U(x,0)=0 implies u_x (x,0)=0 ?

  • Thread starter kingwinner
  • Start date
In summary: So u_x(x,0)=lim h->0 (0-0)/h= 0. So the derivative is 0. In summary, If u(x,0)=0 for all x, then ux(x,0)=0.
  • #1
kingwinner
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Homework Statement


Let u(x1,x2,...,xn,t) be a function of n+1 variables. Let D be an open, bounded, connected set in R^n(with respect to x1,...,xn) and all functions are smooth.
Claim 1:
If u=0 on the boundary of D, then ut=0 on the boundary of D.


Let u(x,y) be a function of 2 variables.
Claim 2:
If u(x,0)=0, then ux(x,0)=0.

I don't understand either of the claims above.

Homework Equations


N/A

The Attempt at a Solution


I really can't think of a reason why these are true. Also, I cannot check it directly by differentiating u with respect to t (or x) because I don't even know what the function u is. I was only given that u is 0 at some very specific points.

Can someone please explain or prove it?
Any help is appreciated! :)
 
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  • #2
kingwinner said:

Homework Statement


Let u(x1,x2,...,xn,t) be a function of n+1 variables. Let D be an open, bounded, connected set in R^n(with respect to x1,...,xn) and all functions are smooth.
Claim 1:
If u=0 on the boundary of D, then ut=0 on the boundary of D.
Let u(x,y) be a function of 2 variables.

Claim 2:
If u(x,0)=0, then ux(x,0)=0.

I don't understand either of the claims above.

Homework Equations


N/A

The Attempt at a Solution


I really can't think of a reason why these are true. Also, I cannot check it directly by differentiating u with respect to t (or x) because I don't even know what the function u is. I was only given that u is 0 at some very specific points.

Can someone please explain or prove it?
Any help is appreciated! :)
If you don't understand a statement in "n" dimensions, start by reducing it to 1 dimension.
"Let u(x,t) be a function of 2 variables. Let D be an open, bounded connected set in R (The open interval (0,1) is a good example).

claim 1: if u(0,t)= 0 and u(1, t)= 0, for all t, then ut(0,t)= 0, and ut(1,t)= 0."

Which is, in fact, just "claim 2"! Can you think of any function for which that is NOT true? If not, why not?
 
  • #3
I cannot think of any function for which that is NOT true, but I also have no idea how to prove it.
Suppose u(x,0)=0 for all x.
Then why is it true that ux(x,0)=0?

ux(x,0) means take the function u(x,y), FIRST take the partial derivative w.r.t. x, THEN evaluate at y=0. But here the trouble for me is that it seems like there is no way to recover what the function u(x,y) is. (we only know u(x,0)=0 for all x)
Is this the same as FIRST evaluating at y=0, and THEN take the resulting function of x and differenaite w.r.t. x? Why are they the same? This is not obvious to me...

Thanks!
 
  • #4
The definition of u_x(x,y)=lim h->0 (u(x+h,y)-u(x,y))/h. If you put y=0...
 

Related to U(x,0)=0 implies u_x (x,0)=0 ?

1. What does the notation "U(x,0)" refer to in this equation?

The notation "U(x,0)" refers to the function U at a specific point (x,0), where 0 represents the initial time or starting point.

2. What does "u_x (x,0)" represent in this equation?

The notation "u_x (x,0)" represents the partial derivative of the function u with respect to x at the point (x,0).

3. Why is it important for U(x,0)=0 to imply u_x (x,0)=0?

This condition is important because it ensures that the function U remains constant at the initial time, meaning there is no change in the function's value. This, in turn, implies that the function's slope (represented by u_x) is also 0 at this point.

4. How does this equation relate to the concept of boundary conditions?

This equation is an example of a boundary condition, specifically an initial condition. It provides information about the function at the starting point (x,0) and helps to determine the behavior of the function for all subsequent points.

5. Can this equation be applied to any type of function?

Yes, this equation can be applied to any type of function, as long as the function has a well-defined initial value and partial derivative with respect to x. It is commonly used in the field of mathematics and physics to model various phenomena.

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