4.1: Differentiation and Integration of Vector Valued Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
The formal definition of the derivative of a vector valued function is very similar to the definition of the derivative of a real valued function.
Let r(t) be a vector valued function, then
r'(t) = \lim_{h \rightarrow 0} \dfrac{r(t+h)-r(t)}{h}. \nonumber
Because the derivative of a sum is the sum of the derivative, we can find the derivative of each of the components of the vector valued function to find its derivative.
\dfrac{d}{dt} (3 \hat{\text{i}} + \sin t \hat{\text{j}}) = \cos t \hat{\text{j}} \nonumber
\dfrac{d}{dt} \left(3t^2\, \hat{\text{i}} + \cos{(4t)}\, \hat{\text{j}} + te^t \, \hat{\text{k}} \right) = 6t \, \hat{\text{i}} -4\sin{(t)}\,\hat{\text{j}} + (e^t + te^t)\, \hat{\text{k}} \nonumber
Properties of Vector Valued Functions
All of the properties of differentiation still hold for vector values functions. Moreover because there are a variety of ways of defining multiplication, there is an abundance of product rules.
Suppose that \text{v}(t) and \text{w}(t) are vector valued functions, f(t) is a scalar function, and c is a real number then
- \dfrac{d}{dt} \left( \text{v}(t) + \text{w}(t) \right) = \dfrac{d}{dt}\text{v}(t) + \dfrac{d}{dt} \text{w}(t),
- \dfrac{d}{dt} c\text{v}(t) = c\, \dfrac{d}{dt} \text{v}(t) ,
- \dfrac{d}{dt}(f(t) \text{v}(t)) = f '(t) \text{v}(t) + f(t) \text{v}(t)',
- \left( v(t) \cdot \text{w}(t) \right)' = \text{v}'(t) \cdot \text{w}(t)+ \text{v}(t) \cdot \text{w}'(t),
- (v(t) \times \text{w}(t))' = \text{v}'(t) \times \text{w}(t) + \text{v}(t) \times \text{w}'(t),
- \dfrac{d}{dt} v(f(t)) = \text{v}(t)'(f(t)) f '(t).
Show that if r is a differentiable vector valued function with constant magnitude, then
r \cdot r' = 0. \nonumber
Solution
Since r has constant magnitude, call its magnitude k,
k^2 = |r|^2 = r \cdot r. \nonumber
Taking derivatives of the left and right sides gives
0 = (r \cdot r)' = r' \cdot r + r \cdot r' \nonumber
= r \cdot r' + r \cdot r' = 2r \cdot r' . \nonumber
Divide by two and the result follows
Integration of vector valued functions
We define the integral of a vector valued function as the integral of each component. This definition holds for both definite and indefinite integrals.
Evaluate
\int (\sin t)\, \hat{\textbf{i}} + 2t\, \hat{\textbf{j}} - 8t^3 \, \hat{\textbf{k}} \; dt. \nonumber
Solution
Just take the integral of each component
\int (\sin t)\,dt \, \hat{\textbf{i}} + \int 2\,t \, dt \, \hat{\textbf{j}} - \int 8\,t^3 \,dt \, \hat{\textbf{k}}. \nonumber
= (-\cos t + c_1)\, \hat{\textbf{i}} + (t^2 + c_2)\, \hat{\textbf{j}} + (2\,t^4 + c_3)\, \hat{\textbf{k}}. \nonumber
Notice that we have introduce three different constants, one for each component.
Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.