Lecture notes on differentiation and integration of vector-valued functions
by Shilpa Patra
Department of Mathematics Narajole Raj College
West bengal, India
Paper:C4T,unit-4 Differentiation Integration Narajole Raj College
Derivative of vector-valued Function
Definition
The derivative of a vector-valued functionr(t) is
r0(t) = lim
∆t→0
r(t+ ∆t)−r(t)
∆t
provided the limit exists. Ifr(t) exists, thenr(t) is differentiable at t . If r0(t) exists for all t in an open interval (a,b) thenr(t) is differentiable over the interval (a,b) . For the function to be differentiable over the closed interval [a,b] , the following two limits must exist as well:
r0(a) = lim
∆t→0+
r(a+ ∆t)−r(a)
∆t and
r0(b) = lim
∆t→0−
r(b+ ∆t)−r(b)
∆t .
Many of the rules for calculating derivatives of real-valued functions can be applied to calculating the derivatives of vector-valued functions as well.
Geometry
Paper:C4T,unit-4 Differentiation Integration Narajole Raj College
Properties of Derivative of vector-valued Function
Theorem
Letrandube differentiable vector-valued functions of t , let f be a differentiable real-valued function of t , and let c be a scalar.
(Scalar multiple) d
dt[cr(t)] =cr0(t).
(Sum and Difference) d
dt[r(t)±u(t)] =r0(t) +u0(t).
(Scalar Product) d
dt[f(t)r(t)] =f0(t)r(t)±f(t)r0(t).
(Dot Product) d
dt[r(t)·u(t)] =r0(t)·u(t) +r(t)·u0(t).
(Cross Product) d
dt[r(t)×u(t)] =r0(t)×u(t) +r(t)×u0(t).
(Chain Rule) d
dt[r(f(t))] =r0(f(t))·f0(t).
Ifr(t)·r(t) =c, then r(t)·r0(t) = 0
Tangent Vector
Recall that the derivative at a point can be interpreted as the slope of the tangent line to the graph at that point. In the case of a vector-valued function, the derivative provides a tangent vector to the curve represented by the function.
Example
Consider the vector-valued function
r(t) = costi+ sintj.
The derivative of this function is
r0(t) =−sinti+ costj.
If we substitute the valuet=π6 into both functions we get
r(π 6) =
√3 2 i+1
2j and
r0(π 6) =−1
2i+
√ 3 2 j.
Paper:C4T,unit-4 Differentiation Integration Narajole Raj College
Sketch of tangent vector at the Plane Curve
The graph of this function appears in below along with the vectorsr(π6) andr0(π6)
Notice that the vectorr0(π6) is tangent to the circle at the point corresponding tot = π6 . This is an example of a tangent vector to the plane curve
Paper:C4T,unit-4 Differentiation Integration Narajole Raj College
Definite and Indefinite integral of a vector-valued function
Let f , g , and h be integrable real-valued functions over the closed interval [a,b].
The indefinite integral of a vector-valued function r(t) =f(t)i+g(t)j+h(t)kis
ˆ h
f(t)i+g(t)j+h(t)ki
dt=hˆ
f(t)dti i+hˆ
g(t)dti j+hˆ
h(t)dti k.
The definite integral of a vector-valued function is ˆ b
a
h
f(t)i+g(t)j+h(t)ki
dt=hˆ b a
f(t)dti i+hˆ b
a
g(t)dti j+hˆ b
a
h(t)dti k.
Arc length for vector function
Formulas: Plane Curve
Given a smooth curve C defined by the functionr(t) =f(t)i+g(t)j, where t lies within the interval [a,b] , the arc length of C over the interval is
s= ˆ b
a
p(f(t))2+ (g(t))2dt= ˆ b
a
kr0(t)kdt.
Space curve
Given a smooth curve C defined by the function
r(t) =f(t)i+g(t)j+h(t)k, where t lies within the interval [a,b] , the arc length of C over the interval is
s= ˆ b
a
p(f(t))2+ (g(t))2+ (h(t))2dt= ˆ b
a
kr0(t)kdt.
Paper:C4T,unit-4 Differentiation Integration Narajole Raj College
Motion vectors in plane and in space
Definition: Speed, Velocity, Acceleration
Letr(t) be a twice-differentiable vector-valued function of the parameter t that represents the position of an object as a function of time.
The velocityv(t) of the object is given by
Velocity = v(t) =r0(t).
The acceleration vectora(t) is defined to be
Acceleration =a(t) =v0(t) =r00(t).
The speed is defined to be
Speed =v(t) =kv(t)k=kr0(t)k.
References:
1. https://www.whitman.edu/mathematics/multivariable/
multivariable_13_Vector_Functions.pdf
2. http://www.ams.sunysb.edu/~jiao/teaching/ams261_fall12/
lectures/LarCalc9_ch12.pdf
3. https://users.math.msu.edu/users/gnagy/teaching/11-fall/
mth234/L09-234.pdf
