gradient operates on only the factor B. Less general but similar is the Hestenes overdot notation in geometric algebra. The above identity is then expressed as: where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. For the remainder of this article, Feynman subscript notation will be used where appropriate. First derivative identities For scalar fields , and vector fields , , we have the following derivative identities. Distributive properties Product rule for multiplication by a scalar We have the following generalizations of the product rule in single variable calculus. In the second formula, the transposed gradient is an n × 1 column vector, is a 1 × n row vector, and their product is an n × n matrix (or more precisely, a dyad); This may also be considered as the tensor product of two vectors, or of a covector and a vector. Quotient rule for division by a scalar Chain rule Let be a one-variable function from scalars to scalars, a parametrized curve, and a function from vectors to scalars. We have the following special cases of the multi-variable chain rule. For a coordinate parametrization we have: Here we take the trace of the product of two n × n matrices: the gradient of A and the Jacobian of . Dot product rule where denotes the Jacobian matrix of the vector field . Alternatively, using Feynman subscript notation, See these notes. As a special case, when , The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule Note that the matrix is antisymmetric. Second derivative identities Divergence of curl is zero The divergence of the curl of any continuously twice-differentiable vector field A is always zero: This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex. Divergence of gradient is Laplacian The Laplacian of a scalar field is the divergence of its gradient: The result is a scalar quantity. Divergence of divergence is not defined Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. Therefore: Curl of gradient is zero The curl of the gradient of any continuously twice-differentiable scalar field