A gradient is a derivative of a scalar field by differentiation with respect to the location vector field or - to put it shorter - the gradient of a scalar field.
In order to distinguish between the gradient as a mathematical operator and the result of its application, some authors denote the vectors of which make up the gradient fields, as Gradientvektoren, others overlooking the potentials from which they derive, as potential vectors.
Analog does not use the vast majority of authors the term potential field for the scalar field, the potential itself, but which derives from it the gradient field.
Definition and properties
A vector field is a gradient field if and only if there is to him a scalar field, so that:
Is often called the associated " scalar " or just " potential " of the gradient field, not to be confused with the physical concept of " potential ", with the there, the ability of a conservative force field is called a field -exposed body perform work to leave.
Vector fields, the gradient of a scalar field are often referred to in reference to the concept of "conservative force field " as " conservative " vector fields - all of them are jointly focused on three mutually equivalent properties:
Bubbling shown, the field of potential energy as shown in the adjacent figure, off to the place where you get the energy gradient, ie a vector field, show the individual vectors while in the direction of the respective strongest increase of at the site. Following the principle of least constraint that this gradient opposite vectors nothing but each in the direction of steepest descent of pointing " repulsive " forces ( gravitational force ) and ( Coulomb force ) are
Dividing the energy gradient by the scalars m and q delivers analogy, the potential gradient ( gravitational potential ) and ( Coulomb potential ), again show their individual vectors while in the direction of each strongest increase in potential at the site, as for the vectors them opposite ( gravitational acceleration ) and ( electric field strength ) applies:
If it is in the underlying scalar also a potential in the physical sense (see above), so it describes an actual physical work capacity, is the resultant of him gradient, as just established, always with one ( the increase of the amount of opposite ) negative sign written. For scalar fields, however, the only mathematically as potentials behave as the flow or velocity potential, which thus represents no potential energy, the sign of its gradient is undefined and is chosen positive for usual:
Is an open and simply connected (for example, star-shaped ) the quantity and continuously differentiable, clearly then a gradient field if the integrability condition
Is satisfied on. The statement is obtained as a special case from the Poincaré lemma.
In two and three dimensions is the integrability condition:
- For addition: Equivalent thereto in both cases the condition.
In areas that are not simply connected, the integrability condition is necessary but generally not sufficient.