Chain rule with multiple variables

[Undoubtedly0]

I was reading over a textbook that stated the following, where $y(s,t) = v(x(s,t),t)$

$$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$

and

$$\frac{\partial^2y}{\partial t^2} = \frac{\partial^2 v}{\partial x^2}\left ( \frac{\partial x}{\partial t} \right )^2 + 2\frac{\partial^2 v}{\partial x \partial t}\frac{\partial x}{\partial t} + \frac{\partial^2 v}{\partial t^2}+ \frac{\partial v}{\partial x}\frac{\partial^2 x}{\partial t^2}$$

I have been having trouble figuring out how they came to the second statement. My logic proceeds as such, using the product rule:

$$\frac{\partial^2y}{\partial t^2} = \frac{\partial}{\partial t}\left [ \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t} \right] = \frac{\partial^2 v}{\partial t \partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial x}\frac{\partial^2 x}{\partial t^2} + \frac{\partial^2 v}{\partial t^2}$$

But certainly my result is not equivalent to the correct one. Where did I go astray? Thanks all!

 

[lugita15]

It looks like the textbook's result is correct. I just came up with a long, messy proof of the formula, and I've attached it as a PDF. You're probably going to need to zoom into see all the details. Tell me if you don't understand something. It seems like I'm making this proof unnecessarily complicated, so let's see whether I can find a way to make it shorter and more intuitive.

 

[qbert]

you have to remember
$$\frac{\partial v}{\partial x},$$
is a function of t explicitly and a function of t through x.
so that
$$\frac{\partial}{\partial t}\left( \frac{\partial v(x(t), t)}{\partial x} \right) = \frac{\partial}{\partial x}\left( \frac{\partial v(x(t), t)}{\partial x} \right) \frac{\partial x}{\partial t}+ \frac{\partial^2 v(x(t), t)}{\partial t \partial x}$$

this is one of those cases where the notation is kind of sloppy.

and of course the same thing for
$$\frac{\partial v}{\partial t}( x(t), t)$$

 

[lugita15]

qbert, are you using total derivatives instead of partial derivatives, but denoting them as partial derivatives? If so, why?

 

[qbert]

all derivatives are partial holding s constant.
you just need to keep the implicit dependence straight.

 

[lugita15]

OK, I got it now. I just needed to keep the partial and total derivatives straight. I've attached a PDF with two relatively compehensible proofs of the formula.

 

[qbert]

i said something wrong which has been bugging me.
i was going to let it slide but then there was an answer
with the phrase "total derivative", which to me needed
commenting on.

this is basically nit-picking, but...

all the derivatives are partial. they're partial and the
notation is ambiguous as to what's "being held constant"
which is why i commented on the sloppiness of the notation.

when you write $\frac{\partial^2 y}{\partial t^2}$where $y = y(s,t)$ is a function
of the two variables it means, keep s constant through both derivatives.

so we need to remember our function always depends on the
two variables s an t.

now if we define $y(s,t)=v(x(s,t), t)$ then v is
a function of the two variables x and t. and x is a function
of the two variables s and t.

so that
$$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$

The symbol $\frac{\partial }{\partial t}$ means different things on each side of
the equation! acting on y(s,t) it means differentiate with respect to t keeping s constant.
When it acts on v(x,t) it means differentiate with respect to t keeping x constant.

It turns out to be convenient here to introduce
$$\frac{\partial v}{\partial x}(x,t) = w(x, t), \text{ and } \frac{\partial v}{\partial t}(x, t) = u(x, t).$$

So that

$$\frac{\partial y}{\partial t}(s,t) = w(x(s,t), t) \frac{\partial x}{\partial t}(s,t) + u(x(s, t), t)$$

Now doing the derivative holding s constant gives
$$\frac{\partial^2 y}{\partial t^2} = \left( \frac{\partial w}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial w}{\partial t} \right) \frac{\partial x}{\partial t} + w \frac{\partial^2 x}{\partial t^2} + \left( \frac{\partial u}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial u}{\partial t} \right)$$

now plugging in for w and u the derivatives of v gives the books result.
notice all the derivatives are partial.