The following are some of the vector identities that are useful for electromagnetic problems.
Assuming that a is a scalar quantity, its gradient is a vector and is represented as grad(a) and symbolically written as the folowing
$$ \vec B = \nabla a $$
Assuming that A is a vector quantity, its divergence is a scalar and is represented as div(A) and sumbolically written as the following
$$ b = \nabla \cdot \vec A $$
Assuming that A is a vector quantity, its curl is a vector and is represented as curl(A) and sumbolically written as the following
$$ \vec B = \nabla \times \vec A $$
interesting fact: Normal component of the derivative is equivalent to the partial derivative in the given direction
$$ \nabla y(x) \cdot \hat n = \frac{\partial y(x)}{\partial n} $$
Some standard vector identities
$$ \nabla (a+b) = \nabla a + \nabla b $$
$$ \nabla (ab) = a \nabla b + b \nabla a $$
$$ \nabla (\frac{a}{b}) = \frac{a \nabla b - b \nabla a }{b^2} $$
$$ \nabla \cdot ( \vec A+ \vec B) = \nabla \cdot \vec A +\nabla \cdot \vec B $$
$$ \nabla \cdot (a \vec A) = a\nabla \cdot \vec A + \vec A \cdot\nabla a $$
$$ \nabla \cdot \frac{ \vec A}{b} = \frac{b\nabla \cdot \vec A - \vec A \cdot\nabla b}{b^2} $$
$$ \nabla \times (\vec A + \vec B) = \nabla \times \vec A + \nabla \times \vec B $$
$$ \nabla (\vec A \cdot \vec B) = (\vec A \cdot\nabla)B + (\vec B \cdot\nabla)A + \vec A \times (\nabla \times \vec B) + \vec B \times (\nabla \times \vec A) $$
$$ \nabla (a \vec A ) = a (\nabla \times \vec A ) + \nabla a \times\vec A $$
$$ \nabla \cdot (\vec A \times \vec B ) = (\nabla \times \vec A)\cdot \vec B - \vec A \cdot (\nabla \times \vec B) $$
$$ \nabla \times(\vec A \times \vec B ) = \vec A (\nabla \cdot \vec B) - \vec B (\nabla \cdot \vec A) + (\vec B \cdot \nabla) A - (\vec A \cdot \nabla)\vec B $$
$$ \vec A \times \vec B = - \vec B \times \vec A $$
$$ \vec A \cdot (\vec B \times \vec C) = \vec B \cdot (\vec C \times \vec A) = \vec C \cdot (\vec A \times \vec B) $$
$$ \vec A \times(\vec B \times \vec C) = (\vec A \cdot \vec C ) \vec B - (\vec A \cdot \vec B)\vec C $$
$$ (\vec A \times \vec B )\times \vec C = (\vec A \cdot \vec C ) \vec B - (\vec B \cdot \vec C)\vec A $$
$$ \nabla \cdot\nabla a = \nabla ^2a $$
$$ \nabla \times\nabla a = 0 $$
$$ \nabla \cdot (\nabla \times \vec A) = 0 $$
$$ \nabla \times(\nabla \times \vec A) = \nabla \times\nabla \times \vec A = \nabla(\nabla\cdot \vec A) - \nabla^2 \vec A $$
$$ \nabla \cdot (\nabla a \times \nabla b)= 0 $$
$$ \nabla \cdot ( a \nabla b)= a \nabla^2b + \nabla a \cdot\nabla b $$
$$ \nabla \cdot ( a \nabla b - b \nabla a)= a \nabla^2 b -b \nabla^2 a $$
Divergence theorem
$$ on Volume \int \nabla \cdot \vec A \space dV = \oint_{S=\partial V} \vec A \cdot \hat n \space dS $$
$$ on Volume \int\nabla a \space dV = \oint_{S=\partial V} a \space dS $$
Strokes theorem
$$ on Volume \int\nabla \cdot \vec A \space dV = \oint_{S=\partial V} \hat n \times \vec A \space dS $$
$$ on Surface \int \nabla \cdot \vec A \space dS = \oint_{L=\partial S} \hat n \times \vec A \space dL $$
$$ on Surface \int \nabla \cdot \vec A \space dS = \oint_{L=\partial S}\vec A \cdot \vec dL $$
$$ on Surface \int \hat n \times \nabla a \space dS = \oint_{L=\partial S} a \space dL $$
Green’s identity
$$ on Volume \int(a\nabla^2b + \nabla a \cdot \nabla b) \space dV = \oint_{S=\partial V} a \nabla b \cdot \hat n \space dS $$
$$ on Volume \int(a\nabla^2b - \nabla a \cdot \nabla b) \space dV = \oint_{S=\partial V} (a \nabla b - b \nabla a)\cdot \hat n \space dS $$
$$ on Volume \int(a\nabla^2b - \nabla a \cdot \nabla b) \space dV = \oint_{S=\partial V} (a \frac{\partial b}{\partial n} - b \frac{\partial a}{\partial n}) \space dS $$
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