Properties -vector calculus formulas

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:

$$1. \ \nabla \cdot (A \times B) = (\nabla \times A)\cdot B-A \cdot (\nabla \times B)$$
$$2. \ \nabla \times (A \times B) = A (\nabla \cdot B) – B (\nabla \cdot A) + (B \cdot \nabla)A- (A \cdot \nabla)B$$
$$= (\nabla \cdot B + B \cdot \nabla)A – (\nabla \cdot A + A \cdot \nabla)B$$
$$= \nabla \cdot (BA^T)- \nabla \cdot (AB^T)$$
$$= \nabla \cdot (BA^T – AB^T)$$
$$3. \ A \times ( \nabla \times B)= \nabla_B (A \cdot B)-(A\cdot \nabla)B$$
$$= A \cdot J_B-(A \cdot \nabla)B= A \cdot \nabla B -(A\cdot \nabla)B$$
$$4. \ (A \times \nabla) \times B)= A \cdot \nabla B – A ( \nabla \cdot B)$$
$$= A \times ( \nabla \times B )+ (A \cdot \nabla) B – A ( \nabla \cdot B)$$