Final answer:
The Jacobian of the transformation for the equations x = 4v + 3u and y = 4u - 4v is computed by taking the determinant of the matrix of partial derivatives, which are calculated by first solving for u and v and then differentiating with respect to x and y.
Step-by-step explanation:
To compute the Jacobian of the transformation given by the equations x = 4v + 3u and y = 4u - 4v, we need to determine the matrix of partial derivatives of the new variables with respect to the old variables . The Jacobian, denoted as J = ∂(u, v)/∂(x, y), is given by:
J = ∂(u, v)/∂(x, y) =
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∂u/∂x ∂u/∂y
∂v/∂x ∂v/∂y
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We need to find the partial derivatives of u and v with respect to x and y. Solving the transformation equations for u and v, we get u = (4y + 4v)/4 and v = (x - 3u)/4. Differentiating u with respect to x and y, and v with respect to x and y, and substituting into the Jacobian determinant gives us the desired Jacobian. It is important to note that the Jacobian is a determinant, so the computation will involve taking the determinant of the matrix of partial derivatives.