I Partial differentiation and explicit functions

[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

 

[fresh_42]

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 $f(x,t)=6x+\sin(t)$ ?

 

[dyn]

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

 

[fresh_42]

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

Apply the chain rule. $\partial_t f(x,t)=\partial_t(6x)+\partial_t g(t)=0+\partial_t g(t)=g'(t)$

 

[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

 

[fresh_42]

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.