Conservative Vector Field

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	vector calculus

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DEFINITION A vector field $\backslash mathbf\{v\}$ is said to be ''conservative'' if there exists a scalar field $\backslash phi$ such that :$\backslash mathbf\{v\}=-$ abla\phi. Here $$ abla\phi denotes the Gradient of $\backslash phi$. When the above equation holds, $\backslash phi$ is called a Scalar Potential for $\backslash mathbf\{v\}$. PATH INDEPENDENCE A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that $S\backslash subseteq\backslash mathbb\{R\}^3$ is some region of three-dimensional space, and that $P$ is a path in $S$ with start point $A$ and end point $B$. If $$ \mathbf{v}=- abla\phi is a conservative vector field then :$\backslash int\_P\; \backslash mathbf\{v\}\backslash cdot\; d\backslash mathbf\{r\}=\backslash phi(A)-\backslash phi(B).$ This holds as a consequence of the Chain Rule and the Fundamental Theorem Of Calculus . An equivalent formulation of this is to say that :$\backslash oint\; \backslash mathbf\{v\}\backslash cdot\; d\backslash mathbf\{r\}=0$ for every closed loop in $S$. The converse of the above statement is also true. That is, if the Circulation of $\backslash mathbf\{v\}$ around every closed loop in $S$ is zero, then $\backslash mathbf\{v\}$ is a conservative vector field. IRROTATIONAL VECTOR FIELDS A vector field $\backslash mathbf\{v\}$ is said to be ''irrotational'' if its Curl is zero. That is, if :$$ abla imes\mathbf{v} = 0. For this reason, such vector fields are sometimes referred to as ''curl-free'' vector fields. It is an identity of vector calculus that for any scalar field $\backslash phi$: :$$ abla imes abla \phi=0. Therefore every conservative vector field is also an irrotational vector field. Provided that $S$ is a Simply-connected region the converse of this is true: every irrotational vector field is also a conservative vector field.

where $G$ is the Gravitational Constant and $\backslash hat\{\backslash mathbf\{r\}\}$ is a unit vector pointing from $M$ towards $m$. In this case $\backslash mathbf\{F\}\_G=-$
abla\phi_G, where

:

is the Gravitational Potential .

In the case of Conservative Forces , ''path independence'' can be interpreted to mean that the Work Done in going from a point $A$ to a point $B$
is independent of the path chosen, and that the work done in going around a closed loop is zero. In other words, the total Energy of a particle moving under the influence of conservative forces is conserved.


REFERENCES

• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press (2005)


• D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press (2005)