Vector Operators
Based on material from (Bird, et. al. 2007).
The vector operators are tensor math operators that are used throughout
continuum mechanics. These identities are prerequisite to many of the
continuum mechanics problems (e.g. the falling film problem). They
should be remembered and used. I personally used flashcards to remember
the expressions, although many of them can be determined through use of
the gradient (\(\nabla\)) operator and the inner product operator
(\(\cdot\)).
Gradient (\(\nabla\))
Table 1: Vector operators for gradient
| Property | Expression |
|---|
| Commutative | $\nabla\left(fg\right)=f\nabla g+g\nabla f$ |
| Cartesian | $\nabla f=\frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}\hat{y}+\frac{\partial f}{\partial z}\hat{z}$ |
| Cylindrical | $\nabla f=\frac{\partial f}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial f}{\partial\theta}\hat{\theta}+\frac{\partial f}{\partial z}\hat{z}$ |
| Spherical | $\nabla f=\frac{\partial f}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial f}{\partial\theta}\hat{\theta}+\frac{1}{r\sin\left(\theta\right)}\frac{\partial f}{\partial\phi}\hat{\phi}$ |
Divergence (\(\nabla \cdot\))
Table 2: Vector operators for divergence
| Property | Expression |
|---|
| Commutative | $\nabla\cdot\left(f\vec{v}\right)=f\left(\nabla\cdot\vec{v}\right)+\vec{v}\cdot\left(\nabla\cdot f\right)$ |
| Cartesian | $\nabla\cdot f=\frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}$ |
| Cylindrical | $\nabla\cdot f=\frac{1}{r}\frac{\partial\left(rv_{r}\right)}{\partial r}+\frac{1}{r}\frac{\partial v_{\theta}}{\partial\theta}+\frac{\partial v_{z}}{\partial z}$ |
| Spherical | $\nabla\cdot f=\frac{1}{r^{2}}\frac{\partial\left(r^{2}v_{r}\right)}{\partial r}+\frac{1}{r\sin\left(\theta\right)}\frac{\partial\left(\sin\left(\theta\right)v_{\theta}\right)}{\partial\theta}+\frac{1}{r\sin\left(\theta\right)}\frac{\partial v_{\phi}}{\partial\phi}$ |
Laplacian (\(\nabla^{2}\))
Table 3: Vector operators for the Laplacian
| Property | Expression |
|---|
| Cartesian | $\nabla^{2}f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}}$ |
| Cylindrical | $\nabla^{2}f=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}f}{\partial\theta^{2}}+\frac{\partial^{2}f}{\partial z^{2}}$ |
| Spherical | $\nabla^{2}f=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}\sin\left(\theta\right)}\frac{\partial}{\partial\theta}\left(\sin\left(\theta\right)\frac{\partial f}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\left(\theta\right)}\frac{\partial f}{\partial\phi}$ |
Bibliography
