Differentiation and Integration of Vector Valued Functions

Calculus 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.

 

The Derivative of a Vector Valued Function Let r(t) be a vector valued function, then

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.

Examples

        d/dt (3i + sintj) = costj

        d/dt (3t² i + cos(4t) j + te^t k)  =  6t i -4sin(t)j + (e^t + te^t) 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.

 

Properties of Vector Valued Functions Suppose that v(t) and w(t) are vector valued functions, f(t) is a scalar function, and c is a real number then 1.  d/dt(v(t) + w(t)) = d/dt(v(t)) + d/dt(w(t)) 2.  d/dt(cv(t)) = c d/dt(v(t)) 3.  d/dt(f(t) v(t)) = f '(t) v(t) + f(t) v'(t) 4.  (v(t) . w(t))' = v'(t) . w(t)+ v(t) . w'(t) 5.  (v(t) x w(t))' = v'(t)  x  w(t)+ v(t) x w'(t) 6.  d/dt(v(f(t))) = v'(f(t)) f '(t)

Example

Show that if r is a differentiable vector valued function with constant magnitude, then

        r ^. r'  =  0

Solution

Since r has constant magnitude, call it k,

        k²  =  ||r||²  = r ^. r

Taking derivatives of the left and right sides gives

        0  =  (r ^. r)'  =  r' ^. r + r ^. r' 

        =  r ^. r' + r ^. r'  =  2r ^. 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

Evaluate

        (sin t)i + 2t j - 8t³ k dt

Solution

Just take the integral of each component

        ( (sin t)dt i) + ( 2t dt  j) -  ( 8t³ dt k)

        =  (-cost + c₁)i  + (t² + c₂)j  +  (2t⁴ + c₃)k 

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

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