본문 바로가기

수학

[공학수학1] 라플라스 정리 (Note taking) 시험에 필요한 내용 및 IVP 풀이에 쓰이는 내용을 압축해서 정리했다. 주요 조건들은 생략한 것들이 많아 사용전 제대로 공부할 필요가 있다. Laplace Transform정의$$F(s) = \mathcal{L} (f(t)) = \int_{0}^{\infty} e^{-st} f(t) \mathrm{d}t$$기본 idea는 함수를 지수를 곱하고 적분함으로써 캡쳐하는 것이다. 이후에 나올 디랙 델타함수를 보면 이 표현이 더 와닿을 것이다.특징Linear 함Shifting / sifting 특징미분, 적분 연산이 곱하기 나누기 연산으로 바뀜주요 Transform (암기 필요)$f$\mathcal{L} (f)$1$$\dfrac{1}{s}$$t$$\dfrac{1}{s^2}$$t^2$$\dfrac{2!}{s^3}$.. 더보기
[Discrete Mathematics] lecture 8 - Graphs IntroductionGraph is a particular class of discrete structure for representing relations. We deal with various types of graphs, their representations and trees.GraphsDefinition 1A simple graph $G = \left$ consists of‣ a set $V$ of vertices(=nodes)‣ a set $E$ of edges(=arcs, links), which are unordered pairs of distinct elements in $V$. Definition 2A multigraph is a graph with multiple edges. For.. 더보기
[Discrete Mathematics] lecture 7 - lattice OverviewThis lecture covers the lattice theory and boolean algebra.Introduction to LatticeDefinition 1 (lattice) A poset $\left$ which every pair of elements in L has lub and glb is called a lattice.lub is denoted as $+$ and glb is denoted as $*$.We say that $a$ and $b$ join at $a+b$ and meet at $a*b$.Example 1.Consider a poset $\left$ where $L = {a, b, c, d}$ and $ \preceq = I_L ∪ {(a, c),(a, d.. 더보기
[Discrete Mathematics] lecture 6 - algebra OverviewThis lecture introduces the concept of an algebraic structure, defined by a carrier set, operations, and constants. It covers foundational ideas like closure, subalgebras, identity and zero elements, and inverses. Key structures are introduced, including semigroups, monoids, and groups, along with the hierarchy among them.The lecture also defines homomorphisms (structure-preserving maps .. 더보기
푸리에 급수와 변환 - Fourier Series/Transform IntroductionTrigonometric Series→ $a_0 + a_1 \cos x + b_1 \sin x + a_2 \cos 2x + b_2 \sin 2x + \cdots$어떠한 함수가 삼각함수들의 합으로 나타내진다고 하자. Example$f(x) = \begin{cases}-k \:\: & -\pi 위 경우 푸리에 수열을 찾으면$$\frac{4k}{\pi} (\sin x + \frac{1}{3} \sin 3x + \frac{1}{5} \sin 5x + \cdots)$$(How는 잠시후에, 지금은 이렇게 표현될 수 있다는 것만!)Fourier Series다음과 같이 벡터공간을 정의하자.$$V=\left\{f:[-\pi, \pi] \rightarrow \R \:| \:f\: \text{is .. 더보기
[Discrete Mathematics] lecture 5 - counting OverviewThis lecture introduces the fundamentals of counting and recurrence relations. It begins by defining finite and infinite sets, emphasizing the concept of cardinality and distinguishing between countable and uncountable sets. Key counting principles such as the Pigeonhole Principle, sum and product rules, and formulas for permutations and combinations are presented, along with proofs and .. 더보기
[Discrete Mathematics] lecture 4 - Functions OverviewThis lecture covers the basics of functions in discrete mathematics, defining them as relations where each input has a unique output. Key terms like domain, codomain, and range are introduced. It explores composite functions, operators, restrictions, and extensions.Different types of functions are discussed: injective (one-to-one), surjective (onto), bijective (both), along with their pr.. 더보기
[Discrete Mathematics] lecture 3 - Relations OverviewThis lecture presents a comprehensive introduction to relations in discrete mathematics, covering foundational definitions, properties, and theorems. It begins by defining binary relations and their operations (complement, inverse, composition, and power), followed by a detailed look at key relational properties such as reflexivity, symmetry, and transitivity. The text explores the graph.. 더보기

티스토리툴바