Find the Jacobian. д(x, y, z) a(u, v, w for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w), y, and z with respect to u, v, and w is then the Jacobian of x, дх дх дх диду ди ду ду ду диду ди дz дz дz ди av aw x = u – V + W, y = 7uv, z = u + V + W д(x, y, z) д(u, v, w) д(x, y, z) a(u, v, w) =
Find the Jacobian. д(x, y, z) a(u, v, w for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w), y, and z with respect to u, v, and w is then the Jacobian of x, дх дх дх диду ди ду ду ду диду ди дz дz дz ди av aw x = u – V + W, y = 7uv, z = u + V + W д(x, y, z) д(u, v, w) д(x, y, z) a(u, v, w) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Finding the Jacobian**
To find the Jacobian, denoted as
\[
\frac{\partial (x, y, z)}{\partial (u, v, w)}
\]
for the indicated change of variables, we have:
If
- \( x = f(u, v, w) \),
- \( y = g(u, v, w) \),
- \( z = h(u, v, w) \),
then the Jacobian of \( x, y, \) and \( z \) with respect to \( u, v, \) and \( w \) is
\[
\frac{\partial (x, y, z)}{\partial (u, v, w)} =
\begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\
\frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}
\end{vmatrix}.
\]
**Example:**
For the functions defined by
- \( x = u - v + w \),
- \( y = 7uv \),
- \( z = u + v + w \),
we need to compute the Jacobian:
\[
\frac{\partial (x, y, z)}{\partial (u, v, w)} = \boxed{\ }
\]
This example will guide students in understanding how to set up and compute the Jacobian for a given transformation of variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa48f7eb-bce8-4ea0-bf8b-7482c9c56a7c%2F928356aa-b849-43ea-ad50-266f7e51d272%2F9vxqpvh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Jacobian**
To find the Jacobian, denoted as
\[
\frac{\partial (x, y, z)}{\partial (u, v, w)}
\]
for the indicated change of variables, we have:
If
- \( x = f(u, v, w) \),
- \( y = g(u, v, w) \),
- \( z = h(u, v, w) \),
then the Jacobian of \( x, y, \) and \( z \) with respect to \( u, v, \) and \( w \) is
\[
\frac{\partial (x, y, z)}{\partial (u, v, w)} =
\begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\
\frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}
\end{vmatrix}.
\]
**Example:**
For the functions defined by
- \( x = u - v + w \),
- \( y = 7uv \),
- \( z = u + v + w \),
we need to compute the Jacobian:
\[
\frac{\partial (x, y, z)}{\partial (u, v, w)} = \boxed{\ }
\]
This example will guide students in understanding how to set up and compute the Jacobian for a given transformation of variables.
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