Potential

In vector calculus, any vector field of a certain type has an associated scalar field called the potential. Potentials find broad applications in physics. In addition to this scalar potential, the vector potential is a related construct.

Mathematical definition

If ec F is an irrotational (aka conservative, curl-free, or potential) vector field with continuous partial derivatives, the potential of ec F with respect to a reference point mathbf r_0 is defined in terms of a line integral:

Related Topics:
Irrotational - Curl - Partial derivative - Line integral

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:V(mathbf r) = int _{mathbf r_0} ^{mathbf r} ec F cdot d mathbf r'       (1).

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where mathbf r' is a dummy variable of integration. It can be shown that such a scalar field exists for any curl-free vector field.

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By the Fundamental Theorem of Calculus, we can alternatively define V as the scalar field that satisfies the following condition:

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:ec F = abla V       (2).

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This does not give a unique definition of V. In terms of definition (1), the ambiguity lies in the choice of the reference point. In terms of definition (2), V can change by a constant value throughout all space without changing its gradient.

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