Find the Jacobian of the transformation.
x = 7v + 7w^2
y = 9w + 9u^2
z = 2u + 2v^2
∂(x, y, z)/∂(u, v, w) ??
To find the Jacobian of the transformation, we need to find the partial derivatives of the transformed variables with respect to the original variables.
First, let's find the partial derivatives for each transformed variable:
∂x/∂v = 7
∂x/∂w = 14w
∂y/∂w = 9
∂y/∂u = 18u
∂z/∂u = 2
∂z/∂v = 4v
Now, we can arrange these partial derivatives into a Jacobian matrix:
J = [∂(x, y, z)/∂(u, v, w)] = [∂x/∂u ∂x/∂v ∂x/∂w]
[∂y/∂u ∂y/∂v ∂y/∂w]
[∂z/∂u ∂z/∂v ∂z/∂w]
Substituting the partial derivatives we found earlier, the Jacobian matrix becomes:
J = [0 7 14w]
[18u 0 9]
[2 4v 0]
Therefore, the Jacobian of the transformation is:
∂(x, y, z)/∂(u, v, w) = [0 7 14w]
[18u 0 9]
[2 4v 0]
To find the Jacobian of the transformation, we need to calculate the partial derivatives of each of the output variables (x, y, z) with respect to each of the input variables (u, v, w).
Let's begin by calculating the partial derivatives:
∂x/∂u: To find ∂x/∂u, we need to differentiate x with respect to u while keeping v and w constant. Since the derivatives of v and w with respect to u are zero, we only need to differentiate the term 7v + 7w^2 with respect to u. The result is 0 since u does not appear in the equation.
∂x/∂v: To find ∂x/∂v, we need to differentiate x with respect to v while keeping u and w constant. Using the same reasoning as before, we only need to differentiate the term 7v + 7w^2 with respect to v. The result is 7 since v appears linearly.
∂x/∂w: To find ∂x/∂w, we need to differentiate x with respect to w while keeping u and v constant. Here, we only need to differentiate the term 7v + 7w^2 with respect to w. The result is 14w since w appears squared.
Now, let's move on to the partial derivatives for y:
∂y/∂u: To find ∂y/∂u, we need to differentiate y with respect to u while keeping v and w constant. Similarly, the term 9w + 9u^2 only contains u as a variable, so the result is 18u.
∂y/∂v: To find ∂y/∂v, we need to differentiate y with respect to v while keeping u and w constant. The term 9w + 9u^2 does not contain v, so the result is 0.
∂y/∂w: To find ∂y/∂w, we need to differentiate y with respect to w while keeping u and v constant. The derivative of the term 9w + 9u^2 with respect to w is 9.
Finally, let's calculate the partial derivatives for z:
∂z/∂u: To find ∂z/∂u, we need to differentiate z with respect to u while keeping v and w constant. The term 2u + 2v^2 only contains u, so the result is 2.
∂z/∂v: To find ∂z/∂v, we need to differentiate z with respect to v while keeping u and w constant. The derivative of the term 2u + 2v^2 with respect to v is 4v.
∂z/∂w: To find ∂z/∂w, we need to differentiate z with respect to w while keeping u and v constant. Since w does not appear in the equation, the result is 0.
Putting it all together, we have:
J = ∂(x, y, z)/∂(u, v, w) =
[0 7 14w]
[18u 0 9]
[2 4v 0]
Thus, the Jacobian of the given transformation is the 3x3 matrix with the elements above.
not a valid transformation, zeros on diagonal
check with
http://www.wolframalpha.com/widgets/view.jsp?id=d6a52679e750c30e320a7094ec975598
well, just plug them in
∂x/∂u = 0
∂x/∂v = 7
∂x/∂w = 14w
∂y/∂u = 18u
∂y/∂v = 0
∂y/∂w = 9
∂z/∂u = 2
∂z/∂v = 4v
∂z/∂w = 0
So J =
(0 7 14w)
(18u 0 9)
(2 4v 0)
|J| = 1008uvw + 126