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In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken ...
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
In 3 dimensions, an exact vector field (thought of as a 1-form) is called a conservative vector field, meaning that it is the derivative of a 0-form (smooth scalar field), called the scalar potential. A closed vector field (thought of as a 1-form) is one whose derivative vanishes, and is called an irrotational vector field.
A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.
Gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force. Other examples of conservative forces are: force in elastic spring, electrostatic force between two electric charges, and magnetic force between two magnetic poles. The last two forces are called central forces as they ...
The fact that the line integral depends on the path C only through its terminal points r 0 and r is, in essence, the path independence property of a conservative vector field. The fundamental theorem of line integrals implies that if V is defined in this way, then F = –∇V, so that V is a scalar potential of the conservative vector field F ...