Vector Differentiation
1 Vector Function
A function ...v : [ t1 , t2 ] → V3 ...where [ t1 , t2 ] is a closed interval in IR, is called a vector function.
The function ...v ...is continuous at ...t0 ∊ [ t1 , t2 ], if given ε ( ε > 0 ) then Ǝ η ( η > 0 ) such that
| v(t0) – v(t) | < ε ...whenever ...| t0 – t | < η .
The function ...v ...is said to tend to a limit, ...l, ...as t tends to t0 ( t0 ∊ [ t1 , t2 ] ), if given ε ( ε > 0 ) then Ǝ η ( η > 0 ) such that
| l – v(t) | < ε ...whenever ...| t0 – t | < η .
In which case we write:
l ......:= ......limit v(t)
...t → t0
2 Derivative Of A Vector Function
If ...v : [ t1 , t2 ] → V3 ...for some given point t0, consider the function
...:= ...Δv Δt v(t) – v(t0) (t – t0)
Now for fixed t0, ...Δv ⁄ Δt ...is a function of t. What happens as t tends to t0? If Δv ⁄ Δt tends to a limit, then we define:
(t0) ...:= ...dv dt limit t → t0 v(t) – v(t0) (t – t0)
This is called the derivative of v at t0.
Note that the derivative can exist at one point but not at another. If the derivative of v exists at every point in the interval [ t1 , t2 ] then v is said to be differentiable on the interval. The derivative is then a vector function on the interval. We will use the notation v' to denote the derivative of v.
3 Differentiation Results
If u and v are vector functions which are differentiable on a given interval, and λ and μ are scalars then
- (λ u + μ v)' ...= ...λ u' + μ v'
- (u · v)' ...= ...u · v' + u' · v
- (u × v)' ...= ...u × v' + u' × v
It follows that
- | u |2 ...= ...(u · u)' ...= ...u · u' + u' · u ...= ...2 ( u · u' )
If φ is a differentiable (scalar) function into the domain of v, then,
- (v(φ))' ...= ...φ' v'(φ)
Note
- 0 ...= ...(( v × u ) · v )' ...= ...(( v × u ) · v' ) + (( v × u )' · v )