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42

Is the point of math and math classes to learn the big-picture concepts of how to apply mathematical tools or is the point to learn the details to the ground level? Both. One is difficult without the other. How are you going to solve equations that Maple can't solve? How are you going to solve it, exactly or numerically? What's the best way to solve ...


23

Many, many reasons... I was in a number theory program in high school that prohibited calculators. You know how many more things you notice about the theory when you have to find ways to figure out the details yourself? Maybe you have to compute something absurd like $3^{24} (\mod 7)$, but you know what? That's actually really easy once you realize the ...


19

Without knowing the details of a process, it is extremely difficult to program tools yourself that compute this process. Put more succinctly, without understanding an algorithm, it is nearly impossible to implement the algorithm. This is not nearly its only justification, but I would wager it is the most relevant, given your background.


18

Is the point of math and math classes to learn the big-picture concepts of how to apply mathematical tools or is the point to learn the details to the ground level? I will second Bennett, the point is both. Consider the analogy that learning mathematics and physics is much like constructing maps. First, you will see maps others have created, how details ...


13

So that someone can teach the computer how to do it better.


9

(First, read my related answer on Why do we still do symbolic math?) Unlike mathematicians of long ago, I don't have to look up logarithm tables whenever I need to calculate $\log(24)$. My calculator or computer can do that for me, and that's a great advantage. They are great tools. Likewise, my tools can tell me what $298379187912 / 81238.235$ is, or ...


9

Fourier analysis, which has applications in fields as diverse as PDEs and algebraic number theory, has its origins in physics (which, at the time, really wasn't as distinct from mathematics as it is now). The study of waves and heat flow lead Fourier and others to the theory of trigonometric series. For example, by working formally you can show that ...


7

Addressing the Turing Machine: The Turing machine was not invented due to any physical or computational science. In fact, they were invented to solve a math problem. Prior to 1930's, computer science (as in theoretical computer science) did not exist. As hard as it may be to believe today, people did not even entirely understand what algorithms were but ...


5

If an object is twice as far away, the angle it subtends (in the small angle approximation) is half as large. The solid angle it covers is then one quarter as large. This reflects the fact that the surface angle of a sphere of radius $R$ is $4\pi R^2$ so as $R$ doubles the portion of the sphere that the object covers is reduced by a factor $4$


5

Looks perfect to me. My only point of comment is that you should have $−h''(x_0)$ everywhere that you have $h''(x_0)$. The skier can only get airborn if he skies over a bump, meaning $h''(x_0)<0$. I'm making the assumption that you've chosen the positive y-axis up. Then, going airborn does not necessarily mean that he's accelerating in the direction of ...


5

The term energy comes from electrostatics. The energy density of electric field $\mathbf E$ is $\frac12 \varepsilon_0\mathbf E^2$; to get the total energy of the field we integrate that. The derivation of $\frac12 \varepsilon_0\mathbf E^2$ is given in the Wikipedia article Electric potential energy. But I'll outline the steps here: the potential energy ...


4

At the simplest level of 'why?' - and a lot of these other answers are very correct - but at the simplest level, so you can do a sanity-check on the result you get back. For instance. Why learn to do basic math, when calculators are so omnipresent? Well, there was a situation a few years back where the register had gone down, so the cashier was having to ...


4

Take this one: $\iint_Sr\cdot ndS$ where S is a closed surface as you wish. Indeed, we then have $$\iint_Sr\cdot ndS=\iiint_V\nabla\cdot r dV=\iiint_V\left(\partial_x\text{i}+\partial_y\text{j}+\partial_z\text{k}\right)\cdot(x\text{i}+y\text{j}+z\text{k})dV=3V$$ where $V$ is the volume enclosed by $S$.


3

Ok, a semi not stupid answer... Wavelets and wavelet transform are invented in the same sense as Newton and Leibniz's calculus, by a physicist and a mathematician at around the same time (though many argues that Newton brusquely pilfered the recognition of Leibniz's contribution to calculus after his early death) In Mallat's book, a Wavelet Tour of Signal ...


3

It’s all about the three kinds of color receptors in the eye. A typical human responds to color through three types of cone cells in the retina, called $L$, $M$, and $S$ cones. Each type has a different response curve as a function of frequency. The sensation of color is distinguished by a tristimulus, which can be considered a vector $(r_L,r_M,r_S)$ of ...


3

A man walks into a restaurant, orders food, eats, then leaves. Did the man have to cook his food? Did he pay? Did he sit down? Did he eat the food he ordered or just eat something else entirely. Your problem is known in Psychology as the interpretation and application of one's schema. http://en.wikipedia.org/wiki/Schema_(psychology) Because people are ...


