Scalar Potential - Integrability Conditions

Integrability Conditions

If F is a conservative vector field (also called irrotational, curl-free, or potential), and its components have continuous partial derivatives, the potential of F with respect to a reference point is defined in terms of the line integral:

where C is a parametrized path from to

The fact that the line integral depends on the path C only through its terminal points and is, in essence, the path independence property of a conservative vector field. The fundamental theorem of calculus for line integrals implies that if V is defined in this way, then so that V is a scalar potential of the conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point

Read more about this topic:  Scalar Potential

Famous quotes containing the word conditions:

    A society which is clamoring for choice, which is filled with many articulate groups, each urging its own brand of salvation, its own variety of economic philosophy, will give each new generation no peace until all have chosen or gone under, unable to bear the conditions of choice. The stress is in our civilization.
    Margaret Mead (1901–1978)