4.1: Differentiation and Integration of Vector Valued Functions

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.

Definition: The Derivative of a Vector 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.

Example $\PageIndex{1}$

$$\dfrac{d}{dt} (3 \hat{\text{i}} + \sin t \hat{\text{j}}) = \cos t \hat{\text{j}} \nonumber$$

$$\dfrac{d}{dt} \left(3t^2\, \hat{\text{i}} + \cos{(4t)}\, \hat{\text{j}} + te^t \, \hat{\text{k}} \right) = 6t \, \hat{\text{i}} -4\sin{(t)}\,\hat{\text{j}} + (e^t + te^t)\, \hat{\text{k}} \nonumber$$

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

1. $\dfrac{d}{dt} \left( \text{v}(t) + \text{w}(t) \right) = \dfrac{d}{dt}\text{v}(t) + \dfrac{d}{dt} \text{w}(t)$,
2. $\dfrac{d}{dt} c\text{v}(t) = c\, \dfrac{d}{dt} \text{v}(t)$,
3. $\dfrac{d}{dt}(f(t) \text{v}(t)) = f '(t) \text{v}(t) + f(t) \text{v}(t)'$,
4. $\left( v(t) \cdot \text{w}(t) \right)' = \text{v}'(t) \cdot \text{w}(t)+ \text{v}(t) \cdot \text{w}'(t)$,
5. $(v(t) \times \text{w}(t))' = \text{v}'(t) \times \text{w}(t) + \text{v}(t) \times \text{w}'(t)$,
6. $\dfrac{d}{dt} v(f(t)) = \text{v}(t)'(f(t)) f '(t)$.

Example $\PageIndex{2}$

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.

Example $\PageIndex{3}$

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.