Properties
For scalar fields ψ, ϕ and vector fields A, B, we have the following derivative identities.
Distributive properties:
$$ 1. \ \nabla (\psi+\phi)=\nabla\psi+\nabla\phi $$
$$ 2. \ \nabla (A+B)=\nabla A+\nabla B $$
$$ 3. \ \nabla \cdot(A+B)=\nabla \cdot A+\nabla \cdot B $$
$$ 4. \ \nabla \times (A+B)=\nabla \times A+\nabla \times B $$
Product rule for multiplication by a scalar:
$$ 1. \ \nabla(\psi\phi)=\phi\nabla\psi+\psi\nabla\phi $$
$$ 2. \ \nabla(\psi A)=(\nabla\psi)^TA+\psi\nabla A = \nabla\psi\otimes A+\psi\nabla A $$
$$ 3. \ \nabla\cdot(\psi A)=\psi\nabla\cdot A+(\nabla\psi)\cdot A $$
$$ 4. \ \nabla×(\psi A)=\psi\nabla\times A+(\nabla\psi)\times A $$
$$ 5. \ \nabla^2(fg)=f\nabla^2g+2\nabla f\cdot\nabla g+g\nabla^2f $$
Quotient rule:
$$ 1. \ \nabla \left( \frac {\psi}{\phi} \right) = \frac {\phi\nabla\psi – \psi\nabla\phi }{\phi^2} $$
$$ 2. \ \nabla \cdot \left( \frac {A}{\phi} \right) = \frac {\phi\nabla \cdot A – \nabla\phi \cdot A }{\phi^2} $$
$$ 3. \ \nabla \times \left( \frac {A}{\phi} \right) = \frac {\phi\nabla \times A – \nabla\phi \times A}{\phi^2} $$
$$ 1. \ \nabla (f \circ g) = (f’ \circ g) \nabla g $$
$$ 2. \ \nabla (f \circ A) = (\nabla f \circ A) \nabla A $$
$$ 3. \ \nabla \cdot (A \circ f) = (A’ \circ f) \cdot \nabla f $$
$$ 4. \ \nabla \times (A \circ f) = -(A’ \circ f) \times \nabla f $$
Dot product rule:
$$ \nabla (A \cdot B) = (A \cdot \nabla)B+ (B \cdot \nabla)A + A \times (\nabla \times B) + B \times (\nabla \times A) $$
$$ =A \cdot J_B+B\cdot J_A = A \cdot \nabla B + B \cdot \nabla A $$
where JA denotes the Jacobian matrix of the vector field A
As a special case, when A = B,
$$ \frac 12 \nabla (A \cdot A) = A \cdot J_A = A \cdot \nabla A = (A \cdot \nabla) A + A \times (\nabla \times A) $$
Cross product rule:
$$ 1. \ \nabla \cdot (A \times B) = (\nabla \times A)\cdot B-A \cdot (\nabla \times B) $$
$$ 2. \ \nabla \times (A \times B) = A (\nabla \cdot B) – B (\nabla \cdot A) + (B \cdot \nabla)A- (A \cdot \nabla)B $$
$$ = (\nabla \cdot B + B \cdot \nabla)A – (\nabla \cdot A + A \cdot \nabla)B $$
$$ = \nabla \cdot (BA^T)- \nabla \cdot (AB^T) $$
$$ = \nabla \cdot (BA^T – AB^T) $$
$$ 3. \ A \times ( \nabla \times B)= \nabla_B (A \cdot B)-(A\cdot \nabla)B $$
$$ = A \cdot J_B-(A \cdot \nabla)B= A \cdot \nabla B -(A\cdot \nabla)B $$
$$ 4. \ (A \times \nabla) \times B)= A \cdot \nabla B – A ( \nabla \cdot B) $$
$$ = A \times ( \nabla \times B )+ (A \cdot \nabla) B – A ( \nabla \cdot B) $$