# Tagged Questions

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### Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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### Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
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### Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
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### solenoid and irrotational vector

Let $V$ be a vector point function. $V$ is solenoid if $\operatorname{div} V =0$ and irrotational if $\operatorname{curl} V =0$. How can one visualize examples of solenoid or irrotational functions? ...
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### Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
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### Implicit Function Theorem (Two Variables)

While reading through a paper I came a cross a result (due to the Implicit Function Theorem) that I cannot derive. Let $u(x,y)$ be the solution of a PDE ($x$ and $y$ are independent variables). We ...
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### Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
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### Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
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### Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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### Divergence Theorem To Calculate Surface Integral

$M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}\leq z$} We are asked to find the surface area of this surface. This is my way: $\partial M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}= z$} so the ...
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### Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
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### Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
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### When is $x\mapsto |x|^{s-1}x$ a diffeomorphism?

Consider the function $f:B^n\rightarrow B^n$ from the disk to itself $$f(x)=\vert x\vert^{s-1}x$$ where $s>0$ and we are considering the euclidean norm (we define the function to be $0$ in the ...
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### Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
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### Lower boundary on hessian eigen values [duplicate]

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
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### Spherical-Coordinate Reference Frame

my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q ...
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### A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
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### curve integral - intersection between plane and sphere

I am going to calculate the line integral $$\int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
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### Find the derived of an implicit given function.

Let $C=\{(x,y,z) \in \mathbb(R)^3| \sin x + \sin^2 y + \sin^4 z=0 \ \text{and} \ (x-z)^2=4\pi^2\}.$ By the implicit function theorem, we have that $C$ can be parametrized as a smooth curve in the ...
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### Gauss's theorem in 2d: how can it be expressed in differential forms?

How do we express the 2d version of Gauss's theorem in the language of differential forms? In 3d, I know it is $$d \left(Fdydz + Gdzdx + Hdxdy\right) = F_x + G_y + H_z dxdydz$$ so by Stokes' ...
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### Prove: If $\int_{\phi}\omega = \int_{\psi}\omega$ whenever $\phi$ and $\psi$ have the same endpoints, then $\omega=df$

This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that $\int_{\phi}\omega$ only depends on the endpoints $\phi(a)$ and $\phi(b)$ where ...
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### Clarification: What does it mean when “$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points”

"$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points" Does this mean: (A) $\phi:[a,b]\to U$ and $\psi:[a,b]\to U$ (B) $\phi:[a,b]\to U$ and $\psi:[c,d]\to U$ s.t. ...
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### Regular Parametrization of a Sphere

Is there a function $f:U→ \mathbb{R^3}$, such that: (1) U is an open connected subset of $\mathbb{R^2}$; (2) f is $C^r , r≥1$; (3) the Jacobian of f is of maximal rank at all points of U; (4) ...
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### Line Integrals in $\mathbb{R}^n$ and Differential 1-Forms

I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, ...
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### Substitution of variables in Laplacian

Suppose we have a function $u\colon \mathbb{R}^n \to \mathbb{R}$. Let $x \in \mathbb{R}^n$ and let $x=cy$ for a given constant $c$. How do I write $\Delta_x u(x) = \Delta_x [u(cy)]$ in terms of ...
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### Parametrisation of a surface.

I'm currently going through my course notes, currently teaching the theory leading up to surface integrals. In particular, I am reading the section on the parametrisation of surfaces. However, there ...
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### Partial derivatives and functions equal to 0

If I have the function (family of curves) $$F(x,y,p)=(px)^2+p=0$$ I am under the impression that $$\frac{\partial F(x,y,p)}{\partial p}=2px^2+1$$ Is not always equal to $0$. Please could you ...
There are two functions, $u(t)$ and $v(t)$, $u:[0,\infty]\to {R}$ , $t=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})$, $v=\frac{-uu'}{2tu'-u}$. They should satisfy the next: $u>0$ and $u+2tv>0$. If we ...