Theorem 1Theorem 1. Suppose that $F(x, y)$ is differentiable and that the equation $F(x, y) = 0$ defines $y$ as a differentiable function of $x$. Then at any point where $\partial_y F \neq 0,$ $$\frac{dy}{dx} = - \frac{\partial_x F}{\partial_y F}.$$Proof. Since $F(x, y) = 0$, the derivative $\frac{dF}{dx}$ must be zero. By the Chain Rule, we find $$0 = \frac{dF}{dx} = \frac{\partial F}{\partial ..

Theorem 1Theorem 1. If $w=f(x, y)$ is differentiable and if $x = x(t), y=y(t)$ are differentiable functions of $t$, then the composition $w=f(x(t), y(t))$ is a differentiable function of $t$ and $$\frac{dw}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}.$$Proof. Let $\Delta x, \Delta y$ and $\Delta w$ be the increments that result from changing $t..

Partial DerivativeDefinition 1. The partial derivative of $f(x, y)$ with respect to $x$ at the point $(x_0, y_0)$ is $$\frac{\partial f}{\partial x} \Bigg|_{(x_0, y_0)} = \lim_{h \to 0} \frac{f(x_0 + h, y_0) - f(x_0, y_0)}{h},$$ provided the limit exists. The partial derivative with respect to $y$ is defined in the same way.특정 변수를 상수로 취급하고 한 변수만 다룬다는 의미에서 편미분이라고 말한다. Clairaut's TheoremTheorem 1...

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 $\textbf{r}(t) = (f(t), g(t), h(t))$ 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 of $\textbf{r}$,(2) $|\textbf{v}|$ is the speed of $\te..

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..

ParametrizationDefinition 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 closed in..

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..