Ciro Santilli @cirosantilli 35

 Incoming links: Derivative

Covariant derivative  Updated 2025-01-01  +Created 1970-01-01

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A generalized definition of derivative that works on manifolds.

TODO: how does it maintain a single value even across different coordinate charts?

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Distribution (mathematics)  Updated 2025-01-01  +Created 1970-01-01

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Generalize function to allow adding some useful things which people wanted to be classical functions but which are not,

It therefore requires you to redefine and reprove all of calculus.

For this reason, most people are tempted to assume that all the hand wavy intuitive arguments undergrad teachers give are true and just move on with life. And they generally are.

One notable example where distributions pop up are the eigenvectors of the position operator in quantum mechanics, which are given by Dirac delta functions, which is most commonly rigorously defined in terms of distribution.

Distributions are also defined in a way that allows you to do calculus on them. Notably, you can define a derivative, and the derivative of the Heaviside step function is the Dirac delta function.

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Linear map  Updated 2025-01-01  +Created 1970-01-01

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A linear map is a function

f : V_{1} (F) \to V_{2} (F)

where

V_{1} (F)

and

V_{2} (F)

are two vector spaces over underlying fields

F

such that:

\forall v_{1}, v_{2} \in V_{1}, c_{1}, c_{2} \in F f (c_{1} v_{1} + c_{2} v_{2}) = c_{1} f (v_{1}) + c_{2} f (v_{2})

(1)

A common case is

F = R

V_{1} = R_{m}

and

V_{2} = R_{n}

One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of

F

Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.

As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.

The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.

For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.

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Linear operator  Updated 2025-01-01  +Created 1970-01-01

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We define it as a linear map where the domain is the same as the image, i.e. an endofunction.

Examples:

a 2x2 matrix can represent a linear map from $R^{2}$ to $R^{2}$ , so which is a linear operator
the derivative is a linear map from $C^{\infty}$ to $C^{\infty}$ , so which is also a linear operator

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Momentum operator  Updated 2025-01-01  +Created 1970-01-01

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One dimension in position representation:

\overset{p}{^} = - i ℏ \frac{\partial}{\partial x}

(1)

In three dimensions In position representation, we define it by using the gradient, and so we see that

\overset{p}{^} = - i ℏ \frac{\partial}{\partial x}

(2)

Video 1.

Position and Momentum from Wavefunctions by Faculty of Khan (2018)

Source. Proves in detail why the momentum operator is

\frac{\partial}{\partial x}

. The starting point is determining the time derivative of the expectation value of the position operator.

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One-form  Updated 2025-01-01  +Created 1970-01-01

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www.youtube.com/watch?v=tq7sb3toTww&list=PLxBAVPVHJPcrNrcEBKbqC_ykiVqfxZgNl&index=19 mentions that it is a bit like a dot product but for a tangent vector to a manifold: it measures how much that vector derives along a given direction.

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SymPy  Updated 2025-01-01  +Created 1970-01-01

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This is the dream cheating software every student should know about.

It also has serious applications obviously. www.sympy.org/scipy-2017-codegen-tutorial/ mentions code generation capabilities, which sounds super cool!

The code in this section was tested on sympy==1.8 and Python 3.9.5.

Let's start with some basics. fractions:

from sympy import *
sympify(2)/3 + sympify(1)/2

outputs:

7/6

Note that this is an exact value, it does not get converted to floating-point numbers where precision could be lost!

We can also do everything with symbols:

from sympy import *
x, y = symbols('x y')
expr = x/3 + y/2
print(expr)

outputs:

x/3 + y/2

We can now evaluate that expression object at any time:

expr.subs({x: 1, y: 2})

outputs:

4/3

How about a square root?

x = sqrt(2)
print(x)

outputs:

sqrt(2)

so we understand that the value was kept without simplification. And of course:

sqrt(2)**2

outputs 2. Also:

sqrt(-1)

outputs:

I is the imaginary unit. We can use that symbol directly as well, e.g.:

I*I

gives:

-1

Let's do some trigonometry:

cos(pi)

gives:

-1

and:

cos(pi/4)

gives:

sqrt(2)/2

The exponential also works:

exp(I*pi)

gives;

-1

Now for some calculus. To find the derivative of the natural logarithm:

from sympy import *
x = symbols('x')
diff(ln(x), x)

outputs:

1/x

Just read that. One over x. Beauty.

Let's do some more. Let's solve a simple differential equation:

y''(t) - 2y'(t) + y(t) = sin(t)

Doing:

from sympy import *
x = symbols('x')
f, g = symbols('f g', cls=Function)
diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x)**4)
print(dsolve(diffeq, f(x)))

outputs:

Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)

which means:

f (x) = C_{1} + C_{2} x e^{x} + c o s (x) / 2

(1)

To be fair though, it can't do anything crazy, it likely just goes over known patterns that it has solvers for, e.g. if we change it to:

diffeq = Eq(f(x).diff(x, x)**2 + f(x), 0)

it just blows up:

NotImplementedError: solve: Cannot solve f(x) + Derivative(f(x), (x, 2))**2

Sad.

Let's try some polynomial equations:

from sympy import *
x, a, b, c = symbols('x a b c d e f')
eq = Eq(a*x**2 + b*x + c, 0)
sol = solveset(eq, x)
print(sol)

which outputs:

FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))

which is a not amazingly nice version of the quadratic formula. Let's evaluate with some specific constants after the fact:

sol.subs({a: 1, b: 2, c: 3})

which outputs

FiniteSet(-1 + sqrt(2)*I, -1 - sqrt(2)*I)

Let's see if it handles the quartic equation:

x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)

Something comes out. It takes up the entire terminal. Naughty. And now let's try to mess with it:

x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(f*x**5 + e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)

and this time it spits out something more magic:

ConditionSet(x, Eq(a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5, 0), Complexes)

Oh well.

Let's try some linear algebra.

m = Matrix([[1, 2], [3, 4]])

Let's invert it:

m**-1

outputs:

Matrix([
[ -2,    1],
[3/2, -1/2]])

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The derivative is the generator of the translation group  Updated 2025-01-01  +Created 1970-01-01

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Take the group of all Translation in

R^{1}

Let's see how the generator of this group is the derivative operator:

\frac{\partial}{\partial x}

(1)

The way to think about this is:

the translation group operates on the argument of a function $f (x)$
the generator is an operator that operates on $f$ itself

So let's take the exponential map:

e^{x_{0} \frac{\partial}{\partial x}} f (x) = (1 + x_{0} \frac{\partial}{\partial x} + x_{0}^{2} \frac{\partial ^{2}}{\partial x ^{2}} + \dots) f (x)

(2)

and we notice that this is exactly the Taylor series of

f (x)

around the identity element of the translation group, which is 0! Therefore, if

f (x)

behaves nicely enough, within some radius of convergence around the origin we have for finite

x_{0}

e^{x_{0} \frac{\partial}{\partial x}} f (x) = f (x + x_{0})

(3)

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!

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