PlanetPhysics/Gradient

The gradient is the vector sum of the resultant rate of increase of a scalar funcion $V$ and is denoted $\mathrm{\nabla }V$. It represents a directed rate of change of $V$. A directed derivative or vector derivative of $V$, so to speak. In cartesian coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial x}\widehat{𝐢}+\frac{\partial V}{\partial y}\widehat{𝐣}+\frac{\partial V}{\partial z}\widehat{𝐤}$

It is common to regard $\mathrm{\nabla }$ as the gradient operator which obtains a vector $\mathrm{\nabla }V$ from a scalar function $V$ of position in space.

$\mathrm{\nabla }V=(\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐢𝐤})V$

Thus it is easy to work with just the gradient operator

$\mathrm{\nabla }=\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐤}$

This symbolic operator $\mathrm{\nabla }$ was introduced by Sir W. R. Hamilton. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated formulas in which $\mathrm{\nabla }$ occurs a number of times no inconvenience to the speaker or hearer arises from the repetition. $\mathrm{\nabla }V$ is read simply as "del $V$."

Coordinate System Independence

Although this operator $\mathrm{\nabla }$ has been defined as

$\mathrm{\nabla }V=\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐤}$

so that it appears to depend upon the choice of the axes, it is in reality independent of them. This would be surmised from the interpretation of $\mathrm{\nabla }$ as the magnitude and direction of the most rapid increase of $V$. To demonstrate the independence take another set of axes, ${\widehat{𝐢}}^{\prime }$, ${\widehat{𝐣}}^{\prime }$, ${\widehat{𝐤}}^{\prime }$ and a new set of variables $x^{\prime }$, $y^{\prime }$, $z^{\prime }$ referred to them. Then $\mathrm{\nabla }$ referred to this system is

${\mathrm{\nabla }}^{\prime }=\frac{\partial }{\partial x^{\prime }}\widehat{{𝐢}^{\prime }}+\frac{\partial }{\partial y^{\prime }}\widehat{{𝐣}^{\prime }}+\frac{\partial }{\partial z^{\prime }}\widehat{{𝐤}^{\prime }}$

(Please Insert PROOF here...)

Leaving behind the proof of coordinate system independence, here is the gradient opertor in the most common coordinate systems.

Cartesian Coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial x}\widehat{𝐢}+\frac{\partial V}{\partial y}\widehat{𝐣}+\frac{\partial V}{\partial z}\widehat{𝐤}$

Cylindrical Coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial V}{\partial \theta }\widehat{𝜽}+\frac{\partial V}{\partial z}\widehat{𝐳}$

Spherical Coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial V}{\partial \varphi }\widehat{𝝓}+\frac{1}{r\sin \varphi }\frac{\partial V}{\partial \theta }\widehat{𝜽}$

References

[1] Wilson, E. "Vector Analysis." Yale University Press, New Haven, 1913.

This entry is a derivative of the Public domain work [1].