# Partial differentiation and explicit functions

• I
dyn
Hi
For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ?

For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ?

Thanks

Mentor
2022 Award
Hi
For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ?

I guess, yes, as long as the ##x## term is there. But this is more a matter of taste, I think than it is a rigor definition. But ##f(x,t)## is an explicit function of ##x## and ##t##.

If we had ##f(x)=g(t,x(t))## then ##f(x)## would not be an explicit function of ##t##.

For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ?

Thanks

No. ##f## still depends on ##t##, except for the case ##g(t)=C## is a constant.
What do you get for ##\dfrac{\partial f}{\partial t}## if ##f(x,t)=6x+\sin(t)## ?

• benorin
dyn
Thanks. But what about the case where g ( t ) is an unknown arbitrary function ? What would ∂f/∂t be then ?

Mentor
2022 Award
Thanks. But what about the case where g ( t ) is an unknown arbitrary function ? What would ∂f/∂t be then ?
Apply the chain rule. ##\partial_t f(x,t)=\partial_t(6x)+\partial_t g(t)=0+\partial_t g(t)=g'(t)##

• benorin
dyn
My question originally comes a question i am looking at regarding Lagrangian mechanics in cylindrical coordinates. The Lagrangian contains a term containing the time derivative of ρ but the book states that the Lagrangian is not an explicit function of time so ∂L/∂t = 0 and so the Hamiltonian is conserved..

I am confused about how ∂L/∂t = 0 when there is a time derivative in the Lagrangian and also L is said to be not an explicit function of time

• benorin
Mentor
2022 Award
My question originally comes a question i am looking at regarding Lagrangian mechanics in cylindrical coordinates. The Lagrangian contains a term containing the time derivative of ρ but the book states that the Lagrangian is not an explicit function of time so ∂L/∂t = 0 and so the Hamiltonian is conserved..

I am confused about how ∂L/∂t = 0 when there is a time derivative in the Lagrangian and also L is said to be not an explicit function of time
I haven't said it is impossible. It depends on the functions involved. You gave a very simple example with separated variables. Physical systems are normally a lot more complicated.

• benorin