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The del/nabla operator
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$$ \vec{\nabla} \equiv \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k} $$
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The gradient of a scalar field \( f \)
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$$ \vec{\nabla}f $$
$$= \Bigg(\frac{\partial }{\partial x} \hat{i} + \frac{\partial }{\partial y} \hat{j} + \frac{\partial }{\partial z} \hat{k} \Bigg) f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} $$
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The divergence of a vector field \( \vec{E} \)
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$$ \vec{\nabla} \circ \vec{E} $$
$$= \Bigg(\frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k}\Bigg) \circ (E_x \hat{i} + E_y \hat{j} + E_z \hat{k}) = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} $$
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The curl of a vector field \( \vec{E} \)
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$$ \vec{\nabla} \times \vec{E} $$
$$= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ E_x & E_y & E_z \end{vmatrix} $$
$$=\Bigg(\frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z}\Bigg)\hat{i} + \Bigg(\frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x}\Bigg)\hat{j} + \Bigg(\frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y}\Bigg)\hat{k} $$
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