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How come $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}$$ when $$ u = x\; cos \theta - y\; sin \theta $$ $$ v = x\; sin \theta + y\; cos\theta $$ It seems like that formula is twice as great on the right side. (Are the definitions of $u$ and $v$ even relevant?)

ANSWER: This is the definition of the chain rule for partial derivatives when $f$ is a function of $x$ and $y.$ It is irrelevant to the definitions of $u(x,y)$ and $v(x,y).$

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  • $\begingroup$ Use the chain rule for a real value function... $\endgroup$ Oct 20, 2014 at 8:57
  • $\begingroup$ Doesn't that play in to my question though about not being equal? $\endgroup$ Oct 20, 2014 at 8:59
  • $\begingroup$ You defined what $u$ and $v$ are, what about $f$? $\endgroup$
    – 5xum
    Oct 20, 2014 at 9:04
  • $\begingroup$ I was told the definition of $f$ is irrelevant. $\endgroup$ Oct 20, 2014 at 9:06
  • $\begingroup$ It is, but $f$ must be a funcion of $u$ and $v$. $\endgroup$
    – 5xum
    Oct 20, 2014 at 9:07

1 Answer 1

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This is not a proof, but a demonstration: Had $f=f(u,v)$ only where u and v were independent variables, you'd get a total change in f from changes in both $u$ and $v$

$$ df = \frac{\partial f}{\partial u} du + \frac{\partial f}{\partial v} dv $$ I suspect you are comfortable with that. As it turns out, $u$ and $v$ are themselves functions of independent variables x and y (assuming $\theta$ is only a parameter here). So

$$ du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy \\ dv = \frac{\partial v}{\partial x} dx + \frac{\partial v}{\partial y} dy $$

Substitute for these values of $du,dv$ $$ df = \frac{\partial f}{\partial u} (\frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy ) + \frac{\partial f}{\partial v} (\frac{\partial v}{\partial x} dx + \frac{\partial v}{\partial y} dy ) $$

Group terms in dx, dy together and you have your result.

$$ df = \left(\frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x} \right) dx + \left(\frac{\partial f}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial y} \right) dy $$

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