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.
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.
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.