Mathematics for Engineers and Scientists (MATH1551)
Partial Differentiation
1. Calculate ∂f /∂x and ∂f /∂y when f (x, y) is given by:
2
2
p
x2
p
−1
2
2
d)
x +y
,
y2,
a) x + y sin(xy),
b) (x + y)/(x − y),
c)
e) xy ,
f) log(x2 + y 2 ),
g) xy + x3 cos(xy),
+
h) xy/(x + y).
2. Calculate all the first order partial derivatives of the following functions of the three variables
x, y and z:
x−y
,
c) xy z,
d) x cos(yz).
a) xy 3 − yz 2 ,
b) z x+y
3. Calculate ∂f /∂x, ∂f /∂y, ∂ 2 f /∂x∂y and ∂ 2 f /∂y∂x when f (x, y) is given by
a) x2 y 3 + ex + log y,
b) x2 cos y + x2 + cos y,
d) tan−1 (y/x),
e) (x2 + y 2 )−1/2 .
4. Let v(r, t) = tn e−r
2 /4t
c) x tan (y 2 ),
. Find a value of the constant n such that v satisfies the equation
∂v
1 ∂
2 ∂v
= 2
r
.
∂t
r ∂r
∂r
5. Let f (x, y) = u(x, y)eax+by , where u(x, y) is a function for which ∂ 2 u/∂x∂y = 0. Find values
of a and b such that
∂f
∂f
∂ 2f
−
−
+ f = 0.
∂x∂y ∂x ∂y
6. The two-dimensional Laplace equation is ∂ 2 f /∂x2 + ∂ 2 f /∂y 2 = 0. Find all solutions of the
form ax3 + bx2 y + cxy 2 + dy 3 , with a, b, c and d constant.
7. For the following functions, find df /dt by (i) using the chain rule, and (ii) substituting for
x, y (and z in (c)) to find f (t) explicitly and then differentiating with respect to t:
(a) f (x, y) = x2 + y 2 and x(t) = cos t, y(t) = sin t.
(b) f (x, y) = x2 − y 2 and x(t) = cos t + sin t, y(t) = cos t − sin t.
(c) f (x, y, z) = x/z + y/z and x(t) = cos2 t, y(t) = sin2 t, z(t) = 1/t.
p
8. The temperature at a point (x, y, z) in space is given by T (x, y, z) = λ x2 + y 2 + z 2 , where
λ is a constant. Use the chain rule to find the rate of change of temperature with respect to t
along the helix r(t) = (cos t) i + (sin t) j + t k.
9. Let f (x, y, z) = x2 e2y cos 3z. Find the value of df /dt at the point (1, log 2, 0) on the curve
x = cos t, y = log(t + 2), z = t.
1
10. If f (x, y, z) is a function of x, y and z, and if x = s2 + t2 , y = s2 − t2 , z = 2st, find ∂f /∂s
and ∂f /∂t in terms of ∂f /∂x, ∂f /∂y, ∂f /∂z, s and t.
11. Repeat the previous question but with x = s + t, y = s − t, z = st.
12. If f (x, y, z) is a function of x, y and z, and if x = s3 + t2 , y = s2 − t3 , z = s2 t3 , find ∂f /∂s
and ∂f /∂t in terms of ∂f /∂x, ∂f /∂y, ∂f /∂z, s and t.
13. Repeat the previous question but with x = s + sin t, y = cos s − t, z = s cos t.
14. Let f (x, y) = xesin y . If x = sin (u2 + v 2 ) and y = u2 + v 2 , write down the appropriate
form of the chain rule to express ∂f /∂u and ∂f /∂v in terms of ∂f /∂x, ∂f /∂y, ∂x/∂u, ∂x/∂v,
∂y/∂u, and ∂y/∂v. To check your result, calculate ∂f /∂u from your chain rule and verify
that it agrees with the result of making the substitution to find f (u, v) explicitly, followed by
calculating ∂f /∂u directly.
