As I'm sure you know, there are several alternative approaches the field of analytical mechanics that don't use forces, but scalar functions and a variational principle.
32.
The gradient ( or gradient vector field ) of a scalar function is denoted or " f " } } where ( the nabla symbol ) denotes the vector differential operator, del.
33.
A magnetic field is a vector field, but if it is expressed in Cartesian components, each component is the derivative of the same scalar function called the " magnetic potential ".
34.
The Jacobian of the gradient of a scalar function of several variables has a special name : the Hessian matrix, which in a sense is the " second derivative " of the function in question.
35.
The output of the network is then a scalar function of the input vector, \ varphi : \ mathbb { R } ^ n \ to \ mathbb { R }, and is given by
36.
Because the electric field is irrotational, it is possible to express the electric field as the gradient of a scalar function, \ phi, called the electrostatic potential ( also known as the voltage ).
37.
In many situations, the electric field is a conservative field, which means that it can be expressed as the gradient of a scalar function, that is, & nabla; V } }.
38.
The gradient of a scalar function ? is the vector field grad " f " that may be defined through the inner product \ langle \ cdot, \ cdot \ rangle on the manifold, as
39.
Only in the simplest case ( coarse-graining, long-wavelength limit, cubic symmetry of the material ), these properties can be considered as ( complex-valued ) scalar functions of the frequency only.
40.
For example, the second order partial derivatives of a scalar function of " n " variables can be organized into an " n " by " n " matrix, the Hessian matrix.
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