ben green
(Ben Green)
#1
Exercises
8-1. Sketch some level curves φ=kfor the given values of kand then find the
gradient vector.
(i) φ= 4x− 2 y, k =− 2 ,− 1 , 0 , 1 , 2
(ii) φ=xy, k =− 2 ,− 1 , 0 , 1 , 2
(iii) φ=x^2 +y^2 , k = 0, 1 , 9 , 25
(iv) φ= 9x^2 + 4y^2 , k = 0, 36 , 72
8-2. Find the gradient vector associated with the given functions and then evaluate
the gradient at the points indicated.
(i) φ= 4x− 2 y, (4 ,9),(0 ,0),(− 4 ,−9)
(ii) φ=xy, (0 ,1),(− 1 ,0),(0 ,−1),(1 ,0),(1 ,1),(− 1 ,1),(− 1 ,−1),(1,−1)
(iii) φ=x^2 +y^2 , (1 ,0),(3 ,4),(0 ,1),(− 3 ,4),(− 1 ,0),(− 3 ,−4),(0 ,−1),(3,−4)
(iv) φ= 9x^2 + 4y^2 , (2 ,0),(0 ,3),(− 2 ,0),(0 ,−3)
8-3. Find a normal vector to the given surfaces at the point indicated and describe
the surface.
(i) 4 x+ 3 y+ 6z= 13 P(1 , 1 ,1)
(ii) x^2 +y^2 +z^2 = 9 P(1 , 2 ,2)
(iii) z−x^2 −y^2 = 0 P(3 , 4 ,25)
(iv) z=xy P (2 , 3 ,6)
8-4. Discuss the critical points associated with the function z=z(x, y ) = xy. Graph
the level curves z=k, where k=− 2 ,− 1 , 0 , 1 , 2 and describe the surface.
8-5. Find the unit tangent vector at the point P(3 , 2 ,6) on the curve of intersection
of the surfaces
x^2 +y^2 +z^2 = 49, and x+y+z= 11.
8-6. Let rdenote the magnitude of the position vector r =xˆe 1 +yeˆ 2 +zˆe 3.
(i) Show that ∇(rn) = nr n−^2 r
(ii) Show that ∇(ln r) = rr 2
(iii) Show that ∇(f(r)) = f′(r)rr,where f is differentiable.
(iv) Does the result in part (iii) check with the solutions given in parts (i) and (ii)?
