How to compute the volume of a region transformed by a matrix?

This is a rewrite of the OP’s question to emphasize what I think are the research level issues here. Let R be a bounded convex body in Rn and let H:Rn→Rr be a surjective linear map for r<n. How can we compute the volume of H(R)? Of course, the answer to this question will depend … Read more

Multivariable higher-order chain rule

I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what is the derivative $\partial^\alpha ( f \circ G )$ where $\alpha$ is a multiindex in the variables $x_1,\dots,x_m$ of … Read more

Complex-doubly periodic function in two variables?

I am looking for a function f:C2→C2 that satisfies the two equations ∂z2f1(z1,z2)+∂z1f2(z1,z2)=0 and  ∂ˉz1f1(z1,z2)−∂ˉz2f2(z1,z2)=0 and in addition, is doubly-periodic in both its complex variables z1,z2. Does such a function exist and if not, why? I would not even know how to start building such a function. In particular, I would like to have f1(z1+1,z2)=f1(z1,z2+1)=f1(z1,z2) and … Read more

Monotonic dependence on an angle of an integral over the nn-sphere

Let v,w∈Sn−1 be two n dimensional real vectors on sphere. Consider the following integral: ∫x∈Sn−1|⟨x,v⟩|⋅|⟨x,w⟩|dx. Since the integration is taking over the sphere, we have rotation invariance and the value of the integration only depends on the value of ⟨v,w⟩. Now the question is to show that the integral is a non-decreasing function w.r.t ⟨v,w⟩, … Read more

Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?

This is a cross-post. Let U⊆Rn be an open subset, and let f:U→R be smooth. Suppose that x∈U is a strict local minimum point of f. Let dfk(x):(Rn)k→R be its k “derivative”, i.e. the symmetric multilinear map defined by setting dfk(x)(ei1,…,eik)=∂i1…∂ikf(x). Assume that dfj(x)≠0 for some natural j. Let k be the minimal such that … Read more

Any hints on how to prove that the function |αsin(A)+sin(A+B)|−|sin(B)|\lvert\alpha\;\sin(A)+\sin(A+B)\rvert – \lvert\sin(B)\rvert is negative over the half of the total area?

I have this inequality with 0<A,B<π and a real |α|<1: f(A,B):=|αsin(A)+sin(A+B)|−|sin(B)|<0 Numerically, I see that regardless of the value of α, the area in which f(A,B)<0 is always half of the total area π2. I appreciate any hints and comments on how I can prove this. Answer Let us assume α∈[0,1) (the case of α∈(−1,0] … Read more

Give an argument for ∫n0xpdx≤1+2p+3p+⋯+np≤∫n+10xpdx\int_{0}^{n} x^p dx \leq 1 +2^{p} + 3^{p} + \cdots+ n^{p}\leq \int_{0}^{n+1} x^p dx

For any n and p≥0 give an argument that the following is true: ∫n0xpdx≤1+2p+3p+⋯+np≤∫n+10xpdx I’m having trouble even beginning this question. My first thought it to somehow meld this with the squeeze theorem, but, again, am not sure how to begin and show any real work. Any insight is very much appreciated. Answer Write the … Read more

Finding a limit with two independent variables: $\lim_{(x,y)\to (0,0)}\frac{x^2y^2}{x^2+y^2}$

I must find the following limit: $$\lim_{(x,y)\to (0,0)}\frac{x^2y^2}{x^2+y^2}$$ Substituting $y=mx$ and $y=x^2$, I have found the limit to be $0$ both times, as $x \to 0$. I have thus assumed that the above limit is $0$, and will attempt to prove it. Let $\varepsilon>0$. We have that: $$\left\lvert\frac{x^2y^2}{x^2+y^2}\right\rvert=\frac{x^2y^2}{x^2+y^2}\leq\frac{(x^2+y^2)(x^2+y^2)}{x^2+y^2}=x^2+y^2$$ However, I must find $\delta>0$ such that … Read more

What will be value of →r⋅∇\vec{r} \cdot \nabla

I was studying on Nabla Operator and saw that ∇⋅→r≠→r⋅∇ So, if I were to find →r⋅∇ how would I calculate it? I know that →r⋅∇ = r1⋅∂∂x+r2⋅∂∂y+r3⋅∂∂z but what I’m confused is on what am i suppose to differentiate? Thanks. Answer The difference is that ∇⋅r is a scalar, but r⋅∇ is another operator. … Read more