3

Assume you have a systematic way of encoding mathematical propositions, so that for any proposition $P$ there is a unique number $n(P)$ encoding it. Variants of Gödel's argument show that, for any total computable function $f(n)$, there is always a provable proposition $P$ in number theory, such that the shortest proof for $P$ is longer than $f(n(P))$ ...


3

If we ignore some context, and instead consider the quote at face value, it is not Gödel's result that would be relevant, but rather the fact that the theorems of just about any interesting theory do not form a recursive set. This means that there is no computing device that can predict which statements are provable in the theory and which are not. The ...


3

The vector field is continuous on its domain, which is $\mathbb{R}^2 - \{y = 0\}$. It's nevertheless quite reasonable to ask how not being defined on the $x$-axis affects a given problem. For integrations of functions along curves, it just means that the paths we consider must trace out curves contained entirely within the domain; in our situation, this just ...


3

Numerical data of more than 3 dimensions are difficult to visualize on screen. Our computational power is still limited. But it IS still faster than analytic solving unless you already solved similar problems analytically. Floating/fixed-point operations even on 64 bit processors are granular, and can't represent arbitrarily large/small numbers. Computers ...


3

Quantum Mechanics problems cannot yet be solved automatically on the computer. We're not even close to that point. There are many computational theories and approximations, but no canned program that can reliably give you solutions to a wide range of Quantum problems. Not even narrow ranges of problems can be solved so easily at this point. Maybe you'll be ...


3

Your presume wrong: if $dl^2=dx^2+dy^2$, you can't conclude $l=\sqrt{x^2+y^2}$. You're talking about differentials here. All that you can say starting from $$dl^2=dx^2+dy^2$$ is that $$dl=\sqrt{dx^2+dy^2}$$ This is the result of considering the differential length of arc $dl$ as the hipotenuse of a triangle made of sides $dx$, $dy$, and $dl$ in the ...


2

You're going to need to know how to solve differential equations, especially if you're interested in quantum mechanics. Even some of the most basic examples of potentials used for quantum mechanics make CAS choke. Examples: Delta potential, finite potential well (a lot easier to use the basic theory of differential equations to get a transcendental equation ...


2

Let us imagine that the graph of $f$ in $\mathbb{R}^3$ represents some sheet of uniform elastic rubber. If we study the idealized case, we find that the force in the horizontal direction is $\Delta f$. (This is reasonable, as when we study Laplace's equation we find that harmonic functions, those hwich satisfy $\Delta f =0$ satsify a property where $f(x)$ is ...


2

You can check that there exists a potential function, $U$, such that $\mathbf{F} = -\nabla U $, since $\nabla \wedge \mathbf{F} = 0$ (then, $\mathbf{F}$ is known as a force field). You can also verify that $U$ is given by: $$ U(x,y) =- \frac{x^2}{y},$$ which is continuous in its domain, which is the same as the domain of $\mathbf{F}$. Therefore, you can ...


2

$$\textbf{v}.\textbf{r}=(\textbf{c}\times\textbf{r}).\textbf{r}=0\tag{1}$$ and $$\textbf{v}.\textbf{c}=0\tag{2}$$ Therefore $\textbf{v}\perp\textbf{r}~~\forall t$ and $\textbf{v}\perp\textbf{c}$. $$\frac{d}{dt}(\textbf{r}.\textbf{c})=\textbf{v}.\textbf{c}=0\implies \textbf{r}.\textbf{c}=constant\tag{3}$$ which is the equation of a plane orthogonal to ...


2

It is not unique, in fact there are infinitely many possible solutions for any choice of $c>0$. Simply take a vector field $\mathbf{F}$ satisfying: $$\int_S\mathbf{F}\cdot\mathbf{n}dS=k\neq0$$ and then define your new vector field by $\mathbf{G}=\frac{1}{k}\mathbf{F}$. Now there are infinitely many such $\mathbf{F}$, since by Stokes theorem you we that: ...


2

When they talk about "closed surfaces $S$ containing $0$" they tacitly mean that such $S$ should bound a compact body $B\subset{\mathbb R}^3$ which contains $0$ in its interior. Now we cannot have arbitrarily tiny such surfaces giving a fixed value $c>0$ for the integral in question unless something terrible happens at $0$. You have remarked that the ...


1

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1

Gauge theory is one example where one sees physics stimulating the development of new mathematics. 15 years ago Seiberg and Witten based on intuitions coming from physics proposed a new powerful way of doing gauge theory and in particular almost trivializing earlier impressive results of Fields medal caliber, such as Donaldson's diagonalisation result for ...



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