Vector Calculus Product Rules
Table of Contents
For scalar functions \(\small f \) and \(\small g \) and vector functions \(\small \mathbf{A} \) and \(\small \mathbf{B} \), the product rules in vector calculus are as follows:
Gradient #
$$\small \nabla (fg) = f(\nabla g) + g(\nabla f) $$ $$\small \nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla) \mathbf{A} $$
Divergence #
$$\small \nabla \cdot (f \mathbf{A}) = f(\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f) $$ $$\small \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) $$
Curl #
$$\small \nabla \times (f \mathbf{A}) = f(\nabla \times \mathbf{A}) + \mathbf{A} \times (\nabla f) $$ $$\small \nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} + \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) $$