CylindricalCoordinateSystem

The cylindrical coordinate system is a three-dimensional coordinate system that extends the two-dimensional polar coordinate system into three dimensions using a z-coordinate. This system is especially beneficial in situations involving cylindrical symmetry.

The three dimensions in cylindrical coordinates are:

1. r (rho): The radial distance from the origin (the z-axis) to the point in question.
2. θ (theta): The azimuthal angle in the x-y plane, measured counter-clockwise from the positive x-axis.
3. z: The height above the x-y plane.

The conversion between Cartesian coordinates (x, y, z) and cylindrical coordinates (r, θ, z) are as follows:

1. From Cartesian to Cylindrical:

   $$r = \sqrt{x^2 + y^2}$$

   $$θ = \text{atan2}(y, x)$$ (atan2 is a variant of the arctangent function that takes two arguments instead of one. It returns values in the interval $(-π, π]$, which makes it useful for finding the angle of a vector in the plane.)

   $$z = z$$

2. From Cylindrical to Cartesian:

   $$x = r \cdot \cos(θ)$$

   $$y = r \cdot \sin(θ)$$

   $$z = z$$

The differential elements for volume (dV), surface area (dS), and length (dl) in cylindrical coordinates are:

1. Differential Volume element (dV):

   $$dV = r \cdot dr \cdot dθ \cdot dz$$

2. Differential Surface Area elements (dS):

   • For the radial surface element, $dS_r$: $$dS_r = r \cdot dθ \cdot dz$$

   • For the azimuthal surface element, $dS_θ$: $$dS_θ = dr \cdot dz$$

   • For the axial surface element, $dS_z$: $$dS_z = r \cdot dr \cdot dθ$$

3. Differential Length elements (dl):

   $$dl_r = dr$$

   $$dl_θ = r \cdot dθ$$

   $$dl_z = dz$$

In cylindrical coordinates, the gradient, divergence, and curl operators are given by:

1. Gradient (∇f):

   $$∇f = \left(\frac{\partial f}{\partial r}\right) \hat{r} + \left(\frac{1}{r} \cdot \frac{\partial f}{\partial θ}\right) \hat{θ} + \left(\frac{\partial f}{\partial z}\right) \hat{z}$$

2. Divergence (∇⋅F):

   $$∇⋅F = \frac{1}{r} \cdot \frac{\partial}{\partial r} (rF_r) + \frac{1}{r} \cdot \frac{\partial F_θ}{\partial θ} + \frac{\partial F_z}{\partial z}$$

3. Curl (∇xF):

   $$∇×F = \left(\frac{1}{r} \cdot \frac{\partial F_z}{\partial θ} - \frac{\partial F_θ}{\partial z}\right) \hat{r} + \left(\frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}\right) \hat{θ} + \left(\frac{1}{r} \cdot \frac{\partial}{\partial r} (rF_θ) - \frac{1}{r} \cdot \frac{\partial F_r}{\partial θ}\right) \hat{z}$$

Here, $\hat{r}$, $\hat{θ}$, and $\hat{z}$ represent the unit vectors in the cylindrical coordinate system in the r, θ, and z directions respectively. $F_r$, $F_θ$, and $F_z$ represent the components of a vector field F in the r, θ, and z directions respectively.

The Laplacian operator in cylindrical coordinates is given by:

Laplacian (Δf):

$$Δf = \frac{1}{r} \cdot \frac{\partial}{\partial r} (r \cdot \frac{\partial f}{\partial r}) + \frac{1}{r^2} \cdot \frac{\partial^2 f}{\partial θ^2} + \frac{\partial^2 f}{\partial z^2}$$

In all the above formulas, the partial derivative $\frac{\partial}{\partial r}$ means differentiation with respect to r, while keeping θ and z constant, and similarly for the others.