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Vanishing net charge density does not require vanishing current. Positive and negative charge carriers can cancel in charge density while their oppositely directed motions add to a nonzero current. Charge conservation only requires

\partial_{t} ρ + \nabla \cdot J = 0;

(1)

for magnetostatics this becomes

\nabla \cdot J = 0

Because

\nabla \cdot B = 0

, one may introduce a magnetic vector potential with

B = \nabla \times A .

(2)

It is not unique:

A + \nabla χ

gives the same field for any scalar

χ

For the stated current,

\nabla \cdot J = 0,

(3)

so it is consistent with stationary charge conservation. Direct calculation gives

\nabla \times J = λ J_{0} (sin λ z, cos λ z, 0) = λ J .

(4)

Thus

J

is a Beltrami field. For

λ \neq = 0

, choose

B = \frac{μ _{0}}{λ} J .

(5)

Then

\nabla \cdot B = 0

and

\nabla \times B = μ_{0} J

. A convenient Coulomb-gauge potential is

A = \frac{μ _{0}}{λ ^{2}} J,

(6)

since

\nabla \times A = B

and

\nabla \cdot A = 0

. Gradient gauge terms may of course be added.

λ = 0

, the current is the constant field

J = (0, J_{0}, 0)

. The formulas involving

1/ λ

do not apply; one valid choice is

B = (0, 0, - μ_{0} J_{0} x), A = (0, - \frac{1}{2} μ_{0} J_{0} x^{2}, 0) .

(7)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 3B / Solution by

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Multiplication by

(1 - x^{2})^{- 1/2}

puts the equation in Sturm-Liouville theory form:

\frac{d}{d x} (1 - x^{2} T_{n}^{'}) + \frac{n ^{2}}{1 - x ^{2}} T_{n} = 0 .

(1)

Multiply the equations for

T_{n}, T_{m}

T_{m}, T_{n}

, subtract and integrate. The boundary term vanishes because

1 - x^{2} = 0

at both endpoints and the polynomial derivatives are bounded. Therefore, for

n^{2} \neq = m^{2}

\int_{- 1}^{1} \frac{T _{n} ( x ) T _{m} ( x )}{1 - x ^{2}} d x = 0 .

(2)

This is the usual Chebyshev polynomial orthogonality.

Differentiating the original equation and writing

U_{n} = T_{n}^{'}

gives

(1 - x^{2}) U_{n}^{''} - 3 x U_{n}^{'} + (n^{2} - 1) U_{n} = 0 .

(3)

Its self-adjoint form is

\frac{d}{d x} ((1 - x^{2})^{3/2} U_{n}^{'}) + (n^{2} - 1) 1 - x^{2} U_{n} = 0 .

(4)

The same subtraction argument now yields, for

n^{2} \neq = m^{2}

\int_{- 1}^{1} U_{n} (x) U_{m} (x) 1 - x^{2} d x = 0 .

(5)

These two relations form the Chebyshev derivative Sturm-Liouville pair.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 2F / Solution by

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A map

h : X \to X

is a contraction if there is a constant

q < 1

such that

d (h x, h y) \leq q d (x, y)

(1)

for every

x, y \in X

The contraction mapping theorem states that a contraction of a nonempty complete metric space has a unique fixed point, and that the iterates from every starting point converge to it. To prove this, choose

x_{0} \in X

and put

x_{n + 1} = h (x_{n})

. Then

d (x_{n + 1}, x_{n}) \leq q^{n} d (x_{1}, x_{0}),

(2)

so, for

m > n

d (x_{m}, x_{n}) \leq \frac{q ^{n}}{1 - q} d (x_{1}, x_{0}) .

(3)

Thus

(x_{n})

is Cauchy and converges, by completeness, to some

x_{*}

. A contraction is continuous, so

h (x_{*}) = lim_{n} h (x_{n}) = lim_{n} x_{n + 1} = x_{*} .

(4)

y_{*}

is another fixed point, then

d (x_{*}, y_{*}) \leq q d (x_{*}, y_{*}),

(5)

forcing

x_{*} = y_{*}

For the Newton map

g (x) = x - \frac{f ( x )}{f ^{'} ( x )},

(6)

one has

g (r) = r

and

g^{'} (x) = \frac{f ( x ) f ^{''} ( x )}{f ^{'} ( x ) ^{2}}, g^{'} (r) = 0 .

(7)

On the given neighbourhood,

∣ g^{'} (x) ∣ \leq \frac{M ∣ f ( x ) ∣}{δ ^{2}} .

(8)

Since

f (r) = 0

, choose a closed interval

U^{'}

centred at

r

and contained in

U

so small that

M ∣ f (x) ∣/ δ^{2} \leq 1/2

there. Then

∣ g (x) - r ∣ \leq \frac{1}{2} ∣ x - r ∣,

(9)

g (U^{'}) \subseteq U^{'}

and

g

is a contraction on the complete interval

U^{'}

. The local contraction proof for Newton iteration therefore shows that

r

is the unique fixed point of

g

U^{'}

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 1E / ii by

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 1E / Solution by

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The Eisenstein criterion says that a primitive polynomial

f (X) = a_{n} X^{n} + \dots + a_{0} \in Z [X]

(1)

is irreducible if there is a prime

p

such that

p ∤ a_{n}, p ∣ a_{i} (0 \leq i < n), p^{2} ∤ a_{0} .

