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Sobolev Space

The Sobolev space W^k, p(U) consists of all locally summable functions u: U to R such that for each multi-index alpha with |alpha| <= k, the weak derivative D^alphau exists and belongs to L^p(U).

Cambridge IA Numbers and Sets

A proof is a finite sequence of statements, each of which is either an axiom, a previously established result, or follows from earlier statements in the sequence by a valid logical rule, such that the

Discrete Mathematics 1

Continuity (Real Analysis)

Consider the polynomial p(x) = x^3 - 2. Since p(1) = -1 0, the function changes sign on [1, 2], so there must be a root somewhere in between — namely sqrt[3]2. This reasoning is the Intermediate Value

Analysis 1

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Continuity (Real Analysis)

Analysis Real Analysis

Consider the polynomial $p(x) = x^3 - 2$. Since $p(1) = -1 < 0$ and $p(2) = 6 > 0$, the function changes sign on $[1, 2]$, so there must be a root somewhere in between — namely $\sqrt[3]{2}$. This reasoning is the Intermediate Value Theorem, and it requires continuity. If we replace $p$ with the discontinuous function \begin{align} f(x) =…

Updated Apr 4

Power Series

Analysis Real Analysis

How do you define $e^x$? One approach: declare $e = \lim_{n \to \infty} (1 + 1/n)^n$ and define $e^x$ by continuity and the laws of exponents. But this requires proving that $e^{p/q} = (e^{1/q})^p$ makes sense for rationals, extending to irrationals by density of $\mathbb{Q}$, and verifying differentiability — a laborious chain of arguments that…

Updated Apr 4

Cauchy Sequence

Analysis Real Analysis

Consider the sequence of decimal approximations to $\sqrt{2}$: \begin{align} 1, \quad 1.4, \quad 1.41, \quad 1.414, \quad 1.4142, \quad 1.41421, \quad \ldots \end{align} Each term is rational, and the terms are getting closer to each other: the distance $|a_n - a_m|$ between any two late terms is less than $10^{-\min(n,m)}$. In $\mathbb{R}$, this…

Updated Apr 4

Supremum and Infimum

Analysis Real Analysis

Does $\sqrt{2}$ exist? The set $S = \{x \in \mathbb{Q} : x^2 < 2\}$ is nonempty ($1 \in S$) and bounded above ($2$ is an upper bound). If the least upper bound of $S$ existed in $\mathbb{Q}$, it would have to be $\sqrt{2}$ — but $\sqrt{2}$ is irrational. In $\mathbb{Q}$, the set $S$ has upper bounds but no least upper bound: there is no smallest…

Updated Apr 4

Uniform Convergence

Analysis Real Analysis

The sequence of functions $f_n(x) = x^n$ on $[0, 1]$ converges pointwise: for each fixed $x \in 0, 1)$, $x^n \to 0$, and $f_n(1) = 1 \to 1$. Every $f_n$ is continuous, yet the pointwise limit \begin{align} f(x) = \begin{cases} 0 & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x = 1 \end{cases} \end{align} is discontinuous at $x = 1$. The limit of…

Updated Apr 4

Cambridge IB Linear Algebra

Algebra Linear Algebra

Let $\mathbb{F}$ be a field. An $\mathbb{F}$-vector space is an abelian group $(V, +)$ equipped with a scalar multiplication map \begin{align} \mathbb{F} \times V &\to V \\ (\lambda, \mathbf{v}) &\mapsto \lambda \mathbf{v}, \end{align} satisfying the following axioms for all $\lambda, \mu \in \mathbb{F}$ and all $\mathbf{u}, \mathbf{v} \in V$: 1. $\lambda(\mu \mathbf{v}) = (\lambda \mu) \mathbf{v}$ (associativity of scalar multiplication) 2. $\lambda(\mathbf{u} + \mathbf{v}) = \lambda \mathbf{u} + \lambda \mathbf{v}$ (distributivity over vector addition) 3. $(\lambda + \mu)\mathbf{v} = \lambda \mathbf{v} + \mu \mathbf{v}$ (distributivity over field addition) 4. $1_{\mathbb{F}} \mathbf{v} = \mathbf{v}$ (identity element of scalar multiplication)

