Find the Jacobian for the given transformation
-x = 4u cosh 8v, y = 4u sinh 8v
A) 128v
B) 256u
C) 128u
D) 256v

Question

Quizplus · Accepted Answer

The Jacobian of the transformation from (u, v) to (x, y) given by x = 4u cosh 8v and y = 4u sinh 8v is calculated as the determinant of the matrix formed by the partial derivatives of x and y with respect to u and v. The partial derivatives are:∂x/∂u = 4 cosh 8v, ∂x/∂v = 32u sinh 8v,∂y/∂u = 4 sinh 8v, ∂y/∂v = 32u cosh 8v.The Jacobian (J) is then:J = (4 cosh 8v)(32u cosh 8v) - (32u sinh 8v)(4 sinh 8v) = 128u(cosh^2 8v - sinh^2 8v).Since cosh^2 x - sinh^2 x = 1 for any x, we have:J = 128u.

Find the Jacobian for the Given Transformation