(2)

By the Gauss lemma for polynomials, irreducibility over

Q

and over

Z

agree for primitive polynomials.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 18H / Solution by

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If the urn contains

g

green balls, it contains

g + 2

red balls: both update rules preserve the difference

R - G = 2

. Until absorption at

g = 0

, the green count is therefore a birth-death chain with

p_{g} = P (g \to g - 1) = \frac{g}{2 g + 2}, q_{g} = P (g \to g + 1) = \frac{g + 2}{2 g + 2} .

(1)

The function

h (g) = \frac{1}{g + 1}

(2)

is harmonic, since

p_{g} h (g - 1) + q_{g} h (g + 1) = \frac{1}{2 ( g + 1 )} + \frac{1}{2 ( g + 1 )} = h (g) .

(3)

Let

τ_{0}

and

τ_{N}

be the hitting times of

0

and

N

. The optional sampling theorem for a supermartingale, applied to the bounded stopped martingale

h (G_{n \land τ_{0} \land τ_{N}})

, gives

h (m) = P_{m} (τ_{0} < τ_{N}) + \frac{1}{N + 1} P_{m} (τ_{N} < τ_{0}) .

(4)

Solving,

P_{m} (τ_{0} < τ_{N}) = \frac{N - m}{N ( m + 1 )} .

(5)

Letting

N \to \infty

, the events on the left increase to eventual termination. The harmonic hitting probability for the balanced-difference urn is therefore

P (the process terminates) = \frac{1}{m + 1} .

(6)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 17D / c by

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 17D / b by

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 17D / a by

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 16C / Solution by

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A gauge transformation is

A_{μ} ⟼ A_{μ} + \partial_{μ} χ .

(1)

Because partial derivatives commute, the added contribution to

F_{μν}

\partial_{μ} \partial_{ν} χ - \partial_{ν} \partial_{μ} χ = 0,

(2)

so the electromagnetic field tensor is gauge invariant.

Define

E = - \nablaΦ - \partial_{t} A, B = \nabla \times A .

(3)

Since

\partial_{0} = c^{- 1} \partial_{t}

and

A_{0} = - Φ/ c

F_{0 i} = - \frac{E _{i}}{c}, F_{i 0} = \frac{E _{i}}{c}, F_{ij} = ϵ_{ijk} B_{k} .

(4)

The identity

\partial_{λ} F_{μν} + \partial_{μ} F_{ν λ} + \partial_{ν} F_{λ μ} = 0

(5)

follows directly from

F = d A

. Its spatial and mixed components are

\nabla \cdot B = 0, \nabla \times E = - \partial_{t} B .

(6)

For the other two equations define the four-current

j^{μ} = (c ρ, J) .

(7)

Then

\partial_{ν} F^{0 ν} = μ_{0} j^{0}

gives

\nabla \cdot E = μ_{0} c^{2} ρ = \frac{ρ}{ϵ _{0}},

(8)

and

\partial_{ν} F^{i ν} = μ_{0} j^{i}

gives

\nabla \times B - \frac{1}{c ^{2}} \partial_{t} E = μ_{0} J .

(9)

This is the Covariant Maxwell equation with the minus-plus-plus-plus metric.

Using

F^{ρ σ} F_{ρ σ} = 2 (B^{2} - \frac{E ^{2}}{c ^{2}}),

(10)

one finds

T^{00} = \frac{1}{2 μ _{0}} (B^{2} + \frac{E ^{2}}{c ^{2}}) = \frac{1}{2} (\frac{B ^{2}}{μ _{0}} + ϵ_{0} E^{2}) .

(11)

This is the electromagnetic energy density.

For a null vector, the trace term in

T^{μν} k_{μ} k_{ν}

vanishes. Put

q^{ρ} = k_{μ} F^{μ ρ} .

(12)

Antisymmetry gives

q \cdot k = 0

. A vector orthogonal to a null vector has nonnegative Minkowski norm, so

T^{μν} k_{μ} k_{ν} = \frac{1}{μ _{0}} q^{ρ} q_{ρ} \geq 0.

(13)

Explicitly, in a frame with

k^{μ} = κ (1, 1, 0, 0)

q^{2} = κ^{2} [(B_{3} - \frac{E _{2}}{c})^{2} + (B_{2} + \frac{E _{3}}{c})^{2}] .

(14)

Hence the null energy condition for the electromagnetic field is strict whenever the contraction is nonzero:

T^{μν} k_{μ} k_{ν} > 0 .

(15)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A / vi by

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A / v by

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A / iv by

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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