Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 5
To be solved during the tutorial session Thursday/Friday, 18/19 February 2016
1. Use appropriate forms of the chain rule to find
z = sin
x
cos 2y ;
2
∂z
∂u
where
x = 2u + 3v , y = u3 − 2v 2 .
2. A function f (x1 , . . . , xn ) is said to be homogeneous of degree k if f (tx1 , . . . txn ) = tk f (x1 , . . . , xn )
for t > 0. Show that it satisfies
n
X
∂f
xi
= kf .
∂xi
i=1
3. Let r =
Pn
i=1
xi ei and r = |r|, where ei form an orthonormal basis of vectors in Rn . Find
∂r
,
∂xi
i = 1, 2, · · · , n ,
and ∇r ,
|∇r| .
4. Let ur be a unit vector whose counterclockwise angle from the positive x-axis is θ, and let
uθ be a unit vector 90% counterclockwise from ur . Show that if z = f (x, y), x = r cos θ,
y = r sin θ, then
1 ∂z
∂z
ur +
uθ .
∇z =
∂r
r ∂θ
5. Consider the surface
p
3
2x2 − 3xy 2 + 3 cos(2x + 3y) − 3y 3 + 18
z = f (x, y) = ln
2
(a) Find an equation for the tangent plane to the surface at the point P (3, −2, z0 ) where
z0 = f (3, −2).
(b) Sketch the tangent plane.
(c) Find parametric equations for the normal line to the surface at the point P (3, −2, z0 ).
(d) Sketch the normal line to the surface at the point P (3, −2, z0 ).
Show the details of your work.
1