Tensor Calculus in Euclidean Spaces - cylindrical coordinates in E^3

7.6. Tensor Calculus in Euclidean Spaces - cylindrical coordinates in $E^{3}$ #

7.6.1. Cylindrical coordiantes and cylindrical coordinates#

Using cylindrical coordinates $(q^{1}, q^{2}, q^{3}) = (r, θ, z)$ and cylindrical base vectors (uniform in space, so that their derivatives are zero), a point in Euclidean vector space $E^{3}$ can be represented as

\vec{r} = r \cos θ \hat{x} + r \sin θ \hat{y} + z \hat{z} .

7.6.2. Natural basis, reciprocal basis, metric tensor, and Christoffel symbols#

7.6.3. Differential operators#

7.6.3.1. Gradient#

Example 7.15 (Gradient of a scalar field)

\begin{array}{r} \begin{aligned} \nabla F & = {\vec{b}}^{a} \frac{\partial F}{\partial q^{a}} = \\ = {\hat{b}}_{a} g^{a a} \frac{\partial F}{\partial q^{a}} = \\ = \hat{r} \frac{\partial F}{\partial r} + \hat{θ} \frac{1}{r} \frac{\partial F}{\partial θ} + \hat{z} \frac{\partial F}{\partial z} . \end{aligned} \end{array}

Example 7.16 (Gradient of a vector field)

\begin{array}{r} \begin{aligned} \nabla F & = {\vec{b}}^{a} \otimes {\vec{b}}_{b} [\frac{\partial F^{b}}{\partial q^{a}} + Γ_{a c}^{b} F^{c}] = \\ = \dots = \\ = {\vec{b}}^{1} \otimes {\vec{b}}_{1} \partial_{1} F^{1} & + {\vec{b}}^{1} \otimes {\vec{b}}_{2} [\partial_{1} F^{2} + Γ_{12}^{2} F^{2}] & + {\vec{b}}^{1} \otimes {\vec{b}}_{3} \partial_{1} F^{3} \\ + {\vec{b}}^{2} \otimes {\vec{b}}_{1} [\partial_{2} F^{1} + Γ_{22}^{1} F^{2}] & + {\vec{b}}^{2} \otimes {\vec{b}}_{2} [\partial_{2} F^{2} + Γ_{21}^{2} F^{1}] & + {\vec{b}}^{2} \otimes {\vec{b}}_{3} \partial_{2} F^{3} \\ + {\vec{b}}^{3} \otimes {\vec{b}}_{1} \partial_{3} F^{1} & + {\vec{b}}^{3} \otimes {\vec{b}}_{2} \partial_{3} F^{2} & + {\vec{b}}^{3} \otimes {\vec{b}}_{3} \partial_{3} F^{3} \\ = \hat{r} \otimes \hat{r} \partial_{r} F_{r} & + \hat{r} \otimes \hat{θ} \frac{1}{r} [\partial_{r} (r F_{θ}) + F_{θ}] & + \hat{r} \otimes \hat{z} \partial_{r} F_{z} \\ + \hat{θ} \otimes \hat{r} \frac{1}{r} [\partial_{θ} F_{r} - r F_{θ}] & + \hat{θ} \otimes \hat{θ} [\partial_{θ} (\frac{F_{θ}}{r}) + \frac{F_{r}}{r}] & + \hat{θ} \otimes \hat{z} \frac{1}{r} \partial_{θ} F_{z} \\ + \hat{z} \otimes \hat{r} \partial_{z} F_{x} & + \hat{z} \otimes \hat{θ} \frac{1}{r} \partial_{θ} F_{y} & + \hat{z} \otimes \hat{z} \partial_{z} F_{z} . \end{aligned} \end{array}

Example 7.17 (Gradient of a $2^{n d}$ -order tensor field)

7.6.3.2. Directional derivative#

7.6.3.3. Divergence#

Example 7.18 (Divergence of a vector field)

\begin{array}{r} \begin{aligned} \nabla \cdot \vec{F} & = \frac{\partial F^{a}}{\partial q^{a}} + Γ_{a c}^{a} F^{c} = \\ = \frac{\partial F_{r}}{\partial r} + \frac{\partial}{\partial θ} (\frac{F_{θ}}{r}) + \frac{F_{θ}}{r} + \frac{\partial F_{z}}{\partial z} . \end{aligned} \end{array}

Example 7.19 (Divergence of a $2^{n d}$ -order tensor field)

7.6.3.4. Laplacian#

Example 7.20 (Laplacian of a scalar field)

Example 7.21 (Laplacian of a vector field)