
2 votes

Finding the area of an ellipse which is the intersection of $z=x^2+y^2$ and $z=1+2x$.

Changing coordinates in the plane $z=1+2x$: $$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}0\\0\\1\end{pmatrix}+X\begin{pmatrix}0\\1\\0\end{pmatrix}+Y\begin{pmatrix}1/\sqrt5\\0\\2/\sqrt5\end{...
Anne Bauval's user avatar
  • 35.9k
3 votes
Accepted

Is divergence of a curl not 0 in this case/problem?

Is divergence of a curl not 0 in this case/problem? The divergence of a curl is always zero where it is defined. More specifically, if a vector field $\mathbf A$ is twice-differentiable at a point $\...
J. Murray's user avatar
  • 1,169
1 vote

Why is $D_i f = \frac{\partial f}{ \partial g_i}$ and not $\frac{\partial f}{ \partial x_i}$?

I think this confusion is largely due to this notation being very ambiguous. Let's make some thing clear: By $f\circ g$ I will mean the function $f\circ g:\Omega\to\mathbb{R}^s$ defined by $(f\circ g)...
Lorago's user avatar
  • 9,679
1 vote

Why is $D_i f = \frac{\partial f}{ \partial g_i}$ and not $\frac{\partial f}{ \partial x_i}$?

$g_i$ is the i-th coordinate of $g(x)$ that lives inside the transient space $\mathbb{R}^m$ The sum index for i is from 1 to m. This is a major hint.
athanos lee's user avatar
1 vote

Why is $D_i f = \frac{\partial f}{ \partial g_i}$ and not $\frac{\partial f}{ \partial x_i}$?

Try thinking about this in terms of a funciton composition: $$\varphi = f \circ g.$$ Let's go through it one step at a time: since $g$ maps an $n$-dimensional vector into an $m$-dimensional space, you ...
Egor Larionov's user avatar
0 votes

Evaluating $\int_0^{\infty} \int_0^{\infty} \frac{\sin(x)\sin(x+y)}{x(x+y)}\,\mathrm dx\,\mathrm dy$

Per the symmetry between $x$ and $z$\begin{align} &\int_0^{\infty} \int_0^{\infty} \frac{\sin x\sin(x+y)}{x(x+y)}\overset{x+y=z}{dy}dx\\ = &\int_0^{\infty} \int_x^{\infty} \frac{\sin x\sin z}{...
Quanto's user avatar
  • 98.2k
4 votes

Unambiguous derivative notation in Spivak's "Calculus on Manifolds"

No, they don't refer to the same function from a strict mathematical point of view: $$ f_{\text{left}}(x,y) = f_{\text{right}}(u(x,y), v(x,y)). $$
md2perpe's user avatar
  • 26.9k
2 votes
Accepted

find the maximum value of the following multivariable function

Using trigonometric identities, $$ \cos{(x-a)}+\cos{(x-b)} = 2\cos{\left(x-\frac{a+b}{2}\right)}\cos{\left(\frac{b-a}{2}\right)}, $$ $$ \cos{(y-b)}-\cos{(y-a)}\big) = 2\sin{\left(y-\frac{a+b}{2}\...
Sam's user avatar
  • 1,585
1 vote
Accepted

'Integrating' a matrix times a gradient

Let $f=x^2-y^2$. And $$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.$$ Then $A \nabla f = \langle -2y,2x \rangle$ which has a non-zero curl, which implies it is not a gradient function. ...
Paul's user avatar
  • 8,186
4 votes

Area of a Quater-Circle with hyperbolic elements

Given the density function $D = xy$ and the quarter-disc in the first quadrant with radius $R$, we'll set up the integral to find the mass.$$\\$$Okay so mainly you can do that in 4 steps that you're ...
Muhammad Al-Mansour's user avatar
0 votes

Why is $ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy}$ defined while $\lim_{(x,y,z) \to (0,0,0)} \frac{\sin(xyz)}{xyz}$ is not?

By definition, we have $sin(t)=t-\frac{t^3}{3!}+\frac{t^5}{5!}-...$ and hence $sin(xyz)=xyz-\frac{(xyz)^3}{3!}+\frac{(xyz)^5}{5!}-...$ Therefore, when $(x,y,z)$ approaches to zero, the behavior of $\...
WHLin's user avatar
  • 224
3 votes
Accepted

Why is $ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy}$ defined while $\lim_{(x,y,z) \to (0,0,0)} \frac{\sin(xyz)}{xyz}$ is not?

