Hi, I'm learning to do double integration by changing variables and wondering about this.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have f(x, y) and want to find the volume under the surface over some bounded area in the xy plane.

Say, I want to change the variables into u and v by:

u = 3x - 2y

v = x + y

I need to find the relations between dxdy and dudv.

Now I have:

du = 3dx - 2dy

dv = dx + dy

So

dudv = (3dx - 2dy)(dx + dy) = 3(dx)^2 + dxdy - 2(dy)^2

Dividing both sides by dxdy, we obtain:

(dudv)/(dxdy) = 3(dx/dy) + 1 - 2(dy/dx)

Since x and y are independent, dx/dy and dy/dx are 0.

Hence I conclude dudv = dxdy.

It's easily to find a counter example to this. The ratio is actually a constant of 5.

Where have I been wrong here? Thank you very much.

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# Change of variable, why can I not multiply the differentials directly

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