1. Find the directional derivative of the function   at the point (2,3) along the direction .

. The unit vector in the given direction is . Thus .

2. In the above problem, what is the directional directive along the direction  which is perpendicular to the direction ?

The given direction being perpendicular to the direction of the gradient, is along the level surface. The directional derivative is zero along a level surface.

3. Sketch the g vector field in the region .

The sketch is as under

 

4. Find a normal to the surface  at the point (2,1,1)

Define . The gradient is . Unit normal to the level surface  is .

5. Evaluate grad r where r is the distance from origin.

6. Find the tangent plane and a normal line to the surface  at the point (1,1,2)

At the point (1,1,2 ) . The tangent plane is 3(x-1)+6(y-1)+7(z-2)=0.  The unit normal, which is along the gradient is parameterized by x=1+3t, y=1+6t, z=2+7t.