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
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
  • 4,921
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
  • 480
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
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
  • 69.4k
2 votes

Help with integration over a triangular region.

The region is bounded by three lines $$ x+y=4,y={1\over 3}x,\ y=3x$$ We can change variables to $$u=x+y,\ v ={y\over x}$$ i.e. $$ x={u\over 1+v},\ y= {uv\over 1+v}$$Then the region can be described by ...
Ryszard Szwarc's user avatar
2 votes

Help with integration over a triangular region.

Method 1: $$\color{red}{\int_{x=0}^1 \int_{y=x/3}^{3x} 2e^{-y-x} \, dy \, dx} + \color{blue}{\int_{x=1}^3 \int_{y=x/3}^{4-x} 2e^{-y-x} \, dy \, dx}.$$ Method 2: $$\color{green}{\int_{y=0}^1 \int_{x=...
heropup's user avatar
  • 137k
2 votes

Help with integration over a triangular region.

Without changing coordinates, for each $x$, we have to visit each $y$ in the region. The lower bound is consistent: $y = \tfrac13 x$, but the upper bound changes depending on which side of $x = 1$, we'...
Sammy Black's user avatar
  • 25.5k
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
  • 11k
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
  • 14.3k
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
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
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
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
  • 2,567
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
1 vote

Approximation of a non differentiable function

If the function $f$ has partial derivatives at $(a,b)$, then the function $g(x)=f(x,b)$ is differentiable at $x=a$ and the function $h(y)=f(a,y)$ is differentiable at $y=b$. Thus, restricting to these ...
Ted Shifrin's user avatar
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
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
  • 90.8k
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
  • 1,831
1 vote

Help with integration over a triangular region.

I like a more general method where you describe any position inside the triangle as a function of two arbitrary variables that vary between 0 and 1. For example, the point $\boldsymbol{P}$ can be ...
John Alexiou's user avatar
  • 13.9k
1 vote

Does limit exists or not?

Consider the path $xy = x^2+y^4+x^2y^4$ (This definitely goes through the origin as $(0,0)$ is contained in the curve). Plotting the curve is irrelevant and algebraically we obtain $$\lim_{(x,y)\to(0,...
Ninad Munshi's user avatar
  • 34.5k

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