In this diagram we intend to define directional derivatives and use them to compute slopes of tangents to the surface at point C in a prescribed direction. The direction will be prescribed by a line L determined by an angle of measure from the x-axis (see the diagram) Since a unit vector v = ( cos , sin ) is also at an angle measure with the x-axis, we can use it in our computation. Point C on the surface is determined by point P in the domain (see the diagram)and the coordinates at P are ( ). The vector v acts as a direction vector for line L and the parametric equations for L are: x = + (cos ) t and y = + (sin ) t

We also can use previous work on differentials, namely, if z = f(x,y) is a differentaible function at

( ) then = ( ) + ( ) ++and the epsilons go to zero as the delta x and delta y go to zero. If we divide each term in the equation by and take in to account that the partial derivatives at ( ) are constants relative to and that dx/dt = cos , dy/dt = sin , then we get the desired result by " sending " to zero , namely, dz/dt = ( ) cos + ( ) sin , ( to see it clearly, write it out ). This is the directional derivative and it will be used to compute the slope of a tangent line to the surface in a plane containing the line L . Let's look at a similar discussion in our text book. If you are confused by that discussion return to this paragraph and use it.

See 13.6 1st item Directional Derivative

In this next example you will seen an alternative symbolism to dz/dt for directional derivative, namely ,

f(x,y) where u refers to a unit vector in the direction you wish to prescribe (we used v before)

See 13.6 Example 1 Finding a directional derivative

In this next example it is most important to "see" the vector prescribing the direction is not a unit vector and must be normalized.

See 13.6 Example 2 Finding a directional derivative

At a point on the graph of a differentiable function we can compute infinitely many directional derivatives, one for each of the directions we can " leave " the point. That would be one directional derivative per vector u ( see f(x,y) ) . One of these directional derivative values is largest and is in a particularly interesting direction. This leads us to our next study of the gradient vector.

Look at the expression  f(x,y) = ( )cos + ( )sin for directional derivative, and " separate " it to form two vectors, namely, ( cos , sin ) and                                    ( ( ) , ( )) . You will notice that the formula for the directional derivative involves a dot product of these two vectors. We recognize the first vector as a unit vector designating the direction for the directional derivative, but what of the second vector. It will be named the gradient vectorfor function f(x,y) at the point ( ) and will be symbolized by using an upside down triangle.

We will learn that this vector is orthogonal (perpendicular) to a level curve for function f(x,y) at the point ( ) and it "points " in the direction of maximum rate of change. Its norm will give us the maximum rate of change.

Look at the following solved problems in the Exercises at the end of 13.6 : 1,3,7,9,13,17,25 Then
do the following unsolved problems as homework: 2,4,8,10,14,18,26