2
votes
0answers
50 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
0
votes
1answer
68 views

Partial derivative and derivative.

I want to show that if $f:\mathbb{R}^n\to \mathbb{R}$ and $df_a$ is the derivative of the function at $a$ then $df_a(v)=\displaystyle\frac{\partial f}{\partial v}(a)$. I saw a few proofs of this ...
3
votes
2answers
66 views

If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse.

Problem: Let $U\subset\mathbb{C}$ be an open set, $f:U\to\mathbb{C}$ an holomorphic function of class $C^1$ and $z_0\in U$. Prove that if $f'(z_0)\neq 0$ then there exists a neighborhood $V$ of $z_0$ ...
2
votes
1answer
79 views

minimum and maximum of $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$

we are asked to find the minimum and maximum of the function$f:A \to A$ $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$ Where $A$ is the triangle bound by $x=0$,$y=0$ and $y=-x+2\pi$ I'd like someone to review ...
2
votes
1answer
34 views

Cauchy inequality proof

I am studying cauchy inequality proof from notes I have from my class$$(\forall\vec{x},\vec{y}\in\mathbb{R}^n):|\sum_{i=1}^{n}x_iy_i|\le||\vec{x}||\cdot||\vec{y}||$$ We choose $\vec{x},\vec{y}$. And ...
0
votes
1answer
37 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
1
vote
2answers
82 views

Causal character of a surface (Lorentz-Minkowski space $\mathbb{L}^3$)

I'm trying to analyze the causal character of the surface $x^2 + y^2 - z^2 = -1$ in Lorentz-Minkowski space $\mathbb{L}^3$, with the convention $\mathrm{diag[1,1,-1]}$, that is $$\langle \left(x_1, ...
2
votes
1answer
49 views

If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
1
vote
1answer
40 views

For $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$, how is this stationary point $\;$ a saddle point? - Question 14

14. a$)$ Find all stationary points of $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$. $b)$ Classify the stationary points of $f$ as local maxima, local minima or saddle points. Provided Solution a $)$ We compute ...
2
votes
1answer
49 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
3
votes
1answer
76 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$ - 2012 9C

Question: 2012 9C. Consider the (cutoff) paraboloid defined by $z= x^2 + y^2 , \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ...
1
vote
1answer
80 views

With Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$

2012 9C. Consider the (cutoff) paraboloid defined by $z=x^2 + y^2, \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ask ...
2
votes
1answer
43 views

Doubt on proof of Implicit function theorem

On The second part of the proof, where it's stated that V is open as it is the inverse image of the open set $V_0$ under the continuous mapping $y \rightarrow (0, y)$. Let $\pi$ be this continuous ...
5
votes
1answer
170 views

With Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable parametrisations for ...
2
votes
1answer
32 views

Partial differentiability with respect to $x$ and $y$ of $\int_0^x z(s,y)ds$ where $\partial_y z \in C^0$

I have the following (not accredited and not mandatory) Exercise: Problem: Let $z : \mathbb{R}^2 \to \mathbb{R}$ be continuous and in respect to its second variable partially differentiable. ...
2
votes
3answers
276 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the ...
1
vote
2answers
52 views

Geometrical Application of Calculus with Speed

Two vehicles are heading for a crossroad (point $C$) and intend to pass straight through. Vehicle $A$ is $100\,\mathrm{km}$ due North travelling at $80\,\mathrm{km}/\mathrm{hr}$ towards $C$ Vehicle ...
4
votes
2answers
109 views

Why is this proof incorrect? - $f(b) - f(a) = [Df(c)](b-a) = \nabla f(c) \cdot (b-a)$

I'm currently a student in a Vector Calculus class, and received my exam back today. I plan to go to the professor's office hours, but I'd like to ask here (in case there's some blatantly obvious ...
2
votes
1answer
65 views

Prove $\nabla f$ is orthogonal to the surface $f$

I'm trying to prove that $\nabla f$ is orthogonal to the surface $f$. I think I have a valid proof but I'm not sure that it is rigorous. To prove $\nabla f$ is normal I am proving that $\nabla ...
2
votes
1answer
39 views

Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: \begin{equation} u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3. \end{equation}I ...
4
votes
1answer
48 views

Find the mistake in my proof of partial derivatives exisiting implying differentiability.

