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PD.6 Directional Derivatives and the Gradient Vector

In this section we will introduce a type of derivative, called a directional derivative, that enables us to find the rate of change of a function of two or more variables in any direction.

Suppose that we wish to find the rate of change of z at $(x_{0},y_{0})$ in the direction of an arbitrary unit vector u = To do this we consider the surface with equation and we let then the point lies on The vertical plane that passes though in the direction of u intersects S in a curve C. The slope of the tangent line T to C at P is the rate of change of z in the direction of u.

The directional derivative of at $(x_{0},y_{0})$ in the direction of a unit vector is

if the limit exists.

When we compute the directional derivative of a function defined by a formula, we generally use the following theorem:

If is a differentiable function of x and y, then has a directional derivative in the direction of any unit vector and:

The Gradient Vector

Notice from the above theorem that the directional derivative can be written as the dot product of two vectors:

The first vector dot product above is used in more than just this equation so it is given a special name, it is called the gradient of and has a special representation: $\nabla f.$ The definition of $\nabla f$ is given below:

Therefore, the directional derivative can be rewritten as the following:

Functions of Three Variables

Using three variables, we can define the directional derivatives in a similar manner:

The directional derivative of f at in the direction of a unit vector is

if the limit exists.

Also there is this formula for finding the gradient:

Just like with two variables, there above equation can be seen as the dot product of two vectors, with one of the vectors being the gradient vector $(\nabla f)$ :

Also, just as before, the directional derivative can be written as follows:

Maximizing the Directional Derivative

Suppose we have a function of two or three variables and we consider all possible directional derivatives of at a given point. These give the rates of changes of in all possible directions. We can then ask the questions: In which of these directions does change fastest and what is the maximum rate of change? The answers are provided by the following theorem:

Suppose is a differentiable function of two or three variables. The maximum value of the directional derivative $D_{u}f(x,y)$ is and it occurs when u has the same direction as the gradient vector $\nabla f(x).$

Tangent Planes to Level Surfaces

Suppose that S is a surface with equations , that is, it is a level surface of a function of three variables, and let be a point on S. Let C be and curve that lies on the surface S and passes through the point P. Recall that the curve C is described by a continuous vector function Let t $_{0}$ be the parameter value corresponding to P; that is, Since C lies on S, any point must satisfy the equation of S, that is,

We can then use the Chain Rule to differentiate both sides of the above equation as follows:

But since and the equation above can be written as a dot product as follows:

In particular, when $t=t_{0}$ we have so

Because the gradient vector at P, is perpendicular to the tangent vector to any curve C on S that passes though P. If it is therefore natural to define the tangent plane to the level surface at that passes through P and has the normal vector of is (in standard equation form):

or using the symmetric equations:

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