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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.