15. If f (x, y) is a function of x and y, and if x = eu cosh v and y = eu sinh v, prove that
∂f
∂f
∂f
=x
+y ,
∂u
∂x
∂y
and
∂ 2f
∂f
−
= xy
∂u∂v ∂v
∂f
∂f
∂f
=y
+x ,
∂v
∂x
∂y
∂ 2f
∂ 2f
+
∂x2
∂y 2
+ (x2 + y 2 )
∂ 2f
.
∂x∂y
16. If f (x, y) is a function of x and y, and if x = eu cos v and y = eu sin v, express ∂f /∂u and
∂f /∂v in terms of ∂f /∂x and ∂f /∂y, and prove that
2
∂ 2f
∂ 2f
∂ 2f
∂ f
2u
+ 2 =e
+ 2 .
∂u2
∂v
∂x2
∂y
17. If f (x, y) is a function of x and y, and if x = eu cosh v and y = eu sinh v, express ∂f /∂u
and ∂f /∂v in terms of ∂f /∂x and ∂f /∂y, and prove that
2
∂ 2f
∂ f
∂ 2f
∂ 2f
2u
− 2 =e
− 2 .
∂u2
∂v
∂x2
∂y
18. Let f (r) be a function of r and assume that r =
p
x2 + y 2 . Prove that
∂ 2f
∂ 2f
d2 f
1 df
+
=
+
.
2
2
2
∂x
∂y
dr
r dr
19. Write Laplace’s equation
∂ 2f
∂ 2f
+
=0
∂x2
∂y 2
in terms of polar coordinates (r, θ) where x = r cos θ, y = r sin θ.
20. Show that r cos θ is a solution of Laplace’s equation.
21. Show that cos θ is not a solution of Laplace’s equation.
22. Find the most general solution of Laplace’s equation of the form
ar2 cos2 θ + br2 sin2 θ + cr2 cos θ sin θ.
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Surfaces
23. Find the directional derivative of f (x, y, z) = x2 + 2y 2 + 3z 2 at (1, 1, 0) in the direction
i − j + 2k.
24. Find the gradient vectors for the following functions:
a) f (x, y) = ex cos y,
b) f (x, y, z) = log(x2 + 2y 2 − 3z 2 ).
√
25. A function f (x, y) has at the point (1, 2) a directional derivative +2 in direction i and − 2
in direction i + j. Determine grad f at (1, 2), and calculate the directional derivative at (1, 2)
in direction 3i + 4j.
26. Find the directional derivative of f (x, y, z) = x + y 2 + xz 2 at (1, 2, −1) in the direction
i + j − 3k.
27. Find the directional derivative of f (x, y, z) = x3 yz − x2 + z 2 at (2, −1, 1) in the direction
2i + j + 2k.
28. Find the directional derivative of f (x, y, z) = x + y 2 + z 3 + x3 y 2 z at (1, −1, −2) in the
direction i + 2j − 3k.
√
29. A function f (x, y, z) has, at the point (1, −1, 2), a directional derivative − 2 in direction
i + k, √12 in direction j + k and 0 in direction i + j + k. Determine grad f at (1, −1, 2), and
calculate the directional derivative at (1, −1, 2) in direction i − 2j + 3k.
√
30. A function f (x, y, z) has, at the point (1, 1, 1) a directional derivative 3 in direction k, 2 3
in direction i + j + k and √12 in direction i − j. Determine grad f at (1, 1, 1), and calculate the
directional derivative at (1, 1, 1) in direction i − j + k.
31. Let f (x, y, z) = axy 2 + byz + cz 2 x3 . Find values of the constants a, b and c such that at
the point (1, 2, −1) the directional derivative of f takes its maximum value in the direction of
the positive z-axis and that value is 64.
32. Write down the equations of the tangent plane and the normal line to the surface x2 +2yz =
2 at the point (a, b, c). Find the equations of the tangent planes to this surface which are parallel
to the plane 4x + y − 7z = 0. Find also the co-ordinates of the point where the normal at
(2, 1, −1) meets the surface again.
33. Let S be the surface with equation xy + 2yz + 3zx = 0 and let P be the point (1, −1, 1)
on the surface. Find the equations of the normal line and the tangent plane to S at P . Find
the point where the normal at P meets S again.
