Thermodynamic potential을 정의하면서 언급했듯이, 일반적으로 엔트로피는 쉽게 측정되거나 조절될 수 있는 값이 아니다. 따라서 엔트로피를 변수로 가지는 내부 에너지나 엔탈피를 르장드르 변환하여서 헬름홀츠 자유 에너지나 깁스 자유 에너지를 얻었듯이 엔트로피와 관계된 편미분을 다른 값들로 변환할 수 있고, 이렇게 얻은 식들을 Maxwell's relation이라고 부른다. Exact differential을 갖는 함수 $f$는 $$df = \left( \frac{\partial f}{\partial x} \right)_y dx + \left( \frac{\partial f}{\partial y} \right)_x dy$$와 같이 쓸 수 있고, $$df = F_x dx + F_y dy$$를 ..

Thermodynamic PotentialsInternal Energy Internal Energy. $$\begin{align*} U & = U(S, V) \\ dU & = TdS - P dV \\ \Longrightarrow T & = \left( \frac{\partial U}{\partial S} \right)_V \quad \text{ and } \quad P = - \left( \frac{\partial U}{\partial V} \right)_S \end{align*}$$열역학 제 1법칙에 의해 다음이 성립한다. $$dU = TdS - P dV$$ 따라서 $U = U(S, V)$이고 이때 $S$와 $V$는 $U$의 natural variable이라고 불린다. Enthalpy엔탈피 $H$는 $..

Sequences in Metric SpacesDefinition. A sequence is a metric space $M$ is a function from $\mathbb{P}$ into $M$, and we denote such a sequence as $\{ a_n \}_{n=1}^{\infty}$. Definition 37.1Definition 37.1. Let \(\{a_n\}\) be a sequence in a metric space \((M, d)\). We say that \(\{a_n\}\) converges to (or has limit) \(L\), where \(L \in M\), and write \[ \lim_{n \to \infty} a_n = L \] if for eve..

MetricDefinition 35.1. Let \( M \) be a set. A metric on \( M \) is a function \( d \) from \( M \times M \) into \( [0, \infty) \) which satisfies (i) \( d(x, y) = 0 \) if and only if \( x = y \) (ii) \( d(x, y) = d(y, x) \) for all \( y \in M \) (iii) \( d(x, z) \leq d(x, y) + d(y, z) \) for all \( z \in M \) Metric SpaceDefinition 35.2. A metric space is an ordered pair \( (M, d) \), where \..

RemarkRemark. We say that $f$ is bounded on a set $X$ if there exists a number $M$ such that $|f(x)| If $f$ is bounded on $X$ and $Y$, then $f$ is bounded on $X \cup Y$. By induction, we see that if $f$ is bounded on $X_1, ..., X_n$, then $f$ is bounded on $X_1 \cup \cdots \cup X_n.$($\because$) Let $f$ be a funciton from $D \subset \mathbb{R}$ into $\mathbb{R}$. If $f$ is bounded on $X$ and $Y$..

ContinuityDefinition 33.1. Let \( f \) be a function from \( X \subset \mathbb{R} \) into \( \mathbb{R} \). We say that \( f \) is continuous at \( a \) if either (i) \( a \) is an accumulation point of \( X \) and \( \lim_{x \to a} f(x) = f(a) \). (ii) \( a \) is not an accumulation point of \( X \).We say that $f$ is continuous on $[a, b]$ if $f$ is continuous at every point of $[ a, b ]$ (at ..

Accumulation PointDefinition 30.1. Let $X \subset \mathbb{R}$ and let \( a \in \mathbb{R} \). We say that \( a \) is an accumulation point of \( X \) if for every \( \delta > 0 \), there exists a number \( x \in X \) such that \( 0 We say that $a$ is a left (right) accumulation point of $X$ if for every $\delta > 0$, there exists a number $x \in X$ such that $0 다른말로, $a$ 근방에 $a$와는 다른 $x \in X$가 ..

Theorem 28.1Theorem 28.1 (Summation by Parts). Let \( \sum_{n=1}^{\infty} a_n \) be an infinite series and let \( \{s_n\} \) be the sequence of partial sums of \( \sum_{n=1}^{\infty} a_n \). Let \( \{b_n\} \) be any sequence. Then for any positive integer \( n \), we have \[ \sum_{k=1}^{n} a_k b_k = \sum_{k=1}^{n} s_k (b_k - b_{k+1}) + s_n b_{n+1}. \]Proof. Let $s_0 = 0$. Then $$\sum_{k=1}^n a_k..

Power SeriesDefinition 27.1. Let \( t \) be a fixed real number. A power series (expanded about \( t \)) is an infinite series of the form \[ \sum_{n=0}^{\infty} a_n (x - t)^n \] where \( \{a_n\}_{n=0}^{\infty} \) is a sequence and \( x \) is a real number. \([(x - t)^0\) is defined to be 1.] Theorem 27.2Theorem 27.2. Let \( \sum_{n=0}^{\infty} a_n (x - t)^n \) be a power series. Let \[ L = \lim..

Theorem 24.1Theorem 24.1. Let $\sum_{n=1}^{\infty} a_n$ be a series with nonnegative terms. Then $\sum_{n=1}^{\infty} a_n$ converges $\iff$ the sequence of partial sums $\{ s_n \}$ is bounded. Proof. $(\Longrightarrow)$ Since $\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} s_n$ converges, $\{ s_n \}$ is bounded by Theorem 13.2.$(\Longleftarrow)$Since $a_n \geq 0, \forall n \in \mathbb{P}$, $\{ s_..