Question

partial diff. equations
my question here : http://www.m5zn.com/newuploads/2015/11/10/png//81192750f1d3b55.png

Answer 1

there's something missing in your question

the second statement, i think

Answer 2

well,I face this in some steps of solving a problem what is your suggest to be right but take care of t=0

Answer 3

But \(\dfrac{\partial F}{\partial t}\) evaluated at t=0 does not tell you anything about \(\dfrac{\partial F}{\partial x}\). Just as an example,
\[
\begin{align*}
f(x)&:=x^3\\
f(x+ct)&=(x+ct)^3\\
\frac{\partial f(x+ct)}{\partial t}&=3(x+ct)^2(c)=3c(x+ct)^2\\
\frac{\partial f(x)}{\partial x}&=3x^2\\
\left. \frac{\partial f(x+ct)}{\partial t}\right|_{t=0}&=3cx^2
\end{align*}
\]

Answer 4

Unless there are some addition conditions imposed on F I don't see why this should be true.

Answer 5

You simply cannot evaluate the function before doing the derivative. For a simpler example,
\[
\begin{align*}
f(x)&=x^2\\
f'(x)&=2x\\
f'(1)&=2\\
{}\\
f(1)&=1\\
\frac{d(f(1))}{dx}&=0
\end{align*}
\]
In the first set of equations, I find the derivative before putting the number into the derivative. In the second set of equations, I put the number into the function before finding its derivative. Clearly they are not equal.

Answer 6

thomas5267 . . . thank you . . . i got what i want :)