Scalar Field, Vector Field, Scalar Potential

참고: http://en.wikipedia.org/wiki/Scalar_field

http://en.wikipedia.org/wiki/Vector_field

http://en.wikipedia.org/wiki/Scalar_potential

In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity.


스칼라 필드는 공간 상의 점에 스칼라 값(1차원 값)을 대응 시킨 것이다.

In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space.


벡터 필드는 벡터를 대응 시킨 것으로 보통 화살표들로 나타낸다.


The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that:

,^[1]

where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x,y,z.^[2] In some cases, mathematicians may use a positive sign in front of the gradient to define the potential.^[3] Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length.

스칼라 필드의 그래디언트가 벡터 필드 F의 마이너스 값이 되면 이 스칼라 필드는 스칼라 포텐셜이라고 하며, 그냥 포텐셜이라고 부르기도 한다.

In order for F to be described in terms of a scalar potential only, the following have to be true:

1. , where the integration is over a Jordan arc passing from location a to location b and P(b) is P evaluated at location b .
2. , where the integral is over any simple closed path, otherwise known as a Jordan curve.
3.