Contents :
1. Vector function of a scalar variable
2. Limit of a vector function
3. Continuity of a vector function
4. Derivative of a vector function
5. Higher order derivatives
6. Properties of derivatives
1. Vector function of a scalar variable :
Let S ⊂ R. Corresponding to each scalar t 𝞊 S, there is associated a unique vector r, then r is said to be a vector ( vector valued ) function . S is called domain of R. We express it as r= f(t), where f denotes the law of correspondence.
let i,j,k be the three mutually perpendicuclar unit vectors in three dimensional space. Then the vector function f(t) may be expressed in the form
r= f(t)= f1(t) I + f2(t) j + f3(t) k.
here f1(t),f2(t),f3(t) are the real valued functions and are called the components of r.
2. Limit of a Vector Function :
let f(t) be a vector function over the domain S and a 𝞊 S. If ther exists a vector L such that for each 𝞊 >0, it is possible to find s>0 , where 0 < | t-a | <s => | f(t) - L | < 𝞊
then the vector L is called the limit of f(t) as t tends to a.
This is denoted by Lt f(t) = L as t tends to a.
Note :
Let Lt A(t) =L and Lt B(t) = M and 𝛌 being any constant then
i) Lt ( A(t) + B(t) = L+M ii) Lt [ 𝛌 A(t)] = 𝛌L iii) Lt [( A(t) . B(t) ] = L. M
iv) L [(A(t) x B(t) ] = L x M , where . and x represents dot and cross product of vectors.
3. Continuity of a Vector Function :
Let f be a vector function on an interval I and a 𝞊 I. Then f is said to be continuous at a if Lt f(t) = f(a) as t tends to a.
The function f is said to be continuous on I if f is continuous at every point of I.
Note :
If f and g are continuous then f ± g, f.g and fxg are also continuous.
4. Derivative of a Vector Function :
Let f be a vector function on an interval I and a 𝞊 I . Then
Lt (f(t) -f (a)) / (t-a) if is exist , it called the derivative of f at 'a' and is denoted by f '(a) or [df/dt] at t=a.
Also it is said that f is differentiable at t=a.
Note :
* If f is differentiable at t=a , then it is continuous at t=a.
* If f is continuous at t=a then it need not be differentiable at that point.
* If f is differentiable on an intervel I and t 𝞊 I then the derivative of f at t is denoted by df/dt.
* As in calculus of real variables, if the changes in r, f are denoted by 𝝳t and 𝝳f respectively then we have
df/dt = Lt (𝝳f/𝝳t) = Lt ( f(t+𝝳t ) - f(t) )/ 𝝳t
5. Higher Order Derivatives :
Let f be differentiable on an interval I and f ' = df/dt of f.
If Lt ( f’(t) – f ’(a) )/ t-a exists for each t 𝞊 I1 ⊂ I, then f ‘ is said to be
differentiable on I1. Also f
is said to possess second derivative on I1 and is denoted by f ‘’
(t) or d2f/ dt2 .
By induction if f(n-1) is differentiable
on In-1 ⊂ In-2 ⊂ … ⊂ I2 ⊂I1
⊂ I then f is said to be possess
nth derivative on In-1 and is denoted by f(n) or dn f / dtn.
Derivative of a Constant Function :
Let f be
constant vector function on an interval I and a 𝞊 I. Then f ‘
(a) =0.
Proof :
Let f be a
constant vector function on an interval .
Suppose
f(t) =c ∀ t 𝞊 I , where c is a constant.
Let a 𝞊 I.
For t tends
to a,
f ‘
(a) = Lt [ (f(t) -f(a)) / (t -a) ]
= Lt
[ ( c-c) / (t-a) ]
( since f(t) = f(a) = c )
=
Lt [0 / (t-a) ]
f ‘ (a)
=0.
Properties of derivatives :
Let
A , B and C be two differentiable vector functions of scalar variable t over
the domain S then
·
d/dt ( A ± B ) = dA/dt
± dB/dt
·
d/dt ( A.B ) = dA/dt . B + A. dB/dt
·
d/dt ( AxB ) = dA/dt x B + A x dB/dt
·
d/dt (
ABC ) = [ dA/dt BC ] +[ A dB/dt C ] + [ AB dC/dt ]
·
d/dt ( AxBxC ) = dA/dt x( BxC ) + Ax (dB/dt xC )
·
d/dt ( ∅ f ) = ∅ (df/dt)
+ (d∅/dt) f , here ∅ is a scalar differentiable function
·
If f = f1
(t)i + f2 ( t)j + f3(t)k , where f1(t), f2(t),
f3(t) are the cartesian components of the vector f
then df/dt = (df1/dt) i + (df2/dt)
j + (df3/dt) k.
·
If A is a
differentiable vector function of a scalar t over a domain S then d/dt (A2)=
2A (dA/dt)
·
The necessary
and sufficient condition that f(t) is a vector of constant magnitude is
f . (df/dt) = 0.
#Derivative #of #a #Vector #Function : #Integral Calculus
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