Talk:PlanetPhysics/Gradient

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The gradient is the \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} sum of the resultant rate of increase of a \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} funcion $V$ and is denoted $\nabla V$. It represents a directed rate of change of $V$. A directed derivative or vector derivative of $V$, so to speak. In cartesian coordinates

\begin{equation} \nabla V = \frac{ \partial V}{\partial x} {\bf \hat{i}} + \frac{ \partial V}{\partial y} {\bf \hat{j}} + \frac{ \partial V}{\partial z} {\bf \hat{k}} \end{equation}

It is common to regard $\nabla$ as the gradient operator which obtains a vector $\nabla V$ from a scalar \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $V$ of \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} in space.

$$ \nabla V = \left ( \frac{ \partial }{\partial x} {\bf \hat{i}} + \frac{ \partial }{\partial y} {\bf \hat{j}} + \frac{ \partial }{\partial z} {\bf \hat{ik}} \right ) V $$

Thus it is easy to \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} with just the gradient operator

\begin{equation} \nabla = \frac{ \partial }{\partial x} {\bf \hat{i}} + \frac{ \partial }{\partial y} {\bf \hat{j}} + \frac{ \partial }{\partial z} {\bf \hat{k}} \end{equation}

This symbolic \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} $\nabla$ was introduced by Sir W. R. Hamilton. It has been found by experience that the monosyllable \emph{del} is so short and easy to pronounce that even in complicated \htmladdnormallink{formulas}{http://planetphysics.us/encyclopedia/Formula.html} in which $\nabla$ occurs a number of times no inconvenience to the speaker or hearer arises from the repetition. $\nabla V$ is read simply as "del $V$."

\subsection{Coordinate System Independence}

Although this operator $\nabla$ has been defined as

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

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 $\nabla$ as the \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} and direction of the most rapid increase of $V$. To demonstrate the independence take another set of axes, ${\bf \hat{i}'}$, ${\bf \hat{j}'}$, ${\bf \hat{k}'}$ and a new set of variables $x'$, $y'$, $z'$ referred to them. Then $\nabla$ referred to this \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} is

$$\nabla' = \frac{ \partial }{\partial x'} {\bf \hat{i'}} + \frac{ \partial }{\partial y'} {\bf \hat{j'}} + \frac{ \partial }{\partial z'} {\bf \hat{k'}} $$

(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

\begin{equation} \nabla V = \frac{ \partial V}{\partial x} {\bf \hat{i}} + \frac{ \partial V}{\partial y} {\bf \hat{j}} + \frac{ \partial V}{\partial z} {\bf \hat{k}} \end{equation}

Cylindrical Coordinates

\begin{equation} \nabla V = \frac{ \partial V}{\partial r} {\bf \hat{r}} + \frac{1}{r}\frac{ \partial V}{\partial \theta} {\bf \hat{\theta}} + \frac{ \partial V}{\partial z} {\bf \hat{z}} \end{equation}

Spherical Coordinates

\begin{equation} \nabla V = \frac{ \partial V}{\partial r} {\bf \hat{r}} + \frac{1}{r}\frac{ \partial V}{\partial \phi} {\bf \hat{\phi}} + \frac{1}{r \sin \phi}\frac{ \partial V}{\partial \theta} {\bf \hat{\theta}} \end{equation}

\subsection{References}

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

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} work [1].

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