Question

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x=8 u+4 v, y=6 u-8 v implies ∂(u, v)/∂(x, y)=

Answer 1

To find ∂(u, v)/∂(x, y), we calculate the partial derivatives of x and y with respect to u and v, arrange them in a matrix, and then find the determinant of this matrix. The Jacobian determinant is -88.

Step-by-step explanation:

The question given asks to find the Jacobian determinant ∂(u, v)/∂(x, y) for the transformation given by x=8u+4v and y=6u-8v. To find this, we need to calculate the partial derivatives of u and v with respect to x and y and arrange them in a matrix, then take the determinant of that matrix. The determinant of the matrix is the Jacobian determinant.

First, we calculate the partial derivatives:

1. ∂x/∂u = 8 and ∂x/∂v = 4
2. ∂y/∂u = 6 and ∂y/∂v = -8

Next, we setup the matrix and calculate its determinant:

| ∂x/∂u ∂x/∂v |
| ∂y/∂u ∂y/∂v |

Substituting the partial derivatives into the matrix gives us:

| 8 4 |
| 6 -8 |

The determinant of this matrix, which is the Jacobian, is calculated as (8 * (-8)) - (4 * 6) = -64 - 24 = -88.

Therefore, ∂(u, v)/∂(x, y) = -88.