Jacobians, changing variables in multiple integration

[xzibition8612]

Homework Statement

Find the Jacobian of the transformation.

1) x=uv, y=u^2+v^2

2)x=v+w, y=u+w, z=u+v

Homework Equations

?

The Attempt at a Solution

The textbook gave the answers for 1) ad-bc and 2) 2
How do I do this type of problem? I don't know how to get these answers. Thanks a lot.

 

[HallsofIvy]

Homework Statement

Find the Jacobian of the transformation.

1) x=uv, y=u^2+v^2

2)x=v+w, y=u+w, z=u+v

Homework Equations

?

The Attempt at a Solution

The textbook gave the answers for 1) ad-bc and 2) 2
How do I do this type of problem? I don't know how to get these answers. Thanks a lot.

I don't know what you could mean by "ad- bc" since there is no a, b, c, or d in the problem. Perhaps you mean that the Jacobian, for a 2 variable problem, is the determinant
$$\left|\begin{array}{cc}\frac{\partial u}{\partial x}& \frac{\partial v}{\partial x} \\ \frac{\partial u}{\partial y} & \frac{\partial u}{\partial y}\end{array}\right|$$

and, of course,
$$\left|\begin{array}{cc}a & b \\ c & d\end{array}\right|= ad- bc$$
but that's a general formula, not an answer to this problem.

 

[Rasalhague]

Jacobian can mean (1) a matrix whose ij entry is the partial derivative of the ith component of the transformation with respect to its jth argument (where i denotes the row and j the column), or (2) the determinant of this matrix. The textbook's answers show that it means (2) the determinant.

In question 1, the first component of the transformation is x. The second is y. So the Jacobian matrix is

$$\begin{pmatrix} \frac{\partial x}{\partial u} & & \frac{\partial x}{\partial v}\\ & & \\ \frac{\partial y}{\partial u} & & \frac{\partial y}{\partial v} \end{pmatrix}$$

The textbook's "answer" which you quote for this one isn't an answer to the specific problem, but a general formula for finding the deterimant of a 2x2 matrix whose entries are labelled a, b, c, d, thus

$$\begin{pmatrix} a & & b\\ & & \\ c & & d \end{pmatrix}$$

(I see from the preview HallsofIvy has already given much the same answer, but I'll post this anyway, in case this more spaced out way of printing the matrix makes it clearer. Halls has followed the convention whereby the Jacobian matrix is defined as the inverse of mine. Check which convention your textbook uses.)

 

[gomunkul51]

Read and learn:

http://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspxand check out:

http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant