Hw 5 Due Mar 8
• 2.5 Exercise: 11, 13, 28, 30, 36, 37, 39, 41
• 2.6 Exercise: 2, 10, 14, 22, 25, 49.
y = y(x)
is given implicitly by (as the intersection of two
z = z(x)
3x2 + 4xy − 5zex−1 = 0
2x2 + y = 0
• Suppose the parameterized curve
surfaces)
Find the tangent line equation to the curve at x = 1. Hint: the tangent line equation has the
following form
∂y y
y(1)
∂x
=
(x − 1) +
∂z
z(1)
z
∂x
x=1
1
156
Chapter 2
Differentiation in Several Variables
2 m/sec2 . What is the perceived frequency φ she
hears at that moment? How fast is it changing? Does
Hermione hear the clarinet’s note becoming higher or
lower?
x z
18. Suppose that w = g
,
is a differentiable funcy y
tion of u = x/y and v = z/y. Show then that
11. Suppose z = f (x, y) has continuous partial deriva-
tives. Let x = e cos θ, y = e sin θ . Show that
then
$ 2 %
2 2
2
∂z
∂z
∂z
∂z
+
= e−2r
+
.
∂x
∂y
∂r
∂θ
r
12. Suppose that z = f (x, y) has continuous partial
derivatives. Let x = 2uv and y = u 2 + v 2 . Show that
then
$ 2 %
∂z ∂z
∂z
∂z 2
∂z ∂z
+
= 2x
.
+ 4y
∂u ∂v
∂x
∂y
∂x ∂y
13. If w = g u 2 − v 2 , v 2 − u 2
has continuous partial
derivatives with respect to x = u 2 − v 2 and y = v 2 −
u 2 , show that
v
∂w
∂w
+u
= 0.
∂u
∂v
14. Suppose that z = f (x + y, x − y) has continuous par-
tial derivatives with respect to u = x + y and v =
x − y. Show that
2 2
∂z
∂z
∂z ∂z
−
.
=
∂x ∂y
∂u
∂v
xy
15. If w = f
is a differentiable function of
x 2 + y2
xy
u= 2
, show that
x + y2
∂w
∂w
+y
= 0.
∂x
∂y
2
x − y2
16. If w = f
is a differentiable function of
x 2 + y2
x 2 − y2
, show that then
u= 2
x + y2
x
x
∂w
∂w
+y
= 0.
∂x
∂y
y−x z−x
,
is a differentiable
xy
xz
y−x
z−x
function of u =
and v =
. Show then that
xy
xz
17. Suppose w = f
∂w
∂w
∂w
x
+ y2
+ z2
= 0.
∂x
∂y
∂z
2
x
r
∂w
∂w
∂w
+y
+z
= 0.
∂x
∂y
∂z
In Exercises 19–27, calculate D(f ◦ g) in two ways: (a) by first
evaluating f ◦ g and (b) by using the chain rule and the derivative matrices Df and Dg.
19. f(x) = (3x 5 , e2x ), g(s, t) = s − 7t
20. f(x) = x 2 , cos 3x, ln x , g(s, t, u) = s + t 2 + u 3
21. f (x, y) = ye x , g(s, t) = (s − t, s + t)
22. f (x, y) = x 2 − 3y 2 , g(s, t) = (st, s + t 2 )
#
!s
y x
3
23. f(x, y) = x y − , + y , g(s, t) =
, s2t
x y
t
24. f(x, y, z) = (x 2 y + y 2 z, x yz, e z ),
g(t) = (t − 2, 3t + 7, t 3 )
25. f(x, y) = x y 2 , x 2 y, x 3 + y 3 , g(t) = (sin t, et )
26. f(x, y) = x 2 − y, y/x, e y ,
g(s, t, u) = (s + 2t +
3u, stu)
27. f(x, y, z) = (x + y + z, x 3 − e yz ),
g(s, t, u) = (st, tu, su)
g: R3 → R2 be a differentiable function
such that g(1, −1, 3) = (2, 5) and Dg(1, −1, 3) =
1 −1 0
. Suppose that f: R2 → R2 is de4
0 7
28. Let
fined by f(x, y) = (2x y, 3x − y + 5).
D(f ◦ g)(1, −1, 3)?
What
is
29. Let g: R2 → R2 and f: R2 → R2 be differentiable
functions such that g(0, 0) = (1, 2), g(1, 2) =
(3, 5), f(0, 0) = (3, 5), f(4, 1) = (1, 2), Dg(0, 0) =
1 0
2 3
,
Dg(1, 2) =
,
Df(3, 5) =
−1 4
5 7
1 1
−1 2
, Df(4, 1) =
.
3 5
1 3
(a) Calculate D(f ◦ g)(1, 2).
(b) Calculate D(g ◦ f)(4, 1).
30. Let z = f (x, y), where f has continuous partial
derivatives. If we make the standard polar/rectangular
substitution x = r cos θ, y = r sin θ , show that
∂z
∂x
2
+
∂z
∂y
2
=
∂z
∂r
2
1
+ 2
r
∂z
∂θ
2
.
2.5
31. (a) Use the methods of Example 6 and formula (10)
in this section to determine ∂ 2 /∂ x 2 and ∂ 2 /∂ y 2
in terms of the polar partial differential operators
∂ 2 /∂r 2 , ∂ 2 /∂θ 2 , ∂ 2 /∂r ∂θ , ∂/∂r , and ∂/∂θ . (Hint:
You will need to use the product rule.)
(b) Use part (a) to show that the Laplacian operator
∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 is given in polar coordinates by
the formula
(b) Use the result of part (a) to find dy/d x when y
is defined implicitly in terms of x by the equation x 3 − y 2 = 0. Check your result by explicitly
solving for y and differentiating.
35. Find dy/d x when y is defined implicitly by the equa-
tion sin(x y) − x 2 y 7 + e y = 0. (See Exercise 34.)
36. Suppose that you are given an equation of the form
∂2
∂2
∂2
1 ∂
1 ∂2
+
=
+
.
+
∂x2
∂ y2
∂r 2
r ∂r
r 2 ∂θ 2
32. Show that the Laplacian operator ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 +
∂ 2 /∂z 2 in three dimensions is given in cylindrical coordinates by the formula
∂2
∂2
∂2
∂2
1 ∂
∂2
1 ∂2
+
+
=
+
+
.
+
∂x2
∂ y2
∂z 2
∂r 2
r ∂r
r 2 ∂θ 2
∂z 2
F(x, y, z) = 0,
for example, something like x 3 z + y cos z +
(sin y)/z = 0. Then we may consider z to be defined
implicitly as a function z(x, y).
(a) Use the chain rule to show that if F and z(x, y) are
both assumed to be differentiable, then
∂z
Fx (x, y, z)
=−
,
∂x
Fz (x, y, z)
33. In this problem, you will determine the formula for the
Laplacian operator in spherical coordinates.
(a) First, note that the cylindrical/spherical conversions given by formula (6) of §1.7 express the
cylindrical coordinates z and r in terms of the
spherical coordinates ρ and ϕ by equations of precisely the same form as those that express x and
y in terms of the polar coordinates r and θ . Use
this fact to write ∂/∂r in terms of ∂/∂ρ and ∂/∂ϕ.
(Also see formula (10) of this section.)
(b) Use the ideas and result of part (a) to establish the
following formula:
∂2
∂2
∂2
+
+
∂x2
∂ y2
∂z 2
∂2
∂2
1 ∂2
1
= 2 + 2 2 +
∂ρ
ρ ∂ϕ
ρ 2 sin2 ϕ ∂θ 2
+
2 ∂
cot ϕ ∂
+ 2
.
ρ ∂ρ
ρ ∂ϕ
34. Suppose that y is defined implicitly as a function y(x)
by an equation of the form
F(x, y) = 0.
(For example, the equation x 3 − y 2 = 0 defines y as
two functions of x, namely, y = x 3/2 and y = −x 3/2 .
The equation sin(x y) − x 2 y 7 + e y = 0, on the other
hand, cannot readily be solved for y in terms of x. See
the end of §2.6 for more about implicit functions.)
(a) Show that if F and y(x) are both assumed to be
differentiable functions, then
dy
Fx (x, y)
=−
dx
Fy (x, y)
provided Fy (x, y) = 0.
157
Exercises
Fy (x, y, z)
∂z
=−
.
∂y
Fz (x, y, z)
(b) Use part (a) to find ∂z/∂ x and ∂z/∂ y where z is
given by the equation x yz = 2. Check your result
by explicitly solving for z and then calculating the
partial derivatives.
37. Find ∂z/∂ x and ∂z/∂ y, where z is given implicitly by
the equation
x 3 z + y cos z +
sin y
= 0.
z
(See Exercise 36.)
38. Let
⎧ 2
⎪
⎨ x y
f (x, y) = x 2 + y 2
⎪
⎩
0
if (x, y) = (0, 0)
.
if (x, y) = (0, 0)
(a) Use the definition of the partial derivative to find
f x (0, 0) and f y (0, 0).
(b) Let a be a nonzero constant and let x(t) =
(t, at). Show that f ◦ x is differentiable, and find
D( f ◦ x)(0) directly.
(c) Calculate D f (0, 0)Dx(0). How can you reconcile
your answer with your answer in part (b) and the
chain rule?
Let w = f (x, y, z) be a differentiable function of x, y, and
z. For example, suppose that w = x + 2y + z. Regarding the
variables x, y, and z as independent, we have ∂w/∂ x = 1 and
∂w/∂ y = 2. But now suppose that z = x y. Then x, y, and z
are not all independent and, by substitution, we have that w =
x + 2y + x y so that ∂w/∂ x = 1 + y and ∂w/∂ y = 2 + x. To
overcome the apparent ambiguity in the notation for partial
derivatives, it is customary to indicate the complete set of independent variables by writing additional subscripts beside
158
Chapter 2
Differentiation in Several Variables
41. Suppose s = x 2 y + x zw − z 2 and x yw − y 3 z + x z
the partial derivative. Thus,
∂w
∂x
= 0. Find
y,z
would signify the partial derivative of w with respect to x,
while holding both y and z constant. Hence, x, y, and z are the
complete set of independent variables in this case. On the other
hand, we would use (∂w/∂ x) y to indicate that x and y alone are
the independent variables. In the case that w = x + 2y + z,
this notation gives
∂w
∂w
∂w
= 1,
= 2, and
= 1.
∂ x y,z
∂ y x,z
∂z x,y
If z = x y, then we also have
∂w
= 1 + y,
∂x y
and
∂w
∂y
x
= 2 + x.
39. Let w = x + 7y − 10z and z = x + y .
2
∂w
∂w
∂w
∂w
,
,
,
,
∂ x y,z
∂ y x,z
∂z x,y
∂x y
∂w
and
.
∂y x
(a) Find
(b) Relate (∂w/∂ x) y,z and (∂w/∂ x) y by using the
chain rule.
40. Repeat Exercise 39 where w = x 3 + y 3 + z 3 and z =
2x − 3y.
2.6
∂s
∂z
and
x,y,w
∂s
∂z
.
x,w
42. Let U = F(P, V, T ) denote the internal energy of a
gas. Suppose the gas obeys the ideal gas law P V = kT ,
where k isa constant.
∂U
.
(a) Find
∂T P
∂U
(b) Find
.
∂T V
∂U
(c) Find
.
∂P V
43. Show that if x, y, z are related implicitly by an equation
In this way, the ambiguity of notation can be avoided. Use this
notation in Exercises 39–45.
2
of the form F(x, y, z) = 0, then
∂x
∂y
∂z
= −1.
∂ y z ∂z x ∂ x y
This relation is used in thermodynamics. (Hint: Use
Exercise 36.)
44. The ideal gas law P V = kT , where k is a constant,
relates the pressure P, temperature T , and volume V
of a gas. Verify the result of Exercise 43 for the ideal
gas law equation.
45. Verify the result of Exercise 43 for the ellipsoid
ax 2 + by 2 + cz 2 = d
where a, b, c, and d are constants.
Directional Derivatives and the Gradient
In this section, we will consider some of the key geometric properties of the
gradient vector
∂f ∂f
∂f
∇f =
,
,...,
∂ x1 ∂ x2
∂ xn
of a scalar-valued function of n variables. In what follows, n will usually be 2
or 3.
The Directional Derivative
Let f (x, y) be a scalar-valued function of two variables. In §2.3, we understood the
partial derivative ∂∂ xf (a, b) as the slope, at the point (a, b, f (a, b)), of the curve
obtained as the intersection of the surface z = f (x, y) with the plane y = b.
The other partial derivative ∂∂ yf (a, b) has a similar geometric interpretation. However, the surface z = f (x, y) contains infinitely many curves passing through
(a, b, f (a, b)) whose slope we might choose to measure. The directional derivative enables us to do this.
2.6
Exercises
173
does not apply. Geometrically, this makes perfect sense, since at the origin the
◆
polar angle θ can have any value.
2.6 Exercises
1. Suppose f (x, y, z) is a differentiable function of three
variables.
(a) Explain what the quantity ∇ f (x, y, z) · (−k) represents.
(b) How does ∇ f (x, y, z) · (−k) relate to ∂ f /∂z?
In Exercises 2–8, calculate the directional derivative of the
given function f at the point a in the direction parallel to the
vector u.
2. f (x, y) = e y sin x, a =
!π
#
3i − j
,0 , u = √
3
10
i + 2j
3. f (x, y) = x 2 − 2x 3 y + 2y 3 , a = (2, −1), u = √
5
1
4. f (x, y) = 2
, a = (3, −2), u = i − j
(x + y 2 )
5. f (x, y) = e x − x 2 y, a = (1, 2), u = 2i + j
6. f (x, y, z) = x yz, a = (−1, 0, 2), u =
7. f (x, y, z) = e−(x
8. f (x, y, z) =
3k
2
+y 2 +z 2 )
2k − i
√
5
, a = (1, 2, 3), u = i + j + k
xe y
, a = (2, −1, 0), u = i − 2j +
3z 2 + 1
9. For the function
⎧
x|y|
⎪
⎨
x 2 + y2
f (x, y) =
⎪
⎩
0
if (x, y) = (0, 0)
,
if (x, y) = (0, 0)
◆
10. For the function
if (x, y) = (0, 0)
if (x, y) = (0, 0)
(a) calculate f x (0, 0) and f y (0, 0).
◆
11. The surface of Lake Erehwon can be represented by a
region D in the x y-plane such that the lake’s depth (in
meters) at the point (x, y) is given by the expression
400 − 3x 2 y 2 . If your calculus instructor is in the water at the point (1, −2), in which direction should she
swim
(a) so that the depth increases most rapidly (i.e., so
that she is most likely to drown)?
(b) so that the depth remains constant?
12. A ladybug (who is very sensitive to temperature) is
crawling on graph paper. She is at the point (3, 7) and
notices that if she moves in the i-direction, the temperature increases at a rate of 3 deg/cm. If she moves
in the j-direction, she finds that her temperature decreases at a rate of 2 deg/cm. In what direction should
the ladybug move if
(a) she wants to warm up most rapidly?
(b) she wants to cool off most rapidly?
(c) she desires her temperature not to change?
13. You are atop Mt. Gradient, 5000 ft above sea level,
equipped with the topographic map shown in Figure 2.74. A storm suddenly begins to blow, necessitating your immediate return home. If you begin heading
due east from the top of the mountain, sketch the path
that will take you down to sea level most rapidly.
14. It is raining and rainwater is running off an ellipsoidal
(a) calculate f x (0, 0) and f y (0, 0). (You will need to
use the definition of the partial derivative.)
(b) use Definition 6.1 to determine for which unit
vectors v = vi + wj the directional derivative
Dv f (0, 0) exists.
T (c) use a computer to graph the surface z = f (x, y).
⎧
xy
⎨
2
x + y2
f (x, y) =
⎩
0
(b) use Definition 6.1 to determine for which unit
vectors v = vi + wj the directional derivative
Dv f (0, 0) exists.
T (c) use a computer to graph the surface z = f (x, y).
,
dome with equation 4x 2 + y 2 + 4z 2 = 16, where
z ≥ 0. Given that gravity will cause the raindrops to
slide down the dome as rapidly as possible, describe
the curves whose paths the raindrops must follow.
(Hint: You will need to solve a simple differential
equation.)
15. Igor, the inchworm, is crawling along graph paper in
a magnetic field. The intensity of the field at the point
(x, y) is given by M(x, y) = 3x 2 + y 2 + 5000. If Igor
is at the point (8, 6), describe the curve along which he
should travel if he wishes to reduce the field intensity
as rapidly as possible.
In Exercises 16–19, find an equation for the tangent plane to
the surface given by the equation at the indicated point (x0 , y0 ,
z 0 ).
174
Differentiation in Several Variables
Chapter 2
You are here
1000
2000
3000
5000
4500
3000
2000
N
1500
W
500
0
E
S
Figure 2.74 The topographic map of Mt. Gradient in Exercise 13.
