Let's break this down into **two main steps** as requested, focusing on (a) and (b) here, with explanations. We'll consider \( f(z) = z^{3/2} \) as a complex transformation. --- ## **Step 1: Jacobian of the Transformation** Let \( z = x + iy \) (Cartesian coordinates). The transformation is \( w = f(z) = z^{3/2} \). First, write \( z \) in polar form: \[ z = re^{i\theta} \] \[ w = z^{3/2} = r^{3/2} e^{i\frac{3}{2}\theta} \] Let \( u = \text{Re}(w) \), \( v = \text{Im}(w) \). So, \[ u = r^{3/2} \cos\left(\frac{3}{2}\theta\right), \quad v = r^{3/2} \sin\left(\frac{3}{2}\theta\right) \] The **Jacobian** is: \[ J = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{vmatrix} \] #### Explanation: - We have expressed the transformation in terms of \( (x, y) \to (u, v) \). - The Jacobian determines where the transformation is valid (invertible) — i.e., where \( J \neq \). #### **Domain:** - The transformation is a coordinate transformation where \( z \neq \), i.e., \( r > \), and where the function is analytic (excluding the branch cut along the negative real axis for the principal value). --- ## **Step 2: Metric Tensors \( g^{ab} \)** The metric tensor in the \( (u, v) \) coordinates is computed by pulling back the Euclidean metric from \( (x, y) \): The Euclidean metric in \( (x, y) \) is: \[ ds^2 = dx^2 + dy^2 \] In \( (u, v) \) coordinates, the metric tensor is: \[ g_{ab} = \begin{pmatrix} \left(\frac{\partial x}{\partial u}\right)^2 + \left(\frac{\partial y}{\partial u}\right)^2 & \frac{\partial x}{\partial u} \frac{\partial x}{\partial v} + \frac{\partial y}{\partial u} \frac{\partial y}{\partial v} \\ \frac{\partial x}{\partial u} \frac{\partial x}{\partial v} + \frac{\partial y}{\partial u} \frac{\partial y}{\partial v} & \left(\frac{\partial x}{\partial v}\right)^2 + \left(\frac{\partial y}{\partial v}\right)^2 \end{pmatrix} \] But since \( f(z) \) is holomorphic and conformal where \( f'(z) \neq \), the metric is multiplied by a conformal factor: \[ g_{ab} = \lambda(u, v) \delta_{ab} \] where \( \lambda(u, v) = |f'(z)|^2 \). #### Explanation: - For a conformal map, the metric tensor is simply the Euclidean metric times the modulus squared of the derivative of the transformation. - Here, \( f'(z) = \frac{3}{2} z^{1/2} \), so \( |f'(z)|^2 = \frac{9}{4} r \). --- ## **Final Answers (a) and (b):** ### **(a) Jacobian:** \[ J = \left| \frac{\partial (u, v)}{\partial (x, y)} \right| = \left| f'(z) \right|^2 = \frac{9}{4} r \] **Domain:** \( z \neq \) (i.e., \( r > \)), excluding the branch cut (typically \( \theta = \pi \)). ### **(b) Metric Tensor:** \[ g_{ab} = \frac{9}{4} r \, \delta_{ab} \] --- If you'd like, we can move to (c) and (d) in the next step!

Let's continue with **Step 2** to address parts (c) and (d) of the problem. --- ## **Step 3: Inverse Metric Tensor \( g^{ab} \)** The inverse metric tensor \( g^{ab} \) is simply the reciprocal of the metric tensor components. Given the metric tensor in \( (u, v) \): \[ g_{ab} = \frac{9}{4} r \, \delta_{ab} \] The components of the inverse metric tensor \( g^{ab} \) can be computed as: \[ g^{ab} = \frac{4}{9 r} \delta^{ab} \] #### Explanation: - The inverse metric tensor is essential for raising indices and is obtained by taking the reciprocal of the components of the original metric tensor. - The factors of \( r \) appear due to the conformal nature of the transformation. --- ## **Step 4: Laplacian \( \Box \) Acting on a Scalar Field \( \phi \)** The Laplacian in curved coordinates can be expressed using the metric tensor as follows: \[ \Box \phi = g^{ab} \nabla_a \nabla_b \phi \] For a scalar function \( \phi \), the Laplacian in our transformed coordinates is given by: \[ \Box \phi = \frac{4}{9 r} \left( \frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial v^2} \right) \] #### Explanation: - The Laplacian operator \( \Box \) combines the metric's inverse and the second derivatives of the scalar field \( \phi \). - The specific form shows how the geometry of the transformation affects the behavior of scalar fields. --- ## **Final Answers (c) and (d):** ### **(c) Inverse Metric Tensor \( g^{ab} \):** \[ g^{ab} = \frac{4}{9 r} \delta^{ab} \] ### **(d) Laplacian \( \Box \phi \):** \[ \Box \phi = \frac{4}{9 r} \left( \frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial v^2} \right) \] --- This concludes the derivation for parts (a), (b), (c), and (d). If you have further inquiries or need additional clarifications, feel free to ask!