Let $\Omega\subseteq \mathbb{R} ^2$ be open. Let the function $f: \mathbb{R} ^2 \mapsto \mathbb{R}$ have partial derivatives $f_x'$ and $f'_y$ at every point in $\Omega$. I will now try to prove that ...
1
vote
1answer
65 views

proving existence of diffeomorphism

In my hand out of manifold, I found the following lemma but there is no proof there: Let $U\subseteq\mathbb{R}^m$ be open and pick some $a\in U$. Suppose that $f:U\mapsto \mathbb{R}^n$ is a smooth ...
2
votes
1answer
154 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
1
vote
1answer
67 views

Applying the implicit function theorem

Let $F:\mathbb{R}^2\to\mathbb{R}$ of class $C^2$ and $\displaystyle\frac{\partial f}{\partial v}(u,v)\neq0\; \forall(u,v)\in\mathbb{R}^2$. If $(x_0,y_0,z_0)\in\mathbb{R}^3$ is such that ...
2
votes
1answer
51 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
0
votes
1answer
48 views

Compute a line integral with Green's theorem

Consider the two-dimensional vector field $\mathbf{F}=(x^2-y, x+y^2)$, with $\mathbf{x}=(x,y)\in \mathbb{R}^2$. Compute the divergence and the curl of $\mathbf{F}$. Compute the line ...
1
vote
1answer
71 views

Calculate $\int_D x^2 dxdydz$ for $D$ an ellipsoid

Let $D$ be the ellipsoid $$x^2/a^2 + y^2/b^2 + z^2/c^2 \leq 1.$$ Compute $\int_D x^2 dx dy dz$. Map to the unit ball by $\varphi: (x,y,z) \mapsto (x/a, y/b, z/c)$. Now $$\int_D \frac{1}{abc}x^2 ...
0
votes
1answer
141 views

Help with Proof of the Chain Rule

I'm trying to prove the chain rule. I started out using the standard Frechet Derivative definition. After getting stuck (and generally disliking the heavy use of inequalities), I adopted an approach a ...
2
votes
1answer
56 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
4
votes
1answer
119 views

Prove that there exists a scalar potential $f( \mathbf{ x} )$ such that $\mathbf{ F } = − \nabla f$ [2012 11c]

Question: If F is an irrotational vector field (i.e. $ \nabla \times \mathbf{ F = 0 }$ everywhere), prove that there exists a scalar potential $f( \mathbf{ x} )$ such that $\mathbf{ F } = − ...
2
votes
2answers
444 views

Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
3
votes
4answers
138 views

Show that $\int_{1}^\infty \dotsb\int_{1}^\infty \frac{dx_1 \dotsb dx_n}{x_1^{\alpha_1}+\dotsb + x_n^{\alpha_n}}<\infty$

Here's my solution to an old qualifier problem. Would you tackle it differently? Is there a flaw in my work? Suppose that $\alpha_1, \dotsc, \alpha_n$ are positive numbers such that ...
2
votes
0answers
68 views

How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?

Let $U\subset\mathbb{R}^n$ be a open set and $f:U\to\mathbb{R}$ a function in $C^\infty_c(U)$. Evans PDE book uses the following result $$\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial ...
0
votes
1answer
50 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
12
votes
1answer
225 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
6
votes
0answers
121 views

Proof of inverse function theorem by approximation property

In proving the inverse function theorem using the approximation characterization of the derivative, we are given $F:\mathbb{R}^n \to \mathbb{R}^n$ such that  $$F(p_0 + h) - F(p_0) = DF_{p_0}(h) + ...
6
votes
1answer
51 views

Is this proof that the vectors are colinear correct?

I was solving the following exercise: "Let $x,y \in \mathbb{R}^n$ be nonzero such that if $z$ is orthogonal to $x$ then $z$ is orthogonal to $y$. Prove that $x$ and $y$ are colinear". My idea was: ...
4
votes
2answers
100 views

Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$

Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$ so I said that the integral we need is $\iint_{D} {x^2 + y^2 dxdy}$. But when I ...