13: Vector Functions

13.1 Vector Functions and Space Curves #

Vector Functions #

  • We can write a vector function as $$\vec r(t) = \langle f(t), g(t), h(t) \rangle = f(t) \hat i + g(t) \hat j + h(t) \hat k$$
  • The domain of .$\vec r$ consists of all values of .$t$ for which each of the terms are defined
  • The limit of a vector function is $$\lim_{t\to a} \vec r(t) = \big\langle \lim_{t\to a} f(t), \lim_{t\to a} g(t), \lim_{t\to a} h(t) \big\rangle$$
  • .$\vec r$ is continuous at time .$a$ if .$\lim_{t\to a} \vec r(t) = \vec r(a)$

Space Curve #

  • The set .$C$ of all points defined by a vector function .$\vec r$ over interval .$I$ is called a space curve.
  • Think of .$C$ as being traced out by a moving particle whose position follows .$\vec r$
  • Space curves are parametrized by a vector function but isn’t necessarily that vector function!
    • E.x. .$\vec r(t) \langle \cos t, \sin t, t \rangle \neq \vec q(t) = \langle \cos 3t, \sin 3t, 3t \rangle$
    • Space curves exist on the same point, but don’t grow at the same rate

13.2 Derivatives and Integrals of Vector Functions #

Derivatives #

$$\frac{d\vec r}{dt} = \vec r'(t) = \lim_{h \to 0} \frac{\vec r(t+h) - r(t)}{h}$$ $$… = \big\langle f'(t), g'(t), h'(t) \big\rangle$$

  • Note that the last equation only works if each component is differentiable
  • The direction is the tangent line and the magnitude is the rate at which the particle is moving at that point
  • If we just want the tangent line, we can write the unit tangent vector: $$\vec T(t) = \frac{\vec r'(t)}{\Vert \vec r ' (t) \Vert}$$
  • Differentiation rules (notice .$f(t)$ is a scalar function) $$ \frac{d}{dt} \big[f(t) \vec u(t) \big] = f'(t) \vec u(t) + f(t) \vec u'(t)$$ $$ \frac{d}{dt}\big[\vec u(t) \cdot \vec v(t)\big] = \vec u'(t) \cdot \vec v(t) + \vec u(t) \cdot \vec v'(t) $$ $$ \frac{d}{dt} \big[\vec u(t) \times \vec v(t)\big] = \vec u' (t) \times \vec v(t) + \vec u(t) + \vec v'(t)$$ $$ \frac{d}{dt} \big[ \vec u(f(t)) \big] = f'(t) \vec u' (f(t))$$

Integrals #

  • Let .$\vec r (t) = \langle f(t), g(t), h(t) \rangle$, then $$\int_a^b \vec r(t)\ dt = \bigg\langle \int_a^b f(t)\ dt, \int_a^b g(t) \ dt, \int_a^b h(t)\ dt \bigg\rangle$$
  • Fundamental theorem: If .$\vec R(t)$ is an anti-derivative of .$\vec r(t)$, (i.e. .$\vec R'(t) = \vec r(t)$), then $$ \int_a^b \vec r(t) \ dt = \Big[\vec R(t)\Big]_a^b = \vec R(b) - \vec R(a)$$