34. Find a unit normal to the surface 2x3 z + x2 y 2 + xyz − 4 = 0 at the point (2, 1, 0).
35. (Harder). Find the equations of the tangent plane and the normal line to the surface
xy + yz + zx = 0 at the point (a, b, c). Let P be a point on the surface and suppose that the
normal at P meets the surface again at Q, and the normal at Q meets the surface again at R.
Prove that the line through R and P passes through the origin O and that the distance from
O to P is 9 times the distance from O to R.
p
36. Find a vector v in terms of x, y and z which is normal to the surface z = x2 + y 2 +
(x2 + y 2 )3/2 at a general point (x, y, z) 6= (0, 0, 0) of the surface. Find the cosine of the angle θ
between v and the z-axis and determine the limit of cos θ as (x, y, z) → (0, 0, 0).
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37. Find div u where u = (cos x, cos y, cos z). Hence find div u at
38. Find curl u where u = (cos x, cos y, cos z). Hence find curl u at
39. Find div u where u = (cos y, cos z, cos x). Hence find div u at
40. Find curl u where u = (cos y, cos z, cos x). Hence find curl u at
π π π
, ,
4 3 2
.
π π π
, ,
4 3 2
π π π
, ,
4 3 2
.
.
π π π
, ,
4 3 2
.
41. Find div u where u = (x2 + y 2 + z 2 , yz + zx + xy, x2 y 2 z 2 ). Hence find div u at (1, 2, −1).
42. Find curl u where u = (x2 + y 2 + z 2 , yz + zx + xy, x2 y 2 z 2 ). Hence find curl u at (1, 2, −1).
43. Find div u where u = (xy 2 − z cos2 x, x2 sin y, xyz). Hence find div u at π4 , π4 , −1 .
44. Find curl u where u = (xy 2 − z cos2 x, x2 sin y, xyz). Hence find curl u at π4 , π4 , −1 .
45. Find div u where u = (sin x, sin y, sin z). Hence find div u at π4 , π3 , π2 .
46. Find curl u where u = (sin x, sin y, sin z). Hence find curl u at π4 , π3 , π2 .
47. Find div u where u = (z sin x, x sin y, y sin z). Hence find div u at π4 , π3 , π2 .
48. Find curl u where u = (z sin x, x sin y, y sin z). Hence find curl u at π4 , π3 , π2 .
49. Find div u where u = (y sin x, z sin y, x sin z). Hence find div u at π4 , π3 , π2 .
50. Find curl u where u = (y sin x, z sin y, x sin z). Hence find curl u at π4 , π3 , π2 .
51. Find div u where u = (xy sin z, yz sin x, zx sin y). Hence find div u at π4 , π3 , π2 .
52. Find curl u where u = (xy sin z, yz sin x, zx sin y). Hence find curl u at π4 , π3 , π2 .
53. Find all the critical points of the following functions and classify each as a local maximum,
minimum or saddle point.
a) x2 + y 4 − 2x − 4y 2 + 5
b) 2x3 − 9x2 y + 12xy 2 − 60y,
c) (x2 + y 2 )2 − 8(x2 − y 2 ),
d) x2 y 2 − x2 − y 2 ,
e) y 2 + xy + x2 + 4y − 4x + 5.
54. Find all the critical points of f (x, y) = y 2 + sin x and classify each as a local maximum,
minimum or saddle point.
55. Find all the critical points of f (x, y) = x2 − sin y and classify each as a local maximum,
minimum or saddle point.
56. Find all the critical points of f (x, y) = cos x sin y and classify each as a local maximum,
minimum or saddle point.
57. Find all the critical points of f (x, y) = sin x sin y and classify each as a local maximum,
minimum or saddle point.
58. On a map of the Lake District (scale 1:100,000 = 1cm : 1 km), the height of a point with
coordinates (x, y) is h(x, y) = 125y 2 − 100x2 + 50xy + 400 metres above sea level. Find the
direction and rate of fastest ascent when at the point with coordinates (2, 1) and walking at one
km per hour. Also find the directions in which one can traverse the slope, i.e. walk horizontally.
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