Mathematics

Vector Fields and Scalar FunctionsDefinition 1. Let $D \subset \mathbb{R}^m$ for $m \in \mathbb{N}$. Then(1) A scalar function on a domain set $D$ is a function $f: D \rightarrow \mathbb{R}$. (2) A vector-valued function, or vector function, or vector field on $D$ is a function $\textbf{f}: D \rightarrow \mathbb{R}^n$ defined by $\textbf{f}(\textbf{x}) = (f_1(\textbf{x}), f_2(\textbf{x}), \cdots..

Curve Definition 1. We call the vector function $\textbf{r}: (a, b) \rightarrow \mathbb{R}^3$ a curve. We can parametrize curves by $\textbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ where $t \in (a, b)$. Velocity, Speed, Unit Tangent VectorDefinition 2. Let $\textbf{r}$ be a curve. Then(1) $\textbf{v}(t) = \frac{d \textbf{r}}{dt}$ is the velocity vector or the tangent vector of $\textbf{r}$,(2)..

CylindersDefinition 1. A cylinder is a surface that is generated by moving a straight line along a given planar curve while holding the line parallel to a given fixed line. The curve is called a generating curve for the cylinder. In solid geometry, where cylinder means circular cylinder, the generating curves are circles, but now we allow generating curves of any kind.RemarkRemark. any curve $f(..

Vector Equation for a LineA vector equation for the line $L$ through $P_0(x_0, y_0, z_0)$ parallel to $\mathbf{v}$ is $$\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}, \text{  } -\infty Vector Equation for a PlaneA vector equation for the plane through $P_0(x_0, y_0, z_0)$ normal to $\mathbf{n} = A \mathbf{i} + B \mathbf{j} + C\mathbf{k}$ is$$ \mathbf{n} \cdot \overrightarrow{P_0P} = 0$$ where $\ma..

Parametrization of CurvesDefinition 1. If $x$ and $y$ are given as functions $$x = f(t), \text{  } y=g(t)$$ over an interval $I$ of $t$-values, then the set of points $(x, y) = (f(t), g(t))$ defined by these equations is a parametric curve. The equations are parametric equations for the curve.The variable $t$ is a parameter for the curve, and its domain $I$ is the parameter interval. If $I$ is a..

Taylor Series함수 $f$를 양수의 수렴 반경을 가지고 $$f(x) = \sum_{n=0}^{\infty} a_n(x-a)^n$$ 이라고 하자. 이 수렴 구간에서 미분하면 $$f'(x) = a_1 + 2a_2(x-a) + 3a_3(x-a)^2 + \cdots + na_n(x-a)^{n-1} + \cdots , \\ f''(x) = 1\cdot 2 a_2 + 2 \cdot 3 a_3 (x-a) + 3\cdot 4 a_4 (x-a)^2 + \cdots , \\ f'''(x) = 1 \cdot 2 \cdot 3 a_3 + 2 \cdot 3 \cdot 4a_4(x-a) + 3\cdot 4 \cdot 5a_5 (x-a)^2 + \cdots , \\ \vdots$$ 임을 알 수 있다. 각 등식에 $x=a$..

Power SeriesDefinition 1. A power series about $x=a$ is a series of the form $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ in which the center $a$ and the coefficients $c_0, c_1, ..., c_n$ are constants.Convergent Theorem for Power SeriesTheorem 1. If the power series $\sum_{n=0}^{\infty} a_n x^n$ converges at $x = c \neq 0$, then it converges absolutely for all $x$ with $|x| |d|$.Corollary. The convergen..

The $n$th-Term Test for a Divergent Series Theorem 1. If $\sum_{n=1}^{\infty} a_n$ converges, then $a_n \rightarrow 0$.Proof. Let $\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} s_n = L$, where $s_n$ is the partial sums of the series and $L$ is the sum of the series. Note that $$\lim_{n \to \infty} s_n = \lim_{n \to \infty} s_{n-1} = L \\ \Longrightarrow \lim_{n \to \infty} (s_n - s_{n-1}) = \lim..

Ordered SetDefinition 1. An order $relation with the following two properties:(1) If$x \in S$and$y \in S$, then one and only one of the statements$$x(2)$S$is transitive.We call$S$an ordered set if an order is defined in$S$.BoundedDefinition 2. Suppose$S$is an ordered set, and$E \subset S$. (1) If there exists a$\beta \in S$such that$x \leq \beta, \forall x \in E$, we say that$E$ ..

Equivalence RelationDefinition 1. Let $R$ be a relation in a set $X$. Then we say that (a) $R$ is reflexive $\iff$ $\forall x \in X, xRx$.(b) $R$ is symmetric $\iff$ $xRy \Longrightarrow yRx$.(c) $R$ is transitive $\iff$ $xRy \wedge yRz \Longrightarrow xRz$. (d) $R$ is an equivalence relation $\iff$ $R$ is reflexive, symmetric, and transitive. Equivalence relation, 즉 동치 관계는 특정한 수학적 관점에서 볼 때 두 원소..