16. x 3 + y 3 + z 3 = 7, (x 0 , y0 , z 0 ) = (0, −1, 2)
17. ze y cos x = 1, (x 0 , y0 , z 0 ) = (π, 0, −1)
18. 2x z + yz − x 2 y + 10 = 0, (x 0 , y0 , z 0 ) = (1, −5, 5)
19. 2x y 2 = 2z 2 − x yz, (x 0 , y0 , z 0 ) = (2, −3, 3)
20. Calculate the plane tangent to the surface whose equa-
tion is x − 2y + 5x z =
two ways:
(a) by solving for z in terms of x and y and using
formula (4) in §2.3
(b) by using formula (6) in this section.
2
2
7 at the point (−1, 0, − 65 ) in
21. Calculate the plane tangent
π to the surface x sin y +
2
yz
x z = 2e at the point 2, 2 , 0 in two ways:
(a) by solving for x in terms of y and z and using a
variant of formula (4) in §2.3
(b) by using formula (6) in this section.
22. Find the point on the surface x 3 − 2y 2 + z 2 = 27
where the tangent plane is perpendicular to the line
given√parametrically as x = 3t − 5, y = 2t + 7, z =
1 − 2t.
23. Find the points on the hyperboloid 9x 2 − 45y 2 +
5z 2 = 45 where the tangent plane is parallel to the
plane x + 5y − 2z = 7.
24. Show that the surfaces z = 7x 2 − 12x − 5y 2 and
x yz 2 = 2 intersect orthogonally at the point (2, 1, −1).
25. Suppose that two surfaces are given by the equations
F(x, y, z) = c
and
G(x, y, z) = k.
Moreover, suppose that these surfaces intersect at the
point (x0 , y0 , z 0 ). Show that the surfaces are tangent at
(x0 , y0 , z 0 ) if and only if
∇ F(x0 , y0 , z 0 ) × ∇G(x0 , y0 , z 0 ) = 0.
26. Let S denote the cone x 2 + 4y 2 = z 2 .
(a) Find an equation for the plane tangent to S at the
point (3, −2, −5).
(b) What happens if you try to find an equation for a
tangent plane to S at the origin? Discuss how your
findings relate to the appearance of S.
27. Consider the surface S defined by the equation x 3 −
x 2 y 2 + z 2 = 0.
(a) Find an equation for the plane tangent to S at the
point (2, −3/2, 1).
(b) Does S have a tangent plane at the origin? Why or
why not?
If a curve is given by an equation of the form f (x, y) = 0, then
the tangent line to the curve at a given point (x0 , y0 ) on it may
be found in two ways: (a) by using the technique of implicit
differentiation from single-variable calculus and (b) by using
a formula analogous to formula (6). In Exercises 28–30, use
both of these methods to find the lines tangent to the given
curves at the indicated points.
√ √
28. x 2 + y 2 = 4, (x 0 , y0 ) = (− 2, 2)
√
3
29. y 3 = x 2 + x 3 , (x 0 , y0 ) = (1, 2)
30. x 5 + 2x y + y 3 = 16, (x 0 , y0 ) = (2, −2)
2.5 Exercise 30
∂z ∂z
∂z ∂z
cos θ −r sin θ
, ]=[ , ]
sin θ r cos θ
∂r ∂θ
∂x ∂y
−1
1 r cos θ r sin θ
cos θ −r sin θ
=
sin θ r cos θ
r − sin θ cos θ
∂z ∂z
1 ∂z ∂z
r cos θ r sin θ
[ , ]= [ , ]
− sin θ cos θ
∂x ∂y
r ∂r ∂θ
2 2
∂z ∂z
∂z ∂z
∂z
+
= [ , ] ∂x
∂z
∂x
∂y
∂x ∂y
∂y
1 ∂z ∂z
r cos θ r sin θ
r cos θ − sin θ
= 2[ , ]
− sin θ cos θ
r sin θ cos θ
r ∂r ∂θ
2
∂z 1 ∂z ∂z
r 0
∂r
= 2[ , ]
∂z
0 1
r ∂r ∂θ
∂θ
2
∂z
1 ∂z 2
=
+ 2
∂r
r
∂θ
[
7
∂z
∂r
∂z
∂θ