Jacobian Matrix
The Jacobian matrix generalises the notion of first derivative
to functions of multivariable input and output.
In general, a function (also called a transformation) of type
where the first definition uses the standard definition of a vector-valued derivative and the second is based on the Matrix transpose of the Multivariable gradient. Hence, the Jacobian can be thought of as combining these two methods into a single concept which can be extended to all multivariable functions. It packages together all the partial derivatives one can take of a function in a logical way, where rows correspond to dimensions of output and columns correspond to dimensions of input.
Perhaps the most productive way to think about the Jacobian Matrix is The Jacobian matrix describes a linear transformation.
A note on notation
The most universally recognisable notation for the jacobian matrix is
, but the subscript tends to be an inconvenience. Another alternative is the operator , which takes inspiration from the standard differential operator, however this is ugly and can confuse the notion of the directional derivative . My preferred notation is the operator , since it is easily distinguished from standard alphanumeric characters (as are used for functions and variables), and includes within it the notion of the Multivariable gradient. However this is slightly controversial, as it means the gradient becomes a Row vector. I would argue this is the correct interpretation though — indeed, the multivariable gradient certainly behaves like a covector.
This was my stance at the beginning of 2022,
but after taking a class in electrodynamics in 2023 I realise the utility of
#state/tidy | #SemBr