Orthogonal curvilinear coordinates are a coordinate system where the coordinate lines are mutually orthogonal. These coordinates are particularly useful in solving problems with symmetry, such as those involving spheres, cylinders, and other complex geometries. The three most common orthogonal curvilinear coordinate systems are cylindrical, spherical, and parabolic coordinates.

Key Concepts in Orthogonal Curvilinear Coordinates

1. Definition and Examples:

   • Cylindrical Coordinates: $(r, \theta, z)$
   • Spherical Coordinates: $(r, \theta, \phi)$
   • Parabolic Coordinates: Defined by specific transformations involving parabola-based relations.

2. Metric Coefficients (Scale Factors): These coefficients, denoted as $h_i$ (for $i = 1, 2, 3$), relate the infinitesimal distance $ds$ to the infinitesimal changes in the coordinates.

3. Differential Length, Area, and Volume Elements:

   • Length: $ds^2 = h_1^2 du_1^2 + h_2^2 du_2^2 + h_3^2 du_3^2$
   • Area: $dA = h_1 h_2 du_1 du_2$
   • Volume: $dV = h_1 h_2 h_3 du_1 du_2 du_3$

4. Laplacian in Curvilinear Coordinates: A crucial operator in various physical problems, expressed as: $\nabla^2 f = \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial}{\partial u_1}\left(\frac{h_2 h_3}{h_1}\frac{\partial f}{\partial u_1}\right) + \frac{\partial}{\partial u_2}\left(\frac{h_1 h_3}{h_2}\frac{\partial f}{\partial u_2}\right) + \frac{\partial}{\partial u_3}\left(\frac{h_1 h_2}{h_3}\frac{\partial f}{\partial u_3}\right) \right]$

Simple Problems Involving Different Shapes

1. Cube:

   • In Cartesian coordinates, the cube can be described by $0 \leq x, y, z \leq a$.
   • Calculate the volume and surface area using simple integrals in Cartesian coordinates.

2. Sphere:

   • In spherical coordinates, a sphere of radius $R$ can be described by $r = R$.
   • Calculate the volume by integrating the volume element $dV = r^2 \sin\theta \, dr \, d\theta \, d\phi$.

3. Rectangular Parallelepiped:

   • Defined by three distinct lengths $a$, $b$, and $c$.
   • In Cartesian coordinates, the volume and surface area can be calculated using $V = abc$ and $A = 2(ab + ac + bc)$.

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