Here are some useful vector identities commonly used in electromagnetic theory:
Gradient of a scalar function:
\[ \nabla (\phi) = \frac{\partial \phi}{\partial x} \mathbf{\hat{i}} + \frac{\partial \phi}{\partial y} \mathbf{\hat{j}} + \frac{\partial \phi}{\partial z} \mathbf{\hat{k}} \] Where \(\phi\) is a scalar function and \(\mathbf{\hat{i}}\), \(\mathbf{\hat{j}}\), and \(\mathbf{\hat{k}}\) are unit vectors along the \(x\), \(y\), and \(z\) axes, respectively.
Divergence of a vector field:
\[ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \] Where \(\mathbf{F}\) is a vector field.
Curl of a vector field:
\[ \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} – \frac{\partial F_y}{\partial z} \right) \mathbf{\hat{i}} + \left( \frac{\partial F_x}{\partial z} – \frac{\partial F_z}{\partial x} \right) \mathbf{\hat{j}} + \left( \frac{\partial F_y}{\partial x} – \frac{\partial F_x}{\partial y} \right) \mathbf{\hat{k}} \] Where \(\mathbf{F}\) is a vector field.
Laplacian of a scalar function: \[ \nabla^2 \phi = \nabla \cdot (\nabla \phi) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} \] Where \(\phi\) is a scalar function. 5. Vector triple product: \[ \nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) – \nabla^2 \mathbf{F} \] Where \(\mathbf{F}\) is a vector field.
These identities are fundamental in solving various problems in electromagnetic theory, such as Maxwell’s equations, electrostatics, magnetostatics, and electromagnetic wave propagation.
Cylindrical Coordinates: In cylindrical coordinates, we have three coordinates: \( (r, \theta, z) \), where \( r \) is the radial distance from the z-axis, \( \theta \) is the azimuthal angle, and \( z \) is the height along the z-axis. Gradient: \[ \nabla (\phi) = \frac{\partial \phi}{\partial r} \mathbf{\hat{r}} + \frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{\hat{\theta}} + \frac{\partial \phi}{\partial z} \mathbf{\hat{z}} \] Where \( \mathbf{\hat{r}} \), \( \mathbf{\hat{\theta}} \), and \( \mathbf{\hat{z}} \) are unit vectors in the radial, azimuthal, and z-directions, respectively. Divergence: \[ \nabla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial (rF_r)}{\partial r} + \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta} + \frac{\partial F_z}{\partial z} \] Curl: \[ \nabla \times \mathbf{F} = \left( \frac{1}{r} \frac{\partial F_z}{\partial \theta} – \frac{\partial F_{\theta}}{\partial z} \right) \mathbf{\hat{r}} + \left( \frac{\partial F_r}{\partial z} – \frac{\partial F_z}{\partial r} \right) \mathbf{\hat{\theta}} + \frac{1}{r} \left( \frac{\partial (rF_{\theta})}{\partial r} – \frac{\partial F_r}{\partial \theta} \right) \mathbf{\hat{z}} \] Laplacian: \[ \nabla^2 \phi = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} \] Spherical Coordinates: In spherical coordinates, we have three coordinates: \( (r, \theta, \phi) \), where \( r \) is the radial distance from the origin, \( \theta \) is the polar angle measured from the positive z-axis, and \( \phi \) is the azimuthal angle measured from the positive x-axis. Gradient: \[ \nabla (\phi) = \frac{\partial \phi}{\partial r} \mathbf{\hat{r}} + \frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{\hat{\theta}} + \frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \phi} \mathbf{\hat{\phi}} \] Where \( \mathbf{\hat{r}} \), \( \mathbf{\hat{\theta}} \), and \( \mathbf{\hat{\phi}} \) are unit vectors in the radial, polar, and azimuthal directions, respectively. Divergence: \[ \nabla \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r \sin \theta} \frac{\partial (F_{\theta} \sin \theta)}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial F_{\phi}}{\partial \phi} \] Curl: \[ \nabla \times \mathbf{F} = \frac{1}{r \sin \theta} \left( \frac{\partial (F_{\phi} \sin \theta)}{\partial \theta} – \frac{\partial F_{\theta}}{\partial \phi} \right) \mathbf{\hat{r}} + \frac{1}{r} \left( \frac{1}{\sin \theta} \frac{\partial F_r}{\partial \phi} – \frac{\partial (r F_{\phi})}{\partial r} \right) \mathbf{\hat{\theta}} + \frac{1}{r} \left( \frac{\partial (r F_{\theta})}{\partial r} – \frac{\partial F_r}{\partial \theta} \right) \mathbf{\hat{\phi}} \] Laplacian: \[ \nabla^2 \phi = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \phi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \phi}{\partial \phi^2} \] These expressions are essential when working with electromagnetic problems in cylindrical and spherical coordinate systems. They allow for the transformation of vectors and scalar fields between different coordinate systems, facilitating the analysis of electromagnetic phenomena in various contexts.