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.



        d/dt (3i + sintj) = costj

        d/dt (3t2 i + cos(4t) j + tet k)  =  6t i -4sin(t)j + (et + tet) 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)



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

        r . r'  =  0



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.




        (sin t)i + 2t j - 8t3 k dt



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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