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A ∂ over a ∂ with parenthesised lists, the determinant of a matrix of partial derivatives.
Also known as: Jacobian, Functional determinant, Jacobian matrix determinant
The Jacobian is written as if it were a derivative of one list with respect to another, and it is the factor by which a change of variables stretches volume — which is why it appears whenever an integral changes coordinates. A negative Jacobian means the change of variables reverses orientation.
Denotes the determinant of the matrix of first partial derivatives of a vector-valued function.
Contexts: Mathematics
Appears as the scaling factor in a change of variables in a multiple integral, where its absolute value is taken because volume has no sign.
Contexts: Mathematics
A curled partial-derivative symbol with a bracketed list of function names above a rule, and a bracketed list of variable names below.