How do I Compute the Second Partial Derivative of u with Respect to s?

[CAF123]

Homework Statement

I have an expression for the partial derivative of u with respect to s, which is \frac{\partial\,u}{\partial\,s} = \frac{\partial\,u}{\partial\,x}x + \frac{\partial\,u}{\partial\,y}y
How do I compute \frac{\partial^2u}{\partial\,s^2} from this?

 

[CAF123]

Is something missing from the question that is required ?( I have the explicit functions x = x(s,t) and y=y(s,t)). Anyone any ideas?

 

[HallsofIvy]

Homework Statement

I have an expression for the partial derivative of u with respect to s, which is \frac{\partial\,u}{\partial\,s} = \frac{\partial\,u}{\partial\,x}x + \frac{\partial\,u}{\partial\,y}y
How do I compute \frac{\partial^2u}{\partial\,s^2} from this?

Use the chain rule:
\frac{\partial^2 u}{\partial s^2}= \frac{\partial}{\partial u}\left(\frac{\partial u}{\partial x}x\right)+ \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial y}y\right)

The derivative of anything with respect to s, assuming that s itself is a function of x and y, is
\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+ \frac{\partial f}{\partal y}\frac{\partial y}{\partial s}

 

[CAF123]

Use the chain rule:
\frac{\partial^2 u}{\partial s^2}= \frac{\partial}{\partial u}\left(\frac{\partial u}{\partial x}x\right)+ \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial y}y\right)

The derivative of anything with respect to s, assuming that s itself is a function of x and y, is
\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+ \frac{\partial f}{\partal y}\frac{\partial y}{\partial s}

Should that be an s above?
Given this, I have \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\right)x + \frac{\partial}{\partial s}\left(x\right) \frac{\partial u}{\partial x} for the first term.
What do I do from here?

 

[SammyS]

Homework Statement

I have an expression for the partial derivative of u with respect to s, which is \frac{\partial\,u}{\partial\,s} = \frac{\partial\,u}{\partial\,x}x + \frac{\partial\,u}{\partial\,y}y
How do I compute \frac{\partial^2u}{\partial\,s^2} from this?

Is something missing from the question that is required ?( I have the explicit functions x = x(s,t) and y=y(s,t)). Anyone any ideas?

Yes, it's important to know that x = x(s,t) and y=y(s,t)

What is the general form of \displaystyle \frac{\partial\,u}{\partial\,s}\,,\ \ \text{ if } \ \ x = x(s,t)\ \text{ and }\ y=y(s,t)\ ?

 

[CAF123]

Yes, it's important to know that x = x(s,t) and y=y(s,t)

What is the general form of \displaystyle \frac{\partial\,u}{\partial\,s}\,,\ \ \text{ if } \ \ x = x(s,t)\ \text{ and }\ y=y(s,t)\ ?

Is this not the equation I wrote in my first post?

 

[SammyS]

Is this not the equation I wrote in my first post?

The equation in your first post is for some particular x = x(s,t) and y = y(s,t).

In general, \displaystyle \frac{\partial u}{\partial s}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial s}\ .

This along with the equation in your first post tells you something about \displaystyle \frac{\partial x}{\partial s} and \displaystyle \frac{\partial y}{\partial s}\ .

 

[CAF123]

Yes, it tells me that, \frac{\partial x}{\partial s} = x \,\,\text{and}\,\,\frac{\partial y}{\partial s} =y I.e the partial derivatives with respect to s are the actual functions.

 

[SammyS]

Yes, it tells me that, \frac{\partial x}{\partial s} = x \,\,\text{and}\,\,\frac{\partial y}{\partial s} =y

Yes. Now proceed with what you said in post #4 regarding: \displaystyle \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\,x\right)\ ,

\displaystyle \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\,x\right)=\frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\right)x + \frac{\partial}{\partial s}\left(x\right) \frac{\partial u}{\partial x}\ ,

which can be written as:

\displaystyle \left(\frac{\partial^2 u}{\partial s\,\partial x}\right)x + \frac{\partial x}{\partial s}\, \frac{\partial u}{\partial x}\ .

 

[CAF123]

Yes. Now proceed with what you said in post #4 regarding: \displaystyle \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\,x\right)\ ,

\displaystyle \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\,x\right)=\frac{\partial}{\partial s}\left(\frac{\partial u}{\partial x}\right)x + \frac{\partial}{\partial s}\left(x\right) \frac{\partial u}{\partial x}\ ,

which can be written as:

\displaystyle \left(\frac{\partial^2 u}{\partial s\,\partial x}\right)x + \frac{\partial x}{\partial s}\, \frac{\partial u}{\partial x}\ .

Where do I go from here?
I have written the above as \frac{\partial u}{\partial x} x + \left(\frac{\partial u}{\partial s}\right)\left(\frac{\partial u}{\partial x}\right) x = \frac{\partial u}{\partial x} x \left( 1 + \frac{\partial u}{\partial s}\right)

 

[SammyS]

I have written the above as \frac{\partial u}{\partial x} x + \left(\frac{\partial u}{\partial s}\right)\left(\frac{\partial u}{\partial x}\right) x = \frac{\partial u}{\partial x} x \left( 1 + \frac{\partial u}{\partial s}\right)

Those are not correct .

\displaystyle \left(\frac{\partial^2 u}{\partial s\,\partial x}\right)x + \frac{\partial x}{\partial s}\, \frac{\partial u}{\partial x}

\displaystyle <br /> =x\frac{\partial u}{\partial x} + x \left(\frac{\partial^2 u}{\partial s\,\partial x}\right)

\displaystyle =x\frac{\partial }{\partial x}\left( u + \frac{\partial u}{\partial s}\right)\,, \ \ \ but I wouldn't put it into this last form. Perhaps only factor out x.​

Where do I go from here?

Use what HallsofIvy stated in post #3:

...

The derivative of anything with respect to s, assuming that f itself is a function of x and y, is
\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}

where he's giving the the result of using the chain rule on, \displaystyle \frac{\partial f}{\partial s}\ .

Use \displaystyle \frac{\partial u}{\partial s}\ in place of the function, f.

 

[CAF123]

Ok, so \frac{\partial}{\partial s} (f) = \frac{\partial}{\partial s} \left(\frac{\partial u}{\partial s} \right) = \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial s}\right) \frac{\partial x}{\partial s} + \frac{\partial}{\partial y}\left(\frac{\partial u}{\partial s} \right) \frac{\partial y}{\partial s}
But before I computed \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial s}\right),\,\, \text{and not} \frac{\partial}{\partial x}\left(\frac{\partial u}{\partial s}\right) ?

 

[SammyS]

But before I computed \frac{\partial}{\partial s}\left(\frac{\partial u}{\partial s}\right),\,\, \text{and not} \frac{\partial}{\partial x}\left(\frac{\partial u}{\partial s}\right) ?

I failed to notice that you were computing the wrong partial derivative.

What you need is the following:

Ok, so \frac{\partial}{\partial s} (f) = \frac{\partial}{\partial s} \left(\frac{\partial u}{\partial s} \right) = \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial s}\right) \frac{\partial x}{\partial s} + \frac{\partial}{\partial y}\left(\frac{\partial u}{\partial s} \right) \frac{\partial y}{\partial s}

Now, plug-in what you were given for \displaystyle \frac{\partial\,u}{\partial\,s} in the statement of the problem.

The resulting expression will have some occurrences of \displaystyle \frac{\partial x}{\partial s} and\displaystyle \frac{\partial y}{\partial s}\ . Don't forget to change these in accordance with your previous results.