Question

If u = x + y and v = xy, then Jacabion of u and v with respect to x and y is​

Answer 1

Answer:

Given two vectors s=(s1,s2) and t=(t1,t2), the area of the parallelogram with corners at 0,s,s+t,t is given by

∣∣∣s1s2t1t2∣∣∣

and of course, this also goes for any translation of this parallelogram. So all we really need to know are the vectors A′B′ and A′D′, and insert their components into the above deteminant.

We can use the definition of derivative to get

B′≈A′+dx⋅(∂u∂x,∂v∂x)D′≈A′+dy⋅(∂u∂y,∂v∂y)

and by using the fact that we are on an "infinitessimal" scale, we may put equality here.

Inserting this into the above area formula, we get

AreaA′B′C′D′=∣∣∣∣dx⋅∂u∂xdx⋅∂v∂xdy⋅∂u∂ydy⋅∂v∂y∣∣∣∣=dx⋅dy⋅∣∣∣∣∂u∂x∂v∂x∂u∂y∂v∂y∣∣∣∣=AreaABCD⋅∣∣∣∣∂u∂x∂v∂x∂u∂y∂v∂y∣∣∣