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GRADIENTS of Scalar Functions

It is instructive to work up to this "one dimension at a time." For simplicity we will stick to using $\phi$ as the symbol for the function of which we are taking derivatives.

The GRADIENT in One Dimension

Let the dimension be x. Then we have no "extra" variables to hold constant and the gradient of $\phi(x)$ is nothing but $\Hat{\imath} {d\phi \over dx}$. We can illustrate the "meaning" of $\Grad{\phi}$ by an example: let $\phi(x)$ be the mass of an object times the acceleration of gravity times the height h of a hill at horizontal position x. That is, $\phi(x)$ is the gravitational potential energy of the object when it is at horizontal position x. Then

$$\Grad{\phi} \, = \, \Hat{\imath} \, {d\phi \over dx} \; = \ . . . . . . x} (mgh) \; = \; mg \left( dh \over dx \right) \Hat{\imath}.$$

Note that ${dh \over dx}$ is the slope of the hill and $-\Grad{\phi}$ is the horizontal component of the net force (gravity plus the normal force from the hill's surface) on the object. That is, $-\Grad{\phi}$ is the downhill force.

The GRADIENT in Two Dimensions

In the previous example we disregarded the fact that most hills extend in two horizontal directions, say x = East and y = North. [If we stick to small distances we won't notice the curvature of the Earth's surface.] In this case there are two components to the slope: the Eastward slope ${\partial h \over \partial x}$ and the Northward slope ${\partial h \over \partial y}$. The former is a measure of how steep the hill will seem if you head due East and the latter is a measure of how steep it will seem if you head due North. If you put these together to form a vector "steepness" (gradient)

$$\Grad{h} \; = \; \Hat{\imath} \, {\partial h \over \partial x} \; + \; \Hat{\jmath} \, {\partial h \over \partial y}$$

then the vector $\Grad{h}$ points uphill - i.e. in the direction of the steepest ascent. Moreover, the gravitational potential energy $\phi = mgh$ as before [only now $\phi$ is a function of 2 variables, $\phi(x,y)$] so that $-\Grad{\phi}$ is once again the downhill force on the object.

The GRADIENT in Three Dimensions

If the potential $\phi$ is a function of 3 variables, $\phi(x,y,z)$ [such as the three spatial coordinates x, y and z - in which case we can write it a little more compactly as $\phi(\Vec{r})$ where $\Vec{r} \equiv x\Hat{\imath} + y\Hat{\jmath} + z\Hat{k}$, the vector distance from the origin of our coordinate system to the point in space where $\phi$ is being evaluated], then it is a little more difficult to make up a "hill" analogy -- try imagining a topographical map in the form of a 3-dimensional hologram where instead of lines of constant altitude the "equipotentials" are surfaces of constant $\phi$. (This is just what Physicists do picture!) Fortunately the math extends easily to 3 dimensions (or any larger number, if that has any meaning in the context we choose).

In general, any time there is a potential energy function $\phi(\Vec{r})$ we can immediately write down the force $\Vec{F}$ associated with it as

$$\Vec{F} \; \equiv \; - \Grad{\phi}$$

A perfectly analogous expression holds for the electric field $\Vec{E}$ [force per unit charge] in terms of the electrostatic potential $\phi$ [potential energy per unit charge]:²

$$\Vec{E} \; \equiv \; - \Grad{\phi}$$

The GRADIENT in N Dimensions

Although we won't be needing to go beyond 3 dimensions very often in Physics, you might want to borrow this metaphor for application in other realms of human endeavour where there are more than 3 variables of which your scalar field is a function. You could have $\phi$ be a measure of happiness, for instance [though it is hard to take reliable measurements on such a subjective quantity]; then $\phi$ might be a function of lots of factors, such as x₁ = freedom from violence, x₂ = freedom from hunger, x₃ = freedom from poverty, x₄ = freedom from oppression, and so on.³ Note that with an arbitrary number of variables we get away from thinking up different names for each one and just call the $i^{\rm th}$ variable "x_i."

Then we can define the GRADIENT in N dimensions as

$$\Grad{\phi} \; = \; \Hat{\imath}_1 \, {\partial \phi \over \p . . . . . . \; + \; \Hat{\imath}_N \, {\partial \phi \over \partial x_N}$$

$$\mbox{\rm or} \quad \Grad{\phi} \; = \; \sum_{i=1}^N \; \Hat{\imath}_i \, {\partial \phi \over \partial x_i}$$

where $\Hat{\imath}_i$ is a UNIT VECTOR in the x_i direction.