Suppose a change of coordinates t: r2→r2 from the uv-plane to the xy-plane is given by x=4v−2u−3, y=−1+5u+4v.

(a) find the absolute value of the determinant of the jacobian for this change of coordinates. ∣∣∂(x, y)/∂(u, v)∣∣= ∣det |=

(b) if a region d∗ in the uv-plane has area 7.04, find the area of the region t(d∗) in the xy-plane. area =

a) ∣∣∂(x,y)/∂(u,v)∣∣ = 28

b)Area = 197.12

Step-by-step explanation:

a) Find the absolute value of the determinant of the Jacobian

Change of coordinates

x = 4v -2u-3

y = -1+5u+4v

Then

∣∣∂(x,y)/∂(u,v)∣∣=∣det | =| |

∣∣∂(x,y)/∂(u,v)∣∣ = ∣(-2)(4)-(5)(4)∣ = 28  

b) The area of the region in the xy -plane is the area of the region in the uv-plane multiplies by the absolute value of the determinant of the Jacobian.

Thus

Area = (7.04)(28)

Area = 197.12    

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