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 .
Proof: Maximum Rate of Change
Simple proof based on the definition of the cross product.
where is the angle between the gradient and .
The largest possible value of is which occurs at . Therefore, the maximum value of is .
The maximum value occurs when the angle between gradient and is ; in other words, when is pointing in the same direction as the gradient, .
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 .
Proof: Gradient Vector Orthogonality
Let’s prove the case. Let be the level surface given by and let . Note that will be on .
Now, let be any curve on that contains . Let be the vector equation for and suppose that be the value of such that . In other words, is the value of that gives .
Because lies on we know that points on must satisfy the equation for :
With the chain rule, we get:
Note that and , so this becomes:
At , this is
This tells us that the gradient vector at , , is orthogonal to the tangent vector, , to any curve that passes through and on the surface and so must also be orthogonal to the surface .