Chapter 9 Vector Functions

• Vector Function and Space Curves
• Derivatives and Integrals of Vector Functions
• Arc Length and Curvature

# Vector Functions and Space Curves

1. $f:R \to R$
2. $f:R \to R^m$
3. $f:R^n \to R$
4. $f:R^n \to R^m$

• Vector Functions (2)型函數 - valued Functions
  $r(t) = <f(t),g(t),h(t)> = f(t)i + g(t)j+h(t)k$，$f,g,h$ 為實函數。
• Space curve：If f, g, h are continuous real–valued functions,
  then the orbit of $r(t) = <f(t),g(t),h(t)>$ is called a space curve.

# Derivatives and Integrals of Vector Function

# Define

Let $r(t) = <f(t),g(t),h(t)>$.
Then $\Delta r(t) = <\Delta f(t),\Delta g(t),\Delta h(t)>$
Here $\Delta = lim,\frac{d}{dt},\int$.

# Arc Length and Curvature

# Arc Length

$r(t) = <x(t),y(t)>$ or $r(t) = <x(t),y(t),z(t)>$
$L = \int^b_a\sqrt{(dx)^2 + (dy)^2} = \int^b_a\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt = \int^b_a|r'(t)|dt$.
$L = \int^b_a\sqrt{(dx)^2 + (dy)^2 + (dz)^2} = \int^b_a\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2} dt = \int^b_a|r'(t)|dt$.

# Arc length function (Distance function)

$s(t) = \int^t_a|r'(u)|du$.
$\Rightarrow \frac{ds}{dt} = |r'(t)|$= 距離變化率 = 速度

# Reference

• 莊重 - 微積分 (二) Calculus II - 103 學年度