for cylindrical coordinates x = r cos theta, y = rsin theta , z=z find d(x,y,z)/d(r,theta,z)

To find the Jacobian determinant \(\frac{\partial(x, y, z)}{\partial(r, \theta, z)}\) for the transformation from cylindrical coordinates \((r, \theta, z)\) to Cartesian coordinates \((x, y, z)\), we start by writing the expressions for \(x\), \(y\), and \(z\) in terms of \(r\), \(\theta\), and \(z\):

\[
x = r \cos \theta
\]
\[
y = r \sin \theta
\]
\[
z = z
\]

Now, the Jacobian matrix \(J\) is given by:

\[
J = \begin{pmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\
\frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z}
\end{pmatrix}
\]

Let's compute each of the partial derivatives:

\[
\frac{\partial x}{\partial r} = \cos \theta, \quad \frac{\partial x}{\partial \theta} = -r \sin \theta, \quad \frac{\partial x}{\partial z} = 0
\]

\[
\frac{\partial y}{\partial r} = \sin \theta, \quad \frac{\partial y}{\partial \theta} = r \cos \theta, \quad \frac{\partial y}{\partial z} = 0
\]

\[
\frac{\partial z}{\partial r} = 0, \quad \frac{\partial z}{\partial \theta} = 0, \quad \frac{\partial z}{\partial z} = 1
\]

Thus, the Jacobian matrix \(J\) is:

\[
J = \begin{pmatrix}
\cos \theta & -r \sin \theta & 0 \\
\sin \theta & r \cos \theta & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

The Jacobian determinant is:

\[
\frac{\partial(x, y, z)}{\partial(r, \theta, z)} = \det(J)
\]

Calculating the determinant of \(J\):

\[
\det(J) = \cos \theta \cdot (r \cos \theta \cdot 1) - (-r \sin \theta \cdot \sin \theta \cdot 1) + 0
\]
\[
= r \cos^2 \theta + r \sin^2 \theta = r (\cos^2 \theta + \sin^2 \theta)
\]

Since \(\cos^2 \theta + \sin^2 \theta = 1\), we have:

\[
\frac{\partial(x, y, z)}{\partial(r, \theta, z)} = r
\]

So, the Jacobian determinant is \(r\).

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the Jacobian for spherical coordinates?
2. What is the significance of the Jacobian determinant in coordinate transformations?
3. How do you calculate volume integrals using the Jacobian in cylindrical coordinates?
4. How does the Jacobian change when transforming from Cartesian to polar coordinates in 2D?
5. Can you explain the relationship between the Jacobian determinant and change of variables in integrals?

**Tip:** The Jacobian determinant can be used to determine how areas (in 2D) or volumes (in 3D) scale under a transformation.

Learn how to calculate the Jacobian determinant for cylindrical coordinates (r, theta, z) to Cartesian coordinates (x, y, z). Understand the transformation process and the Jacobian matrix involved. Suitable for advanced mathematics learners and enthusiasts.

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