The limit does exist though and it is $1$. Let $\epsilon$ be arbitrarily small and let $\delta$ be such that the inequality $|\frac{\sin(\delta')}{\delta'} -1|$ holds for all $\delta'\le \delta$. ...
Mike's user avatar
  • 20.5k
1 vote
Accepted

Computing a surface integral

You need to solve the equation $ 9 \sin v \cos u + 6 \sin v \sin u - 3 \cos v = 1 $ for $ u $ or $ v $. Since $ u $ appears less often, let's solve it for $ u $. Focussing on the terms with $ u $, ...
Toby Bartels's user avatar
  • 4,749
1 vote

Proving that Jacobian of Composition is equal to Composition of Jacobians using epsilon-delta

I think I understand William's topogical argument, which is why I've accepted William's answer. However, I am still replying to my own question because I finally found the breakthrough necessary for ...
Timothy Leong's user avatar
2 votes

If $A \subset \mathbb{R}^n$ Is a countable set prove that $\exists$ a function $f$ such that $\lim_{x\to a}f(x)=\infty \ \forall a \in A'$

I thought I would post an answer in addition to my comment. Although $f(\frac{p}{q})=q$ gives an example on $\mathbb{Q}$, we cannot compose this with a mapping from $A$ to $\mathbb{Q}$ unless this ...
S.L.'s user avatar
  • 1,140
1 vote

Scalar functions under diffeomorphism of domain

Hint: Consider the functions $$ f_1(x)=e^{-x^2} $$ and $$ f_2(x)=e^{-x^4}. $$ Prove (by studying the critical point $x=0$) that there is no diffeomorphism $g: \mathbb R\to \mathbb R$ such that $f_1=...
Moishe Kohan's user avatar
  • 98.4k
4 votes
Accepted

Hessian matrix determinant greater than zero in a saddle point?

The second derivative test (and the Hessian determinant) only works for bivariate functions. For functions of three or more variables, one needs to use the eigenvalues. In this case, as we have both ...
Julio Puerta's user avatar
  • 5,366
0 votes

Apostol Calculus Vol 2 Exercise 8.17 Q No. 3

Following Apostol Example 1 (section 8.15), we want to find the unit tangent vector, $T$, perpendicular to the normal vectors of the two surfaces, $g(x,y,z) = 2x^2 + 2y^2 - z^2$ and $h(x,y,z) = x^2 + ...
Jake Wren's user avatar
3 votes

What goes wrong with Stokes theorem if a surface is not orientable?

Since this has gotten bumped: The problem (as noted in the comments) is not with Stokes' theorem, but defining a flux integral (or the integral of a differential $2$-form) on a non-orientable surface ...
Andrew D. Hwang's user avatar
0 votes
Accepted

Green's Formula for vector fields in the Navier Stokes Weak Formulation

The Green's formula for vector fields u = (u1, u2, u3) and v = (v1, v2, v3) can be derived from the scalar Green's formula as follows: Apply the scalar Green's formula to each component of the vector ...
alfa's user avatar
  • 75
0 votes

Inverse function theorem generalization

In the case where $n = m + k$, for $k$ an integer greater than $1$, we have a similar theorem that garantees (with some assumptions) that $f$ behaves locally as inclusion. This Theorem is Local ...
Guilherme Costa's user avatar
0 votes

Lagrange multipliers on manifolds in Lee's book

Here is my solution for this question. It uses the inverse function theorem whereas the other answer uses the implicit function theorem. Since $\Phi$ is a submersion at $p$, by the rank theorem, ...
Sam Kirkiles's user avatar
  • 2,382
2 votes
Accepted

How can I find a positively oriented parametrization?

You can parameterize the region with spherical coordinates, $$(x,y,z) = (3\cos u \sin v, 3\sin u \sin v, 3\cos v) =: \vec s(u,v)$$ For $u\in[0,2\pi]$, the constraint $x+y+z\ge1$ can be used to solve ...
user170231's user avatar
  • 19.6k
0 votes

How to solve simple second order ODE with RHS $x / \sigma^2$

By inspection, it may be seen that both $$ x_i(t)=e^{t/\sigma_i},\quad x_i(t)=e^{-t/\sigma_i} $$ are solutions to the equation. Hence the general solution is a linear combination of the form $$ x_i(t)=...
CW279's user avatar
  • 799
1 vote
Accepted

Understanding a question in multivariable calculus

By the inverse function theorem, for any point $(x,y)$ such that $$ \partial_y f(x,y)=6y(y+1) \ne 0 $$ The function can be expressed as $$ f(x,y)=f(x,g(x)) $$ for some continuously differentiable ...
Marco's user avatar
  • 2,477
4 votes
Accepted

A simple question about the Hodge star

You don't get uniqueness this way. Take $V=\langle e_1,e_2\rangle$, so $n=2$ and take $k=1$. Then $\text{vol} = e_1\wedge e_2$ and if $\ast e_1 = ae_1 + be_2$ and $\ast e_2 = ce_1 + de_2$, all we ...
Pengin's user avatar
  • 510
0 votes

How do I find the volume of this body?

