Ex Sheet
8
2A Multivariable Calculus 2013
.
Tutorial Exercises
T1
Let r = ( x, y, z), r = |r| and a be a constant vector. Prove the
following results
(a) div r = 3, (b) div (a r) = 0, (c) div (r n a)
2
Ex Sheet
2A Multivariable Calculus 2013
.
Tutorial Exercises
T1
Find all second order partial derivatives of
(a) z = x log(1 + y),
(b) z = sin( xy),
(c) z =
x
y
2
.
Check in each case that z xy = zy
Ex Sheet
2A Multivariable Calculus 2013
Tutorial Exercises
4
.
Lecture 7
Key Points:
By reversing the order of integration, evaluate
T1
1
(a)
0
1
dy
y
e
sinh( x2 ) dx,
(b)
1
1
dx
changing the order
Ex Sheet
6
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 11
Key Points:
T1 Sketch the wedge shaped region W (in the rst octant) enclosed
by the ve planes x = 0, y = 0, z = 0, x = 1 and
Ex Sheet
2A Multivariable Calculus 2013
4
Tutorial Exercises
T1
By reversing the order of integration, evaluate
1
(a)
0
1
dy
y
e
sinh( x2 ) dx,
(b)
1
2
1
dx
log x
ey
dy.
x
Solution
(a) Sketching the t
Tutorial Exercises
Evaluate the following using beta functions:
T1
/2
(a)
0
/2
(d)
0
/2
sin3 x cos2 x dx,
(b)
sin4 x cos2 x dx,
(e)
2
(g)
0
0
0
sin7 x cos3 x dx, (c)
sin5 x dx,
2
sin3 x cos3 x dx,
(h)
Ex Sheet
2
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 3
Key Points:
T1
Find all second order partial derivatives of
(a) z = x log(1 + y),
(b) z = sin( xy),
implicit partial differen
Chapter 0
Revision of dierentiation, integration
and vector algebra from level 1
0.1
Revision of dierentiation
(Stewart (Ed. 7): Chapter 2, p103.)
0.1.1
Three important rules for dierentiation
Product
Ex Sheet
3
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 5
Key Points:
T1 By making the change of variables indicated, nd the general
solution of each of the following partial different
Ex Sheet
3
2A Multivariable Calculus 2013
.
Tutorial Exercises
T1 By making the change of variables indicated, nd the general
solution of each of the following partial differential equations.
f
f
a) x
Ex Sheet
6
2A Multivariable Calculus 2013
Tutorial Exercises
T1 Sketch the wedge shaped region W (in the rst octant) enclosed
by the ve planes x = 0, y = 0, z = 0, x = 1 and y + z = 1. Then
evaluate
x
7
Ex Sheet
2A Multivariable Calculus 2013
.
Tutorial Exercises
T1
Find the directional derivative of xyz2 at the point (1, 5, 1) in
the direction of the vector (1, 1, 2).
Solution
The unit vector in t
Ex Sheet
8
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 15
Key Points:
T1
Let r = ( x, y, z), r = |r| and a be a constant vector. Prove the
following results
(a) div r = 3, (b) div (a
Ex Sheet
9
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 17
Key Points:
T1
Evaluate
P
xy2 dx + x4 y dy,
where P is the arc of the parabola y = 2x2 from A(0, 0) to B(1, 2). (a)
by parame
Tutorial Exercises
10
Ex Sheet
2A Multivariable Calculus 2013
.
Lecture 19
T1
In R3 let S be the part of the plane 4x + 2y z = 37 enclosed
within the innite cylinder with rectangular section dened by
1
Ex Sheet
2A Multivariable Calculus 2013
.
Tutorial Exercises
T1 State the type of surface given by each of the following equations
in three dimensional space.
(a) 4x + 5y 2z = 20,
(d) x2 +
y2
z2
+
=
Ex Sheet
10
2A Multivariable Calculus 2013
.
Tutorial Exercises
T1
In R3 let S be the part of the plane 4x + 2y z = 37 enclosed
within the innite cylinder with rectangular section dened by 0
x 5, 0 y
Ex Sheet
1
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 1
Key Points:
T1 State the type of surface given by each of the following equations
in three dimensional space.
(a) 4x + 5y 2z =
Ex Sheet
7
2A Multivariable Calculus 2013
Tutorial Exercises
.
Lecture 13
T1
Find the directional derivative of xyz2 at the point (1, 5, 1) in
the direction of the vector (1, 1, 2).
Let f be a scalar
Ex Sheet
9
2A Multivariable Calculus 2013
Tutorial Exercises
T1
Evaluate
P
xy2 dx + x4 y dy,
where P is the arc of the parabola y = 2x2 from A(0, 0) to B(1, 2). (a)
by parametrising the curve, (b) usi