Calculus of Vector-Valued Functions

Introduction of Vector-Valued Functions

For example, a function ${\displaystyle \mathbf{F}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}=<f(t),g(t),h(t)>}$ is a classic vector-valued function in three space, where ${\displaystyle \mathbf{i},\mathbf{j},\mathbf{k}}$ are unit vectors in ${\displaystyle x,y,z}$ directions. In this function, ${\displaystyle f,g,h}$ are normal real-valued functions about ${\displaystyle t}$.

Limits and Derivative

We use the ${\displaystyle \delta -ϵ}$ definition to define the limit of vector valued functions again like what we did in real-valued functions.

That is, for each ${\displaystyle ϵ>0}$, no matter how small it is, there exists a ${\displaystyle \delta }$ makes "when ${\displaystyle 0<|t-c|<\delta }$, ${\displaystyle |\mathbf{F}(t)-\mathbf{L}|<ϵ}$" true. Then we say ${\displaystyle \underset{t\to c}{lim}\mathbf{F}(t)=\mathbf{L}}$.

There is also a requirement that ${\displaystyle \underset{t\to c}{lim}\mathbf{F}(t)}$ exists if and only if both ${\displaystyle \underset{t\to c}{lim}f(t)}$, ${\displaystyle \underset{t\to c}{lim}g(t)}$, ${\displaystyle \underset{t\to c}{lim}h(t)}$ exist.

At this moment, we have:

${\displaystyle {\mathbf{F}}^{\prime }(t)=f^{\prime }(t)\mathbf{i}+g^{\prime }(t)\mathbf{j}+h^{\prime }(t)\mathbf{k}=<f^{\prime }(t),g^{\prime }(t),h^{\prime }(t)>}$

That is the derivative of vector-valued functions.

Integration

With the definition of limits and derivative, the integration can be down with the rules of its inverse.

The product rule and the chain rule need to be applied to each item with variable included.