Partial differentiation and explicit functions

In summary, the conversation discusses whether a function f(x,t) = 6x + g(t) where g(t) is an arbitrary function of t can be considered an explicit function of t. It is determined that it is an explicit function of both x and t, but if it was written as f(x) = g(t,x(t)), it would not be explicit in t. Additionally, the question of whether ∂f/∂t = 0 holds true for this function is addressed, with the conclusion that it depends on the specific functions involved. The conversation also touches on the concept of Lagrangian mechanics and how it relates to this topic.
  • #1
dyn
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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
 
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  • #2
dyn said:
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##.

dyn said:
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)## ?
 
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  • #3
Thanks. But what about the case where g ( t ) is an unknown arbitrary function ? What would ∂f/∂t be then ?
 
  • #4
dyn said:
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)##
 
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  • #5
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
 
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  • #6
dyn said:
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.
 
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1. What is partial differentiation?

Partial differentiation is a mathematical concept used to find the rate of change of a function with respect to one of its variables while holding all other variables constant. It is commonly used in multivariable calculus and is denoted by ∂ (partial derivative).

2. How is partial differentiation different from ordinary differentiation?

Ordinary differentiation is used to find the rate of change of a function with respect to a single variable, while partial differentiation is used to find the rate of change with respect to one variable while holding all other variables constant. This allows us to analyze the effects of each variable on the function separately.

3. What are explicit functions?

Explicit functions are mathematical functions where the dependent variable is expressed explicitly in terms of the independent variable. In other words, the dependent variable is directly written in terms of the independent variable without any implicit or hidden relationships.

4. How do you use partial differentiation to find the maximum or minimum of a function?

To find the maximum or minimum of a function using partial differentiation, we first find the partial derivatives of the function with respect to each variable. Then, we set these partial derivatives equal to zero and solve for the values of the variables. These values will give us the critical points of the function, and we can use the second derivative test to determine if they are maximum or minimum points.

5. Can partial differentiation be applied to any type of function?

Yes, partial differentiation can be applied to any type of function, including polynomial, exponential, trigonometric, and logarithmic functions. However, the function must have multiple variables for partial differentiation to be applicable.

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