A cone-shaped water tank with height H=10 m and radius R=5 m is as shown in the figure below. Water is leaking at a rate of π m3/s through a small hole at the bottom of the tank. How fast was the depth of water decreasing (in m/s) when the depth of the water is 5 m.
My answer ends up as a positive 4/25 m/s but i think it should be negative right since the height is DECREASEING
@CroCo The source from which you get the term must have a precise definition. It does not mean that the phrase must be used consistently from source to source.
For example
a characteristic function in real analysis is very different from a characteristic function in probability
As a more relevant example, not every text defines "increasing" and "decreasing" functions in an interval the same way. Some include equality, some don't.
I have never studied physics, and even with that I'd say that plenty of math books are inconsistent with terminology. What matters is that definitions are consistent within a source.
@Clarinetist, in another book I'm reading, the author states "Some books generalize the idea of positive definiteness and negative definiteness to include nonsymmetric matrices."
Here's another example: in probability, there are two ways to define "geometric distribution" and "negative binomial distribution," and also two ways to write the "exponential distribution." In all three of these situations, you gain experience over time to know how to explain these concepts so that it's not ambiguous, or you just state the definition right out
Right, and there's nothing wrong with that
What matters is that the source you're using states the definition somewhere when it uses a fact on positive definite matrices, or it's clear from the context
I'll quit throwing examples at you, but for example, people use $A \subset B$ to mean the same as $A \subseteq B$, but people also use this to mean "$A$ is a subset of $B$, but $A \neq B$"
@Clarinetist, I'm not sure if I totally I agree with you about the subset. While you're making an excellent point, but I feel some times it is plausible to have two definitions for the same notion but sometimes not.
@user6232128, I'm not sure really what to say but I got confused for this definition since linear algebra is mature field and it is every well studied.
CroCo, everything you've typed is consistent with my statement. You say you're reading pure math, but the definition with pivots sounds more applied or numerical. What is the source ?
There are commonly different definitions of Fourier transform and inverse transform. 10% of undergraduate differential geometry texts have the sign wrong on the definition of torsion of a curve. Deal with it.
@TedShifrin, for the first book, Introduction to Linear Algebra by Gilbert Strang, 4th Edition The second book is Optimal State Estimation by Dan Simon
@TedShifrin , am I picking up the wrong book then? When I say wrong I mean it is not written carefully.
@user6232128, I totally agree but my question is really how come someone can come up with a definition where one can come up with counterpart example? This is really weird to me as graduate student in engineering
Hello everyone. I'm trying to prove that every compact subset of a metric space is closed. I wouldn't like to prove it through the openness of its complement. Any hints? What I have (aiming at contradiction) is a $y\in K'$ such that $y\not\in K$. So there's a neighborhood $V_{r}(y)$ that always has an element of $K$ within its arms, whatever positive value $r$ assumes. How can I use Ks compactness to arrive at a contradiction?
@TedShifrin That's how the the author is doing and I'm trying to reprove each theorem by myself, so I feel that if I prove it using openness of the complement I would be too influenced by the author's idea.
You may want to look at Stein and Shakarchi, but it's problem sets can be really hard. It's not too much on the rigour side of things. Some people prefer that, some people don't.
But to be honest, I have never seen a good complex analysis book yet, which doesn't skip important details.
I was reading Apostol's introduction to analytic number theory and found this:
My question is, how did he turn the sum $$\mu(1)+\mu(p_1)+...+\mu(p_k)+\mu(p_1 p_2)+...\mu(p_{k-1}p_k)+...\mu(p_1...p_k)$$ to $$1+\binom{k}{1}(-1)+...+\binom{k}{k}(-1)^k$$
I am sometimes really bad in noticing obvious things so please tell if this is obvious from the definition of the mobius function.
I'm trying to understand if a 3D convolution of the sort performed in a convolutional layer of a CNN is associative. Specifically, is the following true:
$$
X \otimes(W \cdot Q)=(X \otimes W) \cdot Q,
$$
where
$\otimes$ is a convolution,
$X$ is a 3D input to a convolution layer,
$W$ is a 4D weig...
@ShaVuklia Sorry to have been hurtful. To clarify the meaning of the jokes, I thought you were still pursuing physics, so learning alg geo seemed very arcane for what you wanted to know.
Instead you are pursuing it because you want arcane knowledge, which is respectable, but makes you an algebraist. :D
If the origin of a 2-manifold $M$ is $O=(1,1)$ and geodesics through $O$ are $y=x^{\pm n}$ for $x,y,n \in \Bbb R^+$ then what can be said about the curvature of $M?$
@TobiasKildetoft Probably because of irritating conversations like the one we just had, given that apparently people in the UK still say tuition to mean what you call tuition fees
@MikeMiller Words change meaning following popular usage. Since nobody normally uses tuition in the original meaning, it made sense to cut off the word that had become superfluous
@TobiasKildetoft Like I said, the language split happened before the 1970s --- I can find the definition of tuition as "payment for instruction" in a 1910s american dictionary
The less common definition still at the time, but I don't think the narrative that this is about the cost of US university is correct
Does this way od differentiating make sense? For example, if we want to calculate $\frac{d}{dx}e^x$, first calculate: $$\int 1\,de^x$$ by this method:$$\int 1\,de^x=int \ln(e^x)de^x=\int \ln\,y\,dy$$
@MikeMiller Let $M$ be a Riemannian manifold, then the gradient is an operator $\nabla : C^\infty(M) \to \Gamma(M; TM)$. Completing with respect to the $L^2$-inner product on both sides gives $\nabla : L^2(M) \to \Gamma_{L^2}(M; TM)$, I suppose? This must be a bounded linear operator, and $\nabla^* : \Gamma_{L^2}(M; TM) \to L^2(M)$ be the adjoint, so we can define Laplacian $\Delta = \nabla^* \nabla$ (this is like saying $d^* d$ but without going to forms).
Automatically self-adjoint; but why does it have discrete spectrum?
@anakhro There are exactly $4$ rotational symmetries (identity, and 3 of the other ones you described), so that's the symmetry group inside of $SO(3)$. Taking preimage under $O(3) \to SO(3)$, which is a $2$-sheeted covering, gives all symmetries, so that's of order $2 \cdot 4 = 8$. It seems you are missing the antipodal map: Suppose $[-1, 1] \times [-2, 2] \times [-3, 3]$ is your cuboid, then $x \mapsto -x$ gives an isometry, given by composing all three reflections.
To prove that there are exactly $4$ rotational symmetries you can use orbit-stabilizer, orbit of any face can be exactly 2 because there's only one other face which looks like it. Stabilizer is also exactly 2, the rotation through the axis going through it's center.
