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Thread: Jacobian / chain rule

  1. #1
    Senior Member Danneedshelp's Avatar
    Apr 2009

    Jacobian / chain rule


    In my complex variable class my teacher went over the inverse function theroem and got into some advanced calculus concerning the Jacobian. He said it can be used in real variables to produce all forms of the chain rule through matrix multiplication.

    I wanted to try it out with some real valued functions from my calculus book. However, I guess I didn't really understand something, or missed some conditions, because I am trying to use the jacobian to answer the following calc3 questions.

    1) If $\displaystyle z=f(x,y)$ has continuous second order derivatives and $\displaystyle x=r^{2}+s^{2}$ and $\displaystyle y=2rs$, find $\displaystyle dz/dr$ (those d's should be currled).

    2) $\displaystyle z=x^{2}y+3xy^{4}$, where $\displaystyle x=sin(2t)$ and $\displaystyle y=cos(t)$, find $\displaystyle dz/dt$.

    My work so far has been a dead end. I can find them using the chain rule, but now with the jacobian, because I am not sure how to construct two jacobian matrices from the given information. For instance,

    in the first one, $\displaystyle z=f(x,y)$ has only two partial derivatives, one with respect to x and the other with respect to y. I can find a 2 by 2 jacobian for x and y because I can find partials with respect to r and s for both.

    Any help would be great. I dont really need answers, but help with the method would be great

    Thank you
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Mar 2007
    Santiago, Chile
    we have $\displaystyle z(r,s)=f(x(r,s),y(r,s)),$ then $\displaystyle \displaystyle\frac{\partial z}{\partial r}=\frac{\partial f}{\partial x}\cdot \frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\cdot \frac{\partial y}{\partial r}.$

    in the same fashion you can take the second one.
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