13.1 Vector Functions and Space Curves

Vector Function: $f:\R \to \R^3$

• Component Functions: $\vec r (t) = \langle f(t), g(t), h(t) \rangle = f(t)\vec i + g(t) \vec j + h(t) \vec k$

Space Curve: Set of points defined by parametric equations $x=f(t), y=g(t), z=h(t)$

• Helix: E.g. $\langle \cos t, \sin t, t\rangle$

Vector Limit

• $\lim_{t\to a} \vec r (t) = \langle \lim_{t\to a}f(t), \lim_{t\to a}g(t), \lim_{t\to a} h(t) \rangle$
• $\vec r (t)$ is continuous at $a$ if $\lim_{t \to a} r(t) = r(a)$

Vector Derivative

• $\frac{d\vec r}{dt}=\vec r'(t)=\lim_{h \to 0} \frac{\vec r (t + h) - \vec r (t)}{h}$
• $\vec r' (t) = \langle f'(t), g'(t), h'(t) \rangle$
• Unit Tangent Vector: $\vec T(t) = \frac{\vec r (t)}{|\vec r' (t)|}$

Vector Integral

• $\int_a^b\vec r(t) \, dt = \langle \int_a^b f(t) \, dt, \int_a^b g(t) \, dt, \int_a^b h(t) \, dt \rangle$