Let be a function, let be an input point and let be the corresponding output.

# First-order Taylor expansionΒΆ

The first-order Taylor expansion of at the point is the function defined for each marginal function of by the equation:

for which can be written as:

where is the Jacobian matrix evaluated at the point :

for and .

# Second-order Taylor expansionΒΆ

The second-order Taylor expansion of at the point is the function defined for each marginal function of by the equation:

which can be written as:

where is the Hessian tensor of order 3 evaluated at :

for and .

The first and second order Taylor expansions are used in the following cases:

to evaluate the importance factors of the input point on the output: refer to Taylor importance factors,

to get an approximation of the mean and the variance of the output: refer to Taylor expansion moments.