Partial Differentiation of vector functions : vector calculus #Partial #Differentiation #of #vector #functions : #vector calculus
Partial Differentiation of vector function :
In the
previous section we have studied the
differentiation of vector functions in
one variable. But a vector may be a
function of more than one scalar variable.
partial differentiation arise in geometry, physiscs and applied mathematics when the number of independent variables in the problem under consideration is two or more. Under such a situation, any dependent variable will be a function of more than one variable and hence it possesses not ordinary derivatives with respect to a single variable, but partial derivatives with respect to several independent variables .
Let f be the vector function of scalar variables p,q,r over a domain S, then we write f = f ( p,q,r ).
Treating
t as the variable and p,q as constants ; f can be considered as the vector function of the scalar variable t over a
domain S.
If lt ( f(p,q,t+𝜹t ) – f(p,q,t
) ) / 𝜹t exists then f
is said to have partial derivative with respect to ’ t ‘ and is denoted by ∂f
/ ∂t .
Similarly ,
treating q as the variable and taking p,t as constants
∂f/∂q = Lt ( f( p,q+∂q ,t ) – f( p,q,t
) ) / ∂q etc.
Properties of partial derivatives :
Let A,B,
be the functions of more than one
scalar variable. Then the following can be verified.
·
∂/∂t (ⲫ A ) = (∂ ⲫ /∂t) A + ⲫ (∂A/∂t)
·
If 𝝀 is a constant
then ∂/∂t (𝝀A ) = 𝝀 (∂A/∂t).
·
If c is constant vector then ∂/∂t ( ⲫ c ) = (∂ ⲫ /∂t) c.
·
∂/∂t ( A
± B ) = (∂A /∂t) ±
( ∂B / ∂t).
·
∂/∂t ( A
. B ) = (∂A /∂t) . B + A .
( ∂B / ∂t).
·
∂/∂t ( A
x B ) = (∂A /∂t) x B + A x
( ∂B / ∂t).
·
Let f = f1 i + f2 j +f3
k where f1,f2,f3 are differentiable scalar functions
of more then one variable. Then ∂f/∂t =
(∂f1/∂t) i + ( ∂f2/∂t
) j + ( ∂f3/∂t
) k.
Higher order
partial derivatives are defined as in Calculus of real variables.
Thus , for instant ∂2f / ∂t2 = ∂/∂t
( ∂f/∂t) ,
and (∂2f
/ ∂t∂p) = ∂/∂t
( ∂f/∂p) etc.
For example :
Let f = x2y i – 2xy j + xy2 k then
∂f/∂x = 2xy i – 2y j + y2 k and
∂f/∂y = x2 i – 2x j + 2xy k. etc.
#Partial #Differentiation #of #vector #functions : #vector calculus
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