The following product rules are used when calculating derivatives of scalar functions and vector fields in multiple dimensions. They can be shown by breaking both sides of the equalities into components and using the product rule for single variables (as explained above): Let g,h be differentiable scalar functions of x. Then d/dx(gh)=g(dh/dx)+h(dg/dx).

A few words about notation:

- All vector fields are in
**bold**, all scalar functions are not.
- gh is the product of g and h, and g
**A** is the product of g and **A**.
**A**·**B** is the dot product of **A** with **B**.
**A**×**B** is the cross product of **A** with **B**.
**∇**g is the gradient of g.
**∇**·**A** is the divergence of **A**.
**∇**×**A** is the curl of **A**.
- (
**A**·**∇**)**B** is the vector [A_{x1}∂B_{x1}/∂x1, A_{x2}∂B_{x2}/∂x2, ...]

It makes sense that we would need 6 product rules for multivariable differentiation. We have three operators: gradient, divergence and curl. Gradient operates on a scalar function while divergence and curl operate on vector fields. A scalar can be formed by the product of two scalars functions or by the dot product of two vector fields. A vector can be formed from the product of a scalar function and a vector field or by the cross product of two vector fields. Therefore, there are three operations, each of which can operate on one of two possible combinations of scalar functions and vector fields, and three times two is six.

Let g,h be differentiable scalar functions of x_{1},x_{2},...,x_{n}, and let **A**,**B** be differentiable vector fields in **R**^{n}.

**∇**(gh)=g**∇**h+h**∇**g
**∇**(**A**·**B**)=**A**×(**∇**×**B**)+**B**×(**∇**×**A**)+(**A**·**∇**)**B**+(**B**·**∇**)**A**
**∇**·(g**A**)=g(**∇**·**A**)+**A**·(**∇**g)
**∇**·(**A**×**B**)=**B**·(**∇**×**A**)-**A**·(**∇**×**B**)
**∇**×(g**A**)=g(**∇**×**A**)+**A**×(**∇**g)
**∇**×(**A**×**B**)=(**B**·**∇**)**A**-(**A**·**∇**)**B**+**A**(**∇**·**B**)-**B**(**∇**·**A**)

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Sources:

Griffiths, D. *Introduction to Electrodynamics* Third Edition © 1999.

http://www.pas.rochester.edu/~orr/p217/lecture2.pdf