Calculus and Linear Algebra (GSDR2107)
Higher Institute of Transport and Logistics (HITL)
Semester: First Semester
Level: 200
Year: 2017
THE UNIVERSITY OF BAMENDA
INSTITUTE OF TRANSPORT AND LOGISTICS
Tutorials
COURSE: Mathematic backgrounds
Part One: Vectors, differential operator and applications
Exercise 1: In an orthonormal basis
, the vector
is written as:
= 2
+ 3 + 5
1) Calculate the magnitude of
.
2) Find the magnitude of the projection of
on the XOZ plan
3) Let another vector so that 3
+ 2
a) Calculate
.
b) Calculate
c) Find two vectors
(x
A
,y
A
,0),
( 0 , y
B
, z
B
) perpendicular to
E x e r c i s e 2 : L e t u s c o n s i d e r t h r e e v e c t o r s
(x
1
, y
1
, z
1
), (x
2
y
2
, z
2
) and
(x
3
y
3
, z
3
)
a) Demonstrate that
(
) =
. (
) =
.(
)
b) Demonstrate that if
(
)= (
) - (
)
c) Demonstrate that if
=
, then
=
=
E
XERCISE
3:
Let us consider two scalar functions f(x, y, z) and g(x, y, z) defined by:
f(x, y, z) = x
2
y +3 xyz and g(x, y, z) = 5xy
2
z
3
a) Calculate the gradient of f(x, y, z) and g(x, y, z)
b) Demonstrate that grad(f.g) f.g
dg + g g
df
Exercise 4: Let
(x, y, z)a vector function defined by:
(x, y, z)
= 4 x
2
y + 5 x y z
3
+ +zx y
a) Calculate div
b) Calculate curl
c) Calculate
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Exercise 5: Let us consider the orthonormal basis
,
Let us assume = !
,r = "#
$
%
$
&
$
a) Calculate g
dr, div
b) Calculate grad('nr)
c) Calculate curl
b) Calculate div
(
(
)
c) Calculate ('nr)
Exercise 6: A) Let us consider an orthonormal basis
,
and a field of force f defined by
(x, y, z) = *+
,
-+ *
$
.
, y, z) = -2 xyi + (8y - x
2
)j - 3k
A.l) Calculate the work (circulation) of a material point moving on the action of this force from the
point A(1,0, 2) to the B(1,2,4).
A.2) Calculer curl
(x,y,z)
A.3) Is
a conservative force? Justify your answer.
A.4) Let us assume as U, a scalar function (IJ is called scalar potential). Let us assume
(x, y, z) =
—g
dU. Give the expression of U
point moving from point .4(1,0, —2) to point B(—1,2,4).
A. 6) By comparing the work obtained with questions (A.l) et (A.5), what can you conclude?
B) Application #1: Let us consider the field of force (weight) defined by P = mg
where m is
the mass of a material point receiving this force /
, g is a constant (due to the gravitation on the
earth). Let us assume P = —g
dU.
B. l) Give the expression de U
B.2) What name can you give to U?
B.3) Demonstrate that the work done by /
on its displacement from a A(z
A
) to a point B (
Z
B
)
is in
reality, the variation of the potential energy and that it does not depend on the path followed but only on
the term (z
B
z
A
).
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