Gradient Vector

The gradient vector of is defined to be:

Or if we want to use the standard basis vectors the gradient is:

The above definitions are for functions of 3 variables, but of course this extends to any number of variables

Some nice facts about the gradient vector:

Theorem: Maximum Rate of Change

The maximum value of (and hence the maximum rate of change of the function ) is given by and will occur in the direction given by .

Another one:

Fact

The gradient vector is orthogonal to the level curve at the point . Likewise, the gradient vector is orthogonal to the level surface at the point .