Updated Apr 4

Cambridge IA Vectors and Matrices

Algebra Linear Algebra

A complex number is an expression of the form $z = x + iy$, where $x, y \in \mathbb{R}$ and $i$ is a symbol satisfying $i^2 = -1$. The real number $x$ is called the real part, denoted $\operatorname{Re}(z)$, and $y$ is the imaginary part, denoted $\operatorname{Im}(z)$. The set of all complex numbers is denoted $\mathbb{C}$, and $\mathbb{R}$ is identified with the subset $\{x + i0 : x \in \mathbb{R}\} \subset \mathbb{C}$. This definition isolates the minimal algebraic structure needed to solve equations like $x^2 + 1 = 0$. Note that we do NOT require $x, y$ to be integers (as incorrectly stated in some sources); they must be real numbers.

Updated Apr 4

Matched Asymptotic Expansions with Logarithmic Terms

Calculus Differential Equations

Many differential equations in physics and engineering contain a small parameter $\epsilon$ — a dimensionless number measuring the ratio of two disparate scales. For example, $\epsilon$ might be the ratio of the thickness of a fluid boundary layer to the length of an airfoil, or the ratio of a reaction rate to a diffusion rate. When $\epsilon$ is…

Updated Apr 3

Method of Multiple Scales

Calculus Differential Equations

The method of matched asymptotic expansions handles problems where the solution has different structure in different spatial regions — a boundary layer here, a smooth outer solution there. But there is a second, fundamentally different way that a regular perturbation expansion can fail: the small perturbation may have a cumulative effect over long…

Updated Apr 3

Laplace Transform

Calculus Differential Equations

The Laplace transform converts a function of a real variable $t \ge 0$ — typically representing time — into a function of a complex variable $p$, replacing differentiation with multiplication and initial-value problems with algebraic equations. It is the principal tool of operational calculus: the systematic reduction of linear differential…

Updated Apr 3

Cambridge IB Numerical Analysis

Analysis Real Analysis

[course:Part IB Numerical Analysis, Cambridge, Michaelmas Term] Exact solutions to mathematical problems — inverting matrices, evaluating integrals, solving differential equations — are rarely available in practice. When the matrix is $10{,}000 \times 10{,}000$, the integral has no closed form, or the ODE is nonlinear, the only option is to…

Updated Mar 30

Bifurcation theory for one-dimensional continuous dynamical systems

Calculus Differential Equations

Let $y^$ be an equilibrium of $\dot{y} = f(y, \mu)$ at a fixed $\mu$. The equilibrium is: - Stable (or hyperbolic attracting) if $\partial_y f(y^, \mu) < 0$. - Unstable (or hyperbolic repelling) if $\partial_y f(y^, \mu) > 0$. - Non-hyperbolic if $\partial_y f(y^, \mu) = 0$. A non-hyperbolic equilibrium $y^$ is called semi-stable if $f$ does not change sign on either side of $y^$: that is, if $y^*$ is attracting from one side and repelling from the other.

Updated Mar 30

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Continuity (Real Analysis)

Power Series

Cauchy Sequence

Supremum and Infimum

Uniform Convergence

Cambridge IB Linear Algebra

Cambridge IA Vectors and Matrices

Matched Asymptotic Expansions with Logarithmic Terms

Method of Multiple Scales

Laplace Transform

Cambridge IB Numerical Analysis

Bifurcation theory for one-dimensional continuous dynamical systems

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Explore Mathematics

Create New Bundle

Browse by Area

Trending This Week

Sobolev Space

Cambridge IA Numbers and Sets

Continuity (Real Analysis)

Recently Updated

All Pages

Continuity (Real Analysis)

Power Series

Cauchy Sequence

Supremum and Infimum

Uniform Convergence

Cambridge IB Linear Algebra

Cambridge IA Vectors and Matrices

Matched Asymptotic Expansions with Logarithmic Terms

Method of Multiple Scales

Laplace Transform

Cambridge IB Numerical Analysis

Bifurcation theory for one-dimensional continuous dynamical systems