My notes on graphing functions of two independent variables did not include mention of level curves . The focus was on traces in the coordinate planes or in a plane parallel to a coordinate plane ( these are sometimes referred to as " sections" ). The next idea uses the trace in the xy-plane and the projection of the sections parallel to the xy-plane into the xy-plane ( these are called contours or level curves ) to give a "picture" of the graph of the function of two independent variables. In this context perhaps you recognize  contour maps, isobars, and isotherms.

See Level Curves in 13.1

Let's look at an example of a contour map for a hyperbolic paraboloid. Two important ideas for you to "catch " are : ( 1.) The values of c that are selected can be thought of a heights above the xy-plane at which we are taking a section. ( 2.) By using equal spacings for the c values we are able to " see " the three dimensional figure because it is steeper when contours are closer and more level when the contours are farther apart. Are you able to see it in the following example ? ( Please don't let the authors discussion of transverse axis confuse you. They are just referring to which way the hyperbola " runs " in the xy-plane when the c's are positive versus when the c's values are negative.)

See Example 4 in 13.1

The next important idea has to do with the gradient vector relative to the level curves. Since the directional derivative will be zero if we " leave " a point in the direction of a level curve through that point ( at a point a function has a value, z , and if we use that value as a choice for c for a level curve then that curve is the one we are referring to as the level curve for the function at that point ). Since the directional derivative is the dot product of the gradient vector and a unit vector in the direction of the level curve, the gradient vector must be orthogonal ( perpendicular ) to the level curve for the function at the point under discussion.

Finally, the gradient is perdendicular to a curve and it " points " in the direction of maximum rate of change ( that is, the direction in which we get the largest value of the directional derivative ) and the norm of the gradient is the largest value of the directional derivative.

I think we've said a " mouth full " in the previous two paragraphs and we may need a little time to take it all in . Let's look at it again as it is expressed in your text :

See 13.6 Applications of the Gradient , pg. 935

Because these ideas can be generalized (that is, the ideas we've introduced for directional derivatives and gradients hold true if we are working with functions of three independent variables , we can get the tangent plane to a surface and finish our introduction to Functions of Several Variables with Differential Calculus. First, look at a computation for a function of three independent variables.

See 13.6 Example 8

To write the equation for a plane tangent to a surface graph for a function z = f(x,y) at a point on the surface we need only find a normal vector for the plane ( since we know a point on the plane because the plane is to contain the point on the surface ). If we rearrange the equation for the surface so as to have terms on one side we get z - f(x,y) = 0 or f(x,y) - z = 0 . We do not have the graph of z - f(x,y) because it is in three independent variables but we do have the graph of z - f(x,y) = 0 which is a level surface for the function of three variable (after all , we know that z - f(x,y) is zero-valued for all points on the surface) We know that the gradient vector is perpendicular to the level surface (the surface for which we want a tangent plane) The three partial derivatives for z - f(x,y) are easy to get and provide us with the normal vector for a plane. Note:

( -( ) , - ( ), 1 ) if we started with z - f(x,y)

( ( ) , ( ) , - 1 ) if we started with f(x,y) - z

These two vectors differ only in sense and consequently are both perpendicular to the surface. Either can be used as a normal vector for our tangent plane. For some examples:

See 13.7 Example 2

See 13.7 Example 3

See 13.7 Example 4

Look at the following solved problems in the Exercises at the end of 13.7 : 1,3,5,11,13,21,25,45 Then
do the following unsolved problems as homework: 2,4,6,12,16,18,26,46(don't do the computer graphics work)