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# 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

$r'(t) = \lim_{h \rightarrow 0} \dfrac{r(t+h)-r(t)}{h}.$

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 $$\PageIndex{1}$$

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

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

### 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

$r \cdot r' = 0.$

Solution

Since $$r$$ has constant magnitude, call its magnitude $$k$$,

$k^2 = |r|^2 = r \cdot r.$

Taking derivatives of the left and right sides gives

$0 = (r \cdot r)' = r' \cdot r + r \cdot r'$

$= r \cdot r' + r \cdot r' = 2r \cdot r' .$

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

$\int (\sin t)\, \hat{\textbf{i}} + 2t\, \hat{\textbf{j}} - 8t^3 \, \hat{\textbf{k}} \; dt.$

Solution

Just take the integral of each component

$\int (\sin t)\,dt \, \hat{\textbf{i}} + \int 2\,t \, dt \, \hat{\textbf{j}} - \int 8\,t^3 \,dt \, \hat{\textbf{k}}.$

$= (-\cos t + c_1)\, \hat{\textbf{i}} + (t^2 + c_2)\, \hat{\textbf{j}} + (2\,t^4 + c_3)\, \hat{\textbf{k}}.$

Notice that we have introduce three different constants, one for each component.

### Contributors

• Integrated by Justin Marshall.