Suppose $(x_1, x_2, x_3) \in D$. That's if and only if $ 0\le x_i \le t$. Then, $0 \le x_1 \cdot x_2 \cdot x_3 \le t^3.$ Therefore $(x_1,x_2,x_3) \in B$. But, by definition of $B$, if $(x_1,x_2,x_3) \...
Snared's user avatar
  • 828
2 votes
Accepted

Confusion about divergence theorem for flux computation

Just getting this out of the unanswered pile: the vector field is not defined at the origin, and it doesn’t have a continuous extension to the origin, let alone a $C^1$ extension to the origin, which ...
peek-a-boo's user avatar
  • 56.1k
1 vote

How do I prove local minima are global?

Here is a sketch, I let you check/complete the details. Show that $f(x,y)\to \infty$ when $||(x,y)||\to\infty$. This means that, there exists $K>0$ such that, if $||(x,y)||>K$, then $f(x,y)>...
Taladris's user avatar
  • 11.4k
5 votes
Accepted

Why is $\iiint_B(12x^2+2z)dxdydz=\iiint_B4(x^2+y^2+z^2)dxdydz$?

How are they equal? Odd terms: What is $\iiint_B z \ dx\ dy\ dz$? Without evaluating it, $z$ is odd and is being evaluated over a region that is symmetric in the $\pm z$ directions. In fact any odd ...
user317176's user avatar
  • 11.3k
2 votes

How do I prove local minima are global?

Another way... (informal) Critical points are found from $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0$ which is the system $4x^3-4y=4y^3-4x=0$ as $(0,0),(1,1), (-1,-1)$ with function ...
Bob Dobbs's user avatar
  • 11.1k
3 votes
Accepted

How do I prove local minima are global?

Find a suitable sum of squares. You actually came close to it. $$ x^4 + y^4 - 4xy + 2 $$ Hence, the expression is $ \geq 0 $. Equality holds iff $ x^2 = y^2, xy = 1$ or that $(x,y) = (1, 1), (-1, -1)$...
Calvin Lin's user avatar
  • 69.5k
1 vote

Argue whether the generalized double integral $\int \int_D \frac{x^2-y^2}{x^2+y^2}dxdy$ where $D$ is $x\geq1 , y\geq1$ converge or diverge.

First, we can show that $f$ is not absolutely integrable over $D$. Defining $E = \{(x,y): x\geqslant 1, y\geqslant 1, x^2+y^2 \geqslant 1\}$, we have $E \subset D$. Since $|f|$ is nonnegative we can ...
RRL's user avatar
  • 90.9k
0 votes

Local maxima of $f(x,y) = x^2 + y^2 - xy$ subject to $x^2 = 16$, $y^2 = 16$

Let's examine the objective function $f(x,y)=x^2+y^2-xy$ First derivative would be $\begin{bmatrix}2x-y \\ 2y-x\end{bmatrix}$ The second derivative would be $\begin{bmatrix}2 & -1 \\ -1 & 2 \...
Siong Thye Goh's user avatar
0 votes

Argue whether the generalized double integral $\int \int_D \frac{x^2-y^2}{x^2+y^2}dxdy$ where $D$ is $x\geq1 , y\geq1$ converge or diverge.

The last integral is probably divergent. An informal reasoning show that even with $y$ bounded (i.e. $\int_1^M \cdots dy$) and $x\gg 1$ (you can as well take $\int_{2M}^\infty\cdots dx$) then you get $...
zwim's user avatar
  • 28.6k
1 vote
Accepted

Confusion about the units of divergence

You didn't use the correct expression for divergence in spherical coordinates. Moreover, radians are considered dimensionless so you should ignore them when checking the units. The divergence in ...
Conreu's user avatar
  • 1,928
0 votes

Finish this proof of gradient in polar coordinates

But I think we still need to prove that $\nabla g(r,\theta)=\nabla f(r\cos\theta,r\sin\theta)$ which looks wrong to me. It looks wrong because it is wrong. Let $h(r,\theta)= {[r\cos\theta,r\sin\theta]...
2 votes
Accepted

Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$

Your notation suffers from a number of problems. (the least severe) $\varphi^{-1}$ can be replaced by the diffeomorphism $\psi$ to simplify notation (much more severe) the notation $\nabla$ does not ...
Kurt G.'s user avatar
  • 14.5k

Top 50 recent answers are included