### Author Topic: Chain rule  (Read 1008 times)

#### Jingxuan Zhang

• Elder Member
•     • Posts: 106
• Karma: 20 ##### Chain rule
« on: January 15, 2018, 04:41:28 PM »
Pray what kind of chain rule gives us the following result, which I found in the PDF version (p.35, strangely not online) with little rephrasing:
Quote
$x=\frac{1}{2}(\xi+\eta), t=\frac{1}{2}(\xi-\eta)$ and therefore due to chain rule $v_{\xi}=\frac{1}{2c}(cv_{x}+v_{t})$ and $v_{\eta}=\frac{1}{2c}(cv_{x}-v_{t}).$
And also what is $v$? is it the same one as in the previous chapter, viz., a factor of the operator?

#### Victor Ivrii ##### Re: Chain rule
« Reply #1 on: January 15, 2018, 05:20:36 PM »
For chain rule see Calculus II, $v$ is any function

#### Ioana Nedelcu

• Full Member
•   • Posts: 29
• Karma: 3 ##### Re: Chain rule
« Reply #2 on: January 15, 2018, 11:30:28 PM »
It's the chain rule for multivariable functions. Defining a random function $v(\xi, \eta)$, then $$\frac{\partial v}{\partial x} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial \xi} + \frac{\partial v}{\partial t}\frac{\partial t}{\partial \xi}$$ and using the given x and t functions, you get the resulting partial derivatives of v

#### Victor Ivrii ##### Re: Chain rule
« Reply #3 on: January 16, 2018, 01:42:26 AM »
Defining a random function $v(\xi, \eta)$
"arbitrary function"--yes, "random function"––no, because in mathematics (probability theory and its derivatives) "random" has a very specific mathematical meaning; in particular, there is a theory of PDE and ODE with random coefficients.