What are gradients and how would I use them? I keep seeing this symbol $\nabla$ around and I know enough to understand that it represents the term "gradient." But what is a gradient? When would I want to use one mathematically?
 A: The ∇ (pronounced "del") is an operator, more technically.  In 3D, it (more or less) means the vector
< df/dx, df/dy, df/dz >

So, if f(x,y,z) = x^2 + y^3*z + sin(z), ∇f = < 2x, 3y^2*z, y^3 + cos(z) >
It's actually a bit more subtle than that; technically it means
< d/dx, d/dy, d/dz >

And when you do ∇f, it's sort of like a "multiplication" of ∇ and f;
< d/dx, d/dy, d/dz > f = < d/dx f, d/dy f, d/dz f >

Only, not multiplication, but operation.
There are some neat properties about the del operator.  Here are a couple:


*

*The most famous is that ∇f yields the gradient of f.  That is, at any point (x,y,z), ∇f(x,y,z) is the vector pointing in the direction where it is most increasing.  The magnitude of it is the magnitude of the increase.
This is easier to understand with, say, a 2D f(x,y).  If f(x,y) represents the height of a point at (x,y), then ∇f(x,y) represents the steepest incline from that point.  Or rather, if you placed a ball on that point, it would start rolling in the opposite direction of the gradient vector.

*Normally, for multi-dimensional functions, it is easiest to find the derivative along an axis (x, y, z, etc.).  With ∇, you can find the derivative along any arbitrary direction by using ∇f * u, where * is the dot product and u is the unit vector along the direction you are calculating.

*∇ is also used to calculate divergence (amount that vectors are "spreading out") and curl (amount that vectors are "curling up") of a vector field.
Divergence is ∇ * f (dot product), and curl is ∇ x f (cross product)
They aren't truly "products" in the sense.  Rather, when you are calculating divergence and curl and you must do d/dx * (something), you are actually doing d/dx (something) or d(something)/dx.
