Tangent PlaneDefinition 1. The tangent plane to the level surface $f(x, y, z) = c$ of a differentiable function $f$ at a point $P_0$ where the gradient is not zero is the plane through $P_0$ normal to $\nabla f |_{P_0}$. The normal line of the surface at $P_0$ is the line through $P_0$ parallel to $\nabla f |_{P_0}$. 위와 같이 정의되는 tangent plane, 즉 접평면은 정의에 따라 다음과 같이 계산되고, normal line도 마찬가지다. $$\nab..

Directional DerivativeDefinition 1. The derivative of $f$ at $P_0 (x_0, y_0)$ in the direction of the unit vector $\mathbb{u} = u_1 \mathbb{i} + u_2 \mathbb{j}$ is the number $$\left( \frac{df}{ds} \right)_{\mathbb{u}, P_0} = \lim_{s \to 0} \frac{f(x_0 + su_1, y_0 + su_2) - f(x_0, y_0)}{s},$$ provided the limit exists. It is also denoted by $$D_{\mathbb{u}}f(P_0) \text{  or  } D_{\mathbb{u}}f \b..

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. 기호로는 $\frac{\partial f}{\par..

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 of $\textbf{r}$,(2) $|\textbf{v}|$ is the..

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..