Tag Info

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 ...
• 11.3k
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 ...
• 4,921
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 ...
• 480

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 ...
• 78.4k
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 $(x,y) = (1, 1), (-1, -1)$...
• 69.4k

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 ...
• 30.7k

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: $$\color{green}{\int_{y=0}^1 \int_{x=... • 137k 2 votes Help with integration over a triangular region. Without changing coordinates, for each x, we have to visit each y in the region. The lower bound is consistent: y = \tfrac13 x, but the upper bound changes depending on which side of x = 1, we'... • 25.5k 2 votes How do I prove local minima are global? Another way... (informal) Critical points are found from \frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0 which is the system 4x^3-4y=4y^3-4x=0 as (0,0),(1,1), (-1,-1) with function ... • 11k 2 votes Accepted Change of coordinates on \nabla(h\circ\varphi^{-1}) where h,\varphi:\mathbb{R}^n\to\mathbb{R}^n Your notation suffers from a number of problems. (the least severe) \varphi^{-1} can be replaced by the diffeomorphism \psi to simplify notation (much more severe) the notation \nabla does not ... • 14.3k 2 votes Accepted Confusion about divergence theorem for flux computation Just getting this out of the unanswered pile: the vector field is not defined at the origin, and it doesnâ€™t have a continuous extension to the origin, let alone a C^1 extension to the origin, which ... • 56k 2 votes Accepted How can I find a positively oriented parametrization? You can parameterize the region with spherical coordinates,$$(x,y,z) = (3\cos u \sin v, 3\sin u \sin v, 3\cos v) =: \vec s(u,v)$$For u\in[0,2\pi], the constraint x+y+z\ge1 can be used to solve ... • 19.6k 2 votes If A \subset \mathbb{R}^n Is a countable set prove that \exists a function f such that \lim_{x\to a}f(x)=\infty \ \forall a \in A' I thought I would post an answer in addition to my comment. Although f(\frac{p}{q})=q gives an example on \mathbb{Q}, we cannot compose this with a mapping from A to \mathbb{Q} unless this ... • 1,140 1 vote Scalar functions under diffeomorphism of domain Hint: Consider the functions$$ f_1(x)=e^{-x^2} $$and$$ f_2(x)=e^{-x^4}. $$Prove (by studying the critical point x=0) that there is no diffeomorphism g: \mathbb R\to \mathbb R such that f_1=... • 98k 1 vote Accepted Understanding a question in multivariable calculus By the inverse function theorem, for any point (x,y) such that$$ \partial_y f(x,y)=6y(y+1) \ne 0 $$The function can be expressed as$$ f(x,y)=f(x,g(x)) $$for some continuously differentiable ... • 2,567 1 vote How do I prove local minima are global? Here is a sketch, I let you check/complete the details. Show that f(x,y)\to \infty when ||(x,y)||\to\infty. This means that, there exists K>0 such that, if ||(x,y)||>K, then f(x,y)>... • 11.4k 1 vote Approximation of a non differentiable function If the function f has partial derivatives at (a,b), then the function g(x)=f(x,b) is differentiable at x=a and the function h(y)=f(a,y) is differentiable at y=b. Thus, restricting to these ... • 115k 1 vote Proving that Jacobian of Composition is equal to Composition of Jacobians using epsilon-delta I think I understand William's topogical argument, which is why I've accepted William's answer. However, I am still replying to my own question because I finally found the breakthrough necessary for ... 1 vote Argue whether the generalized double integral \int \int_D \frac{x^2-y^2}{x^2+y^2}dxdy where D is x\geq1 , y\geq1 converge or diverge. First, we can show that f is not absolutely integrable over D. Defining E = \{(x,y): x\geqslant 1, y\geqslant 1, x^2+y^2 \geqslant 1\}, we have E \subset D. Since |f| is nonnegative we can ... • 90.8k 1 vote Accepted Confusion about the units of divergence You didn't use the correct expression for divergence in spherical coordinates. Moreover, radians are considered dimensionless so you should ignore them when checking the units. The divergence in ... • 1,831 1 vote Help with integration over a triangular region. I like a more general method where you describe any position inside the triangle as a function of two arbitrary variables that vary between 0 and 1. For example, the point \boldsymbol{P} can be ... • 13.9k 1 vote Does limit exists or not? Consider the path xy = x^2+y^4+x^2y^4 (This definitely goes through the origin as (0,0) is contained in the curve). Plotting the curve is irrelevant and algebraically we obtain$$\lim_{(x,y)\to(0,...
• 34.5k

Only top scored, non community-wiki answers of a minimum length are eligible