Dated: 30-10-2024
Limits of Vector Valued Functions
\[\lim_{t \to a} \vec{r}(t) = \lim_{t \to a} x(t) \hat i + \lim_{t \to a} y(t) \hat j\]
If limit1 of any component does not exist then we say that limit1 for \(\vec{r}(t)\) does not exist.
Continuity of Vector Valued Functions
- \(\vec{r}(t_0)\) is defined.
- \(\lim_{t \to t_0} \vec r (t)\) exists.
- \(\lim_{t \to t_0} \vec r (t) = \vec r (t_0)\)
Derivative2
Similar to real valued functions,3
\[\vec r ^{\prime}(t) = \frac d {dt} \vec r (t) = \lim_{h \to 0} \frac{\vec r (t + h) + \vec r (t)}{h}\]
Theorem
\[\frac d {dt} \vec r (t) = x^\prime(t) \hat i + y^\prime(t) \hat j + z^\prime(t) \hat k\]
Tangent Vectors
if \(\vec r (t)\) and \(\vec r ^ \prime (t)\) exist and are non zero, then \(\vec r ^ \prime(t_0)\) is called the tangent vector to \(\vec r (t)\) at \(\vec r (t_0)\).
Properties of Derivatives2
Vector valued functions have same properties as for real valued functions.2
There are 2 additional ones
-
\[\frac d {dt} \left(\vec {r_1}(t) \cdot \vec {r_2}(t) \right) = \vec{r_1} \cdot \frac{dr_2}{dt} + \frac{dr_1}{dt} \cdot \vec{r_2}\]
-
\[\frac d {dt} \left(\vec {r_1}(t) \times \vec {r_2}(t) \right) = \vec{r_1} \times \frac{dr_2}{dt} + \frac{dr_1}{dt} \times \vec{r_2}\]
Theorem
\[\vec r (t) \cdot \frac d {dt} \vec r (t) = 0\]
Since both are perpendicular.
Integrals4 For Vector Valued Functions
\[\int_a^b \vec r (t) = \int_a^b (x(t) \cdot dt) \hat i + \int_a^b (y(t) \cdot dt) \hat j\]
Properties of Integrals4
They are similar to properties of integrals4 for the real valued functions.2
References
Read more about notations and symbols.