if i have $f:\mathbb{R}^2\rightarrow \mathbb{R}$ and $w:\mathbb{R}^3\rightarrow \mathbb{R}$ and for example if we have $f(x,y)=x^3+3xy^2$ $$ f(x,y)=x^3+3xy^2 $$ $$ x^3+3xy^2-f(x,y)=0 $$we can mark this as function of $w(x,y,f(x,y))$ $$ w(x,y,z)=x^3+3xy^2-f(x,y) $$ Is it true that:
$$ \nabla w(x,y,z)=\begin{bmatrix} \frac{\delta}{\delta x} \\ \frac{\delta}{\delta y} \\ \frac{\delta}{\delta z} \end{bmatrix} $$ in every situation $\nabla(x,y,z)$ is normal vector for tangent plane of $f(x,y)$ ??
Problem
if i have $f:\mathbb{R}^2\rightarrow \mathbb{R}$ and $w:\mathbb{R}^3\rightarrow \mathbb{R}$ and for example if we have $f(x,y)=x^3+3xy^2$ $$ f(x,y)=x^3+3xy^2 $$ $$ x^3+3xy^2-f(x,y)=0 $$we can mark this as function of $w(x,y,f(x,y))$
$$ w(x,y,z)=x^3+3xy^2-f(x,y) $$ Is it true that:
$$ \n...
I was just curious if you would explain to me the fundamental differences between the logic courses offered in most philosophy departments and the courses offered in math departments, if you knew?
Can someone explain why a 4x4 rank 1 matrix with an eigenvalue of 0 multiplicity 3, has to be diagonalizable? I understand that rank 1 would mean that the dimension of the Eigenspace(0) is 3 but where does the other eigenvalue come from?
hey, I'm looking at an exercise about solving a differential equation using the Laplace transform. I get the (correct) result, that $$\mathcal{L}[y] = \frac{1}{(1+s^2)^2} e^{-\pi s} + \frac{1}{(1+s^2)^2}.$$ now, the standard solution suggests to "know" the value of $$\mathcal{L}^{-1} \left[\frac{1}{(1+s^2)^2}\right]$$ which I don't. Instead, I rewrote my result as $$\mathcal{L}[\sin(t)] \cdot \mathcal{L}[\sin(t-\pi)] + \mathcal{L}[\sin(t)]^2,$$ which is $$\mathcal{L}[\sin(t)] (\mathcal{L}[\sin(t-\pi)] + \mathcal{L}[\sin(t)]).$$
this is just about Laplace transform and why my approach leads to 0, even though the $\mathcal{L}[y]$ expression is the same as the standard solution provides
In an experiment, n coins are tossed, with each one showing up heads with probability p independently of the others. Each of the coins which shows up heads is then tossed again. What is the probability of observing 5 heads in the second round of tosses, if we toss 15 coins in the first round and p = 0.4? (Hint: First find the mass function of the number of heads observed in the second round.)
Uh, I hope I'm not interrupting some important discussion, but can someone take a look at my question? https://math.stackexchange.com/questions/2613617/a-recursive-divisor-function
@CausingUnderflowsEverywhere limit doesn't exist if you can evaluate limit to two different values when approaching different paths ?
if you think of $\epsilon \delta$ definition of limits this would mean you need to get arbitrary close to the limit in order it to exists. If not this would mean limit doesn't exists
Problem
if i have $f:\mathbb{R}^2\rightarrow \mathbb{R}$ and $w:\mathbb{R}^3\rightarrow \mathbb{R}$ and for example if we have $f(x,y)=x^3+3xy^2$ $$ f(x,y)=x^3+3xy^2 $$ $$ x^3+3xy^2-f(x,y)=0 $$we can mark this as function of $w(x,y,f(x,y))$
$$ w(x,y,z)=x^3+3xy^2-f(x,y) $$ Is it true that:
$$ \n...
(This came about when I was thinking about how people don't like to think about vectors as having "direction" and "magnitude", but here the direction and magnitude do represent different things)
@Akiva Not sure what the normal 1-forms are. The 2-form is locally like $\omega = n_1 dx \wedge dy + n_2 dy \wedge dz + n_3 dz \wedge dx$ where $n = (n_1, n_2, n_3)$ is the outward pointing unit normal.
exhausting. too much work. but I decided to learn a bit about Fourier series, transform and Laplace transform, so I picked up some lecture notes from a course at my uni and started solving some exercises. pretty fun stuff.
i read on a text that $cot(\pi /2 + i\pi y ) = tanh(\pi y)$ but im getting the inverse of $tanh$ , someoene can explain this equality so i will see my error?
so is $0/0$ defined in math? I thought it was indeterminate :/ still a bit confused about how we could determine $\lim_{h \to 0} \frac{k-k}{h}$ to be 0 if we couldn't factor out the h from the denominator
Suppose $f:A\rightarrow \Bbb R $ is Lipschitz continuous and its derivative $Df$ is continuous on a closed subset $E$, $E\subset A$. Define $$\eta_\delta(a)=\text{sup}_{0<|x-a|<\delta \text{ and }x\in E}\frac{|f(x)-f(a)-Df(a)(x-a)|}{|x-a|}$$. I would like to proof the pointwise convergence for $\...
Suppose that $X$ is a well-ordered set in the order topology, and that $A,B \subseteq X$ are nonempty sets satisfying $\overline{A} \cap B = \emptyset$ and $A \cap \overline{B} = \emptyset$. Does it follow that $\overline{A} \cap \overline{B} =\emptyset$?
@user193319 Let $X$ be $\{-1,-1/2,-1/3,-1/4,\dots\}\cup\{0\}$. This is well-ordered, and has order type $\omega+1$ (if you know what that means).
Let $A$ be $\{-1,-1/3,-1/5,\dots\}$, i.e. the negative inverse odd numbers, and let $B$ be $\{-1/2,-1/4,-1/6,\dots\}$, i.e. the negative inverse even numbers.
$\bar A\cap B=A\cap\bar B=\emptyset$, but $\bar A\cap\bar B=\{0\}$.
(Another way to do this is to have $X$ be $\{0,1,2,3,4,\dots,\omega\}$ where you define $\omega$ to be greater than any other element. Then you can take $A$ and $B$ to be the even and odd numbers, and then $\bar A\cap\bar B$ would be $\{\omega\}$.)
@AkivaWeinberger BTW, I think your $A^2 = A_{xy}^2 + A_{yz}^2 + A_{xz}^2$ thing is a consequence of $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ where $\alpha, \beta, \gamma$ are the angles the normal to the plane $A$ belongs in makes with the $x, y, z$-axes.
Problem
if i have $f:\mathbb{R}^2\rightarrow \mathbb{R}$ and $w:\mathbb{R}^3\rightarrow \mathbb{R}$ and for example if we have $f(x,y)=x^3+3xy^2$ $$ f(x,y)=x^3+3xy^2 $$ $$ x^3+3xy^2-f(x,y)=0 $$we can mark this as function of $w(x,y,f(x,y))$
$$ w(x,y,z)=x^3+3xy^2-f(x,y) $$ Is it true that